deliverable 5.1: state estimation and forecasting algorithms on … v 1.0.pdf · 2015-02-27 ·...

IDE4L is a project co-funded by the European Commission

Project no: 608860

Project acronym: IDE4L

Project title: IDEAL GRID FOR ALL

Deliverable 5.1: State Estimation and Forecasting Algorithms on MV & LV Networks

Due date of deliverable: 01.03.2015

Actual submission date: 01.03.2015

Start date of project: 01.09.2013 Duration: 36 months

Lead beneficiary name: Dansk Energi, Denmark

Authors:

Dansk Energi (DE), Universidad Carlos III de Madrid (UC3M), Tampere University of Technology (TUT)

Project co-funded by the European Commission within the Seventh Framework Programme (2013-2016)

Dissemination level

PU Public X

PP Restricted to other programme participants (including the Commission Services)

RE Restricted to a group specified by the consortium (including the Commission Services)

CO Confidential, only for members of the consortium (including the Commission Services)

IDE4L Deliverable 5.1

2 IDE4L is a project co-funded by the European Commission

Track Changes

Version Date Description Revised Approved

0.1 16.2.2015 First draft Daniel Olmeda

0.2 17.2.2015 Second draft Antti Mutanen

0.2 20.2.2015 Second draft Fannar Thordarson

0.2 20.2.2015 Second draft Sami Repo

0.2 20.02.2015 Second draft Jasmin Mehmedalic

0.4 27.02.2015 Fourth draft Daniel Olmeda

0.5 27.02.2015 Fifth draft Antti Mutanen

0.5 27.02.2015 Fifth draft Fannar Thordarson

0.5 27.02.2015 Fifth draft Jasmin Mehmedalic

1.0 27.02.2015 Final version Zaid Al-Jassim Zaid Al-Jassim



TABLE OF CONTENTS

1 INTRODUCTION ......................................................................................................................................... 6

2 STATE OF THE ART ..................................................................................................................................... 6

2.1 State Estimation ................................................................................................................................ 7

2.1.1 Methods..................................................................................................................................... 7

2.1.2 IDE4L Concept ............................................................................................................................ 9

2.1.3 Recommendations in regard to the IDE4L concept ................................................................... 9

2.2 Load Forecasting .............................................................................................................................. 10

2.2.1 Methods................................................................................................................................... 10

2.2.2 IDE4L Concept .......................................................................................................................... 11

2.2.3 Recommendations in regard to the IDE4L concept ................................................................. 11

2.3 Production forecasting .................................................................................................................... 12

2.3.1 Methods................................................................................................................................... 12

2.3.2 IDE4L Concept .......................................................................................................................... 13

2.3.3 Recommendations in regard to the IDE4L concept ................................................................. 13

3 Design Specifications ............................................................................................................................... 14

3.1 State Estimation .............................................................................................................................. 14

3.2 Load forecasting .............................................................................................................................. 16

3.2.1 Low Voltage Load Forecasting ................................................................................................. 16

3.2.2 Medium Voltage Load Forecasting .......................................................................................... 18

3.3 Production Forecasting .................................................................................................................... 19

3.3.1 Low Voltage Production Forecasting ....................................................................................... 19

3.3.2 Medium Voltage Production Forecasting ................................................................................ 20

4 Algorithms ............................................................................................................................................... 22

4.1 State Estimation .............................................................................................................................. 22

4.1.1 Formulation for Branch Current Based State Estimation ........................................................ 22

4.1.2 Algorithm Steps ....................................................................................................................... 30

4.1.3 Inputs and Outputs .................................................................................................................. 34

4.1.4 Coordination between MVSE and LVSE ................................................................................... 36

4.1.5 Optional (not implemented) Functionalities ........................................................................... 37

4.1.6 Performance tests ................................................................................................................... 39

4.2 State Forecasting ............................................................................................................................. 41

4.3 Load Forecasting .............................................................................................................................. 42



4.3.1 Real Time Measurement Reading and Filtering ...................................................................... 42

4.3.2 Load Baselines ......................................................................................................................... 43

4.3.3 Short Term Model Training ..................................................................................................... 45

4.3.4 Very Short Term Model Training ............................................................................................. 46

4.3.5 Forecasting .............................................................................................................................. 48

4.4 Production Forecasting .................................................................................................................... 49

4.4.1 Real Time Measurement Reading and Filtering ...................................................................... 49

4.4.2 Production Baselines ............................................................................................................... 50

4.4.3 Short Term Model Training ..................................................................................................... 50

4.4.4 Very Short Term Model Training ............................................................................................. 52

4.4.5 Forecasting .............................................................................................................................. 53

4.5 Interfaces ......................................................................................................................................... 54

4.5.1 Medium Voltage Network Load and Production Forecaster ................................................... 54

4.5.2 Low Voltage Network Load and Production Forecaster .......................................................... 56

REFERENCES..................................................................................................................................................... 58



EXECUTIVE SUMMARY

In the IDE4L project a distribution network automation system that is prepared for smart grids is defined,

developed and demonstrated. In order for an automation system to be prepared for smart grids, it has to

be able to handle a wide array of challenges – these challenges include large amounts of distributed

generation and flexible load in the low voltage networks.

A core element of an automation system that is prepared for smart grids is monitoring. In order to control a

large amount of small generation units and loads, an accurate picture of the network is required. Due to

the nature of distribution networks, it is not economically feasible to install the measurements necessary to

monitor all the different electrical variables of an entire distribution network – the number of

measurements would simply be too large. As a result the voltages, currents and powers in the network are

estimated from a small number of measurements through the use of a state estimator.

In task 5.1 of work package 5, a state estimator suitable for smart grids has been developed. This state

estimator is specifically tailored for distribution networks and can take advantage of smart meters, which

are traditionally only used for billing purposes.

In order for the state estimator to better support the requirements for network power control, it has been

expanded with a state forecaster. The state forecaster is dependent on load and production forecasting

modules, which have also been developed within the task. The use of forecasting allows network power

control to plan ahead and optimize the control actions.

The algorithms developed within task 5.1 of work package 5 are built on the newest knowledge in the field

of state estimation and forecasting, and give a robust approach that acquires the requested network states

rapidly. All the algorithms are developed to run in a decentralized manner, which aligns with the physical

structure of distribution networks and allows for more efficient computation.



1 INTRODUCTION

The future of the power network is being challenged due to increased integration of decentralized power

generation in the lower levels of the distribution grid. Also, the traditional load is shifting course towards

more flexibility due to utilisation of distributed energy resources (DERs), e.g. where local production from

photovoltaics and consumption related to electric vehicles and heat pumps have significant influence on

the power distribution. This calls for a more strategic management for the distribution network, both for

network planning purposes and real-time operation.

In the IDE4L project the objective is to define, develop and demonstrate a distribution network automation

system that accounts for the DERs, which can be arbitrarily allocated in the distribution network. This

implies that for the distribution network, a new approach for the system automation is required that can, in

general, comprehend and manage the wide variety of distribution networks. For the purpose of

establishing an approach, where the traditional one-directional power flow is replaced with a bi-directional

one, the core elements of the automation system have to be updated accordingly.

To update the core elements of the automation system so it can meet the coming challenges, algorithms

for state estimation and state forecasting are developed in WP 5.1 of the IDE4L project, as well as

forecasting modules for the load and the production in the networks. The load and production forecasting

modules are considered as support to the state forecasts. These algorithms are essential in the IDE4L

project, as they are utilized in the majority of the work packages and demonstrations of the project. The

state estimator is the core in congestion management where the real-time estimation of the network

provides the relevant information to the power control to take the right actions. Also, with the increase of

intermittent power generation in the low-voltage and medium-voltage grids, these control actions are fairly

volatile in real-time. The ability to accurately forecast the relative load and production in the networks,

several hours ahead, will limit this volatility, as it enables generation of adequate forecasts of the network

states, which can be utilized in scheduling the power control.

The algorithms developed in WP 5.1 are described in this report. The algorithms involve the state

estimation and the load and production forecasts, each focusing on the low voltage network and the

medium voltage network, separately. The development has required some preliminary work from the

development teams to clarify the scopes and procedures of the algorithm functionalities. This preliminary

work has resulted in state-of-the-art documentation and a design specification describing the key aspects

and functionality of each of the basic algorithms. A short summary of this work is presented in chapters 2

and 3, respectively. Extensive documentation has been written for these parts of the development in the

IDE4L project and, therefore, it is not crucial to give the exhaustive analysis here, but instead the essential

conclusions from these documents are revisited. The development and functionalities of the algorithms are

explained in chapter 4, where the algorithms are described both individually and how they can be coupled

to provide the required state forecasts and schemes for the power control in the case of congestion.



2 STATE OF THE ART

2.1 State Estimation The purpose of distribution system state estimation (DSSE) is to obtain the best possible estimate of the

network state by processing available information. In this case, the network state means node voltages, line

power flows and line current flows. The available information used in DSSE includes network topology,

network configuration, line parameters, measurements and load profiles. Nowadays DSSE relies mainly on

primary substation measurements and load profiles. The substation measurements include real-time

measurements of busbar voltages and feeder current or power flows. With these measurements it is

possible to adjust the feeder loads accurately, but the load distribution inside the feeders remains

uncertain.

The advent of smart grids will change how the distribution networks are operated. There will be a need for

more accurate DSSE because the amount of distribution automation and active control is constantly

increasing. Smart distribution network management functions such as voltage level management, control

of distributed generation, reactive power regulation, feeder reconfiguration and restoration, and demand

side management require accurate real-time estimates of network voltages and line flows. Especially the

increase of distributed generation is an important driver for state estimation development [Cobelo2007].

Luckily, smart grids do not only require better state estimation but also provide tools for enhancing the

state estimation accuracy. First of all, the amount of real-time measurements will increase substantially.

Real-time measurements will be available on selected locations along the medium voltage (MV) feeders

and in secondary substations. The measurements in secondary substations will make the estimation of low

voltage (LV) networks possible for the first time. The smart meter infrastructure can also be used to

improve the state estimation accuracy either by reading them in (near) real-time or by using the data

collected from customer level electricity usage to improve the load profiles that are commonly used as

pseudo measurements in state estimation.

2.1.1 Methods

In order to utilise all the new measurements, new state estimation methods are needed. During the last 20

years, countless new DSSE methods have been proposed in the literature. Many of them are based on the

weighted least squares (WLS) method.

The objective of state estimation is to determine the most likely state of the system based on the quantities

that are measured. On way to accomplish this is by the maximum likelihood estimation, a method widely

used in statistics. If the measurement errors are assumed to be normally distributed, the likelihood

maximation corresponds to minimizing the weighted sum of squares of the measurement residuals. The

weighting of measurements depends on the measurement accuracy. Accurate measurements have large

weights and inaccurate measurements have small weights. [Abur2004]

If the network topology and parameters are perfectly known, the network state can be defined, for

example, with node voltage magnitudes and angles or with branch current magnitudes and angles. In state

estimation these variables are called state variables and all other measurable network variables; node

voltages, loads, line power flows and line current flows can be defined as function of these variables. In



literature, the selection of state variables varies. Some are using node voltages whereas others have chosen

to use branch currents.

2.1.1.1 Node Voltage Based Methods

The traditional transmission system state estimators use WLS estimation where node voltage magnitudes

and phase angles are selected as state variables. These types of state estimators have existed since the

1970’s. To speed up the calculation, the traditional transmission system state estimators use fast decoupled

state estimation where the dependencies between active power and voltage magnitude and reactive

power and voltage angle have been eliminated. The fast decoupled method assumes that line resistances

are substantially smaller than line reactance’s. This assumption is not valid for distribution networks and

the decoupling cannot be used to speed up DSSE. [Abur2004]

Another common assumption that is made to speed up the calculation is that the Jacobian matrix stays

constant during the iteration. This assumption is invalid if the network contains current measurements that

are very common in distribution networks. Moreover, current measurements can cause multiple possible

solutions and slow down the convergence of the state estimation algorithm. In transmission networks,

current measurements can be handled as supplementary measurements since the measurement

redundancy is high and amount of current measurements is small. In distribution networks, measurement

redundancy is low and it is important to utilize all available measurements, including the current

measurements.

Despite the above mentioned problems, the transmission system state estimation principle has been

successfully applied to distribution systems in many studies, for example [Cobelo2007], [Wan2003],

[Baran1994], [Lu1995], [Lin1996]. Work has also been done to improve the current measurement handling

capabilities and computational speed [Baran1994], [Lu1995], [Lin1996], [Handschin1995].

2.1.1.2 Branch Current Based Methods

Since voltage based state estimators have problems with the current measurements needed in DSSE, a new

branch current based state estimation method was presented in [Baran1995]. Compared with the node

voltage based methods the branch current method has several benefits; it is faster, it is not affected by the

line R/X-ratio, it is easy to use current measurements with it and equations are simpler. Moreover the new

method handles power measurements efficiently. This is important since load pseudo measurements have

a vital role in DSSE.

The original branch current based state estimation method has some defects. It could not handle voltage

measurements and was able to calculate only weakly meshed networks. In later publications, the branch

current method has been improved. The calculation speed has been further enhanced [Lin2001] and the

ability to use voltage measurements has been added [Teng2002], [Wang2004]. Additionally, it has been

proposed that current magnitudes and angles could be used as state variables instead of real and imaginary

current components [Wang2004]. The benefit of using current magnitudes and angles is that there is no

need to make an initial guess for the current angle, instead it is automatically estimated based on other

measurements. Also, current magnitude measurements correspond directly to state variables and this

simplifies equations. Capability to utilize phasor measurement units (PMU) was added to branch current

based DSSE in [Pau2013].



2.1.1.3 Other methods

Several non-conventional methods have also been proposed for solving DSSE. The variety of proposed

methods is wide and they all aim to utilize available measurement information more efficiently. These

methods either work independently or are combined as a part of some previously known DSSE method.

Interior point optimization has been applied to state estimation in [Dzafic2011]. In this method, the size of

the estimation problem has been reduced by dividing the network into several measurement areas. The

proposal is designed for radial network as well as for networks with some meshes. The power mismatches

at the boundary between neighboring areas are eliminated by equality constraints.

Fuzzy logic based DSSE algorithms have been developed in [Saric2003] and [Pereira2004]. These state

estimators incorporate information affected by uncertainty by using fuzzy set theory. For example,

historical data can be used to derive “typical” load curves defining a band of possible values. Using these

typical load curves, it is possible to obtain fuzzy assessments for active and reactive loads. Furthermore,

one can obtain fuzzy assessments as a translation of natural language propositions from experienced

operators. Typically they have a lot of qualitative information expressed in a non-mathematical way. These

expressions from human language are transformed into fuzzy numbers and used as fuzzy measurements.

A hybrid particle swarm optimization for distribution system state estimation has been proposed in

[Naka2001] and [Naka2003]. Conventional WLS methods assume that the objective functions to be

minimized are differentiable and continuous. However, certain equipment in distribution systems have

non-linear characteristics and this causes non-linearity to the objective functions of DSSE. Particle swarm

optimization (PSO) can be applied to non-linear and non-continuous optimization problems. A hybrid PSO

(HPSO) adds a selection mechanism of evolutionary computation to PSO and it can generate high-quality

solutions in short calculation time. Another similar method based on combination of Nelder-Mead simplex

search and particle swarm optimization (PSO-NM) is proposed in [Niknam2009].

A probabilistic approach to distribution system state estimation is presented in [Ghosh1997]. Conventional

WLS estimation assumes that measurement errors as normally distributed. Since load profiles are used as

pseudo measurements, this implies that also loads are normally distributed. This is not true for distribution

network loads. Distribution network loads are actually closer to Beta or Lognormal distributions than

normal distributions. The probabilistic DSSE method accounts for non-normally distributed loads and also

incorporates load correlations. Real-time measurements are assumed to be perfectly accurate and are

handled as solution constraints.

2.1.2 IDE4L Concept

In IDE4L concept, the amount of real-time measurements available for distribution network state

estimation will grow immensely. In order to utilize these measurements in the best possible way, a new

modern state estimator is needed.

2.1.3 Recommendations in regard to the IDE4L concept

When selecting the DSSE method to be used in IDE4L project some practical issues should be considered.

Firstly, IDE4L project contains several real-life demonstrations so the selected state estimation method

should have a proven track record, be easy to implement and understand, be computationally efficient and

robust. This rules out the methods presented in chapter 2.1.1.3. These methods have been presented only

in a few academic papers while the WLS methods presented in Chapters 2.1.1.1 and 2.1.1.2 have been

applied in tens of different studies.



WLS estimation can be based either on node voltages or branch currents. Both choices have their relative

strengths. The node voltage method is more established, calculates strongly meshed networks and handles

voltage measurements efficiently. The branch current method has been designed specifically for

distribution networks, is faster [Baran1995], and handles current measurements efficiently. Both methods

are applicable to calculating three-phase MV and LV networks and either could be selected. All the desired

functionalities can be achieved with both methods and the state variable selection affects only the state

estimator and is invisible to other algorithm.

Ultimately, our choice tilts towards the branch current based DSSE method due to historical reasons. The

research team working on IDE4L project has prior experience on branch current based DSSE methods

[Mutanen2008a], [Mutanen 2008b], [Mutanen2011], [Mutanen2013].

2.2 Load Forecasting Load forecasting is a topic of great interest for electric utilities. By making load forecasting an integral part

of planning and operation, utilities are able to address crucial decisions on generation and purchasing of

electric power and future infrastructure development.

Load forecasting involves the accurate prediction of the electric load in a geographical area within a

planning horizon. Based on this horizon, load forecasting is usually classified in three categories: short-term

load forecasting (STLF) for a horizon within one day ahead, medium-term load forecasting (MTLF) for one

day to one year ahead and long-term load forecasting (LTLF) for one year to ten years ahead planning. The

scope of this review is the STLF methods.

2.2.1 Methods

In their most general form load forecasting models can be classified into two broad categories: statistical

approaches and artificial intelligence-based (AI-based) models. Statistical models forecast the load based

on historical data and/or exogenous variables such as weather, day of the week, and the date. Classical

statistical approaches include similar-day, regression, exponential smoothing and time series based models

[Brockwell2002].

On the contrary, AI-based (or non-parametric) models are more flexible and can cope with complexity.

Those systems enable the mapping of the inputs of the model with the outputs, fitting a non-linear curve

through a learning process. Among those inputs, the most relevant exogenous variable is the temperature.

Other parameters that should also be taken into account include user’s behaviour, electricity price,

geographic location and whether or not the forecast horizon includes non-working days. Several AI-

methods have been proposed for STLF, such as Artificial Neural Networks [Winters1960], fuzzy logic

[Peng1992] and expert systems [Hippert2001]. Support vector regression (SVR) [Ho1990] has been

proposed as a feasible alternative to ANNs.

In [Agrawal2013] an introductory study on time series modelling and forecasting is presented. An overview

of load forecasting methods may be found in [Papalexopoulos1994].

In this section, a brief overview of the most relevant methods for load demand forecasting is presented. For

a more detailed review of the state of art, please refer to state of art document of the IDE4L project.

The Similar-Day method or Naive Approach is based on searching for days in the historic data that

shares some common characteristic with the forecasted day. Those common characteristics may

include day of the week, day of the year, kind of day (holidays) and weather. The forecast is either



the load of a single similar-day or a linear combination of several similar-days. Similar-day methods,

though simple, usually achieve low forecast errors and are used as a benchmark to compare new

forecasting approaches

Regression is one of the most widely used statistical techniques for predicting electricity demand.

These regression methods use weighted least squares techniques to model the statistical

relationship between the load and other factors such as temperature, light intensity, wind speed,

humidity, type of day, and demand response. The regression coefficients are calculated by equally

or exponential weighted least squares, using a range of historical measures.

Load forecasting techniques based in stochastic time series assume, in their simplest form, that the

future load is only a function of the previous loads. The input of the algorithm is the load pattern of

historic data. The most common techniques in this field are known as Auto-Regressive (AR) and

Moving Average (MA). These approaches can be combined (i.e. ARMA) and expanded to non-

stationary processes by using the integrated moving average (i.e. ARIMA). Not considering the

effect of weather or other external variables, such as sociological variables, may result in less

robust forecasts [Alfares2002]. If the load is also dependent on external variables, variations of the

previous techniques may be applied (e.g. ARMAX).

Artificial Neural Networks (ANNs) are biologically inspired mathematical models, which learn a

possibly non-linear mapping between inputs and outputs of a system. The network structure

consists in three layers of neurons. The first layer has the same number of neurons as the number

of inputs. The second layer is hidden and encompasses an arbitrary number of neurons. The third

layer has the same number of neurons as there are outputs. Rather than explicitly modelling the

system as a mathematical function, ANNs learn the relation between input and output by example.

The result is a non-linear curve fitting.

Support Vector Machines (SVMs) were introduced by Vapnik [Vapnik2000] as a supervised learning

method for solving classification, and later regression, problems. The original definition of SVM

considered the separation of the feature space as a linear function, thus having poor results on

highly non-linear systems. With the addition of the kernel mapping concept, SVMs are able to map

the feature space into a higher (possibly infinite) dimension space, where the samples can be

separated using simple linear functions to create linear decision boundaries.

2.2.2 IDE4L Concept

In the IDE4L project load and production forecasting will be used in the following WPs:

WP5: Congestion management: mainly in:

o Task 5.1 State estimation where load estimation algorithms will be applied to

forecast the load demand of non-telemetered customers (MV, LV). This information

will be used as pseudo-measurements in the state estimation algorithm.

o Task 5.2 Power Control Algorithm will use load forecasting in the secondary

controller (MV) and (LV) and also in the tertiary controller for performing

congestion management, voltage control and network reconfiguration.


Stochastic time series analysis has proven to be robust, and to require small computational demands. Due

to its characteristics, the adoption of an autoregressive model with exogenous inputs forecaster is

suggested. Autoregressive models present the advantage of flexibility, since they may represent different



order time series. Moreover, in general, adding exogenous inputs, such as weather forecasts, improves

forecast results.

2.3 Production forecasting Short term forecasting techniques of wind or photovoltaic energy can be classified into physical models,

statistical models and Artificial Intelligence-based models (AI).

2.3.1 Methods

In this section, a brief overview of the most relevant methods for generation forecasting is presented. For a

more detailed review of the state of art, please refer to state of art document of the IDE4L project.

Physical models are based on physically modeling the location of the renewable sources whose

power is to be forecasted. In the case of wind turbines, models try to forecast wind reaching each

of the turbines, in order to be able to give a forecast of wind power. The physical models use as

input information global or regional weather forecasts (speed and direction of the wind, or sun

irradiance). Such data is adapted to the installation under study using a meso-scale or micro-scale

model to estimate the wind or irradiance at the position and height where wind turbines or

photovoltaic panels are located. Subsequently, in the case of wind turbines, this wind speed is

transformed into a value of power by using the power curves of installed machines, and a similar

process is used for photovoltaic panels.

The model Prediktor, developed by Landberg in the Risø National Laboratory in Denmark for Elkraft

electrical system operator, is an example of a physical model [Landberg1994]. It makes use of WAsP

(Wind Atlas Analysis and Application Program) to estimate the wind reaching the park turbines,

based on the weather forecasts of the atmospheric model HIRLAM. This information is then used to

forecast power output through the use of the power curve. The effect of turbine wake is

considered through the use of PARK, which models this effect based on the relative position of the

turbines inside the park. Other effects that physical models do not consider are corrected with a

statistical model or MOS, adjusting the results with historical power measures.

Among statistical models the family of time series is especially relevant to the task of generation

forecasting. Generally those methods consider that the output of the forecast system depends only

on historical states of the variables, which are used as inputs to the model. In addition to historical

data, meteorological forecasts of atmospheric models may be used as exogenous inputs. Examples

of statistical models are: Seasonality analysis, Box-Jenkins or Autoregressive Integrated Moving

Average (ARIMA), Multiple Regressions and Exponential Smoothing [Agrawal2013].

A forecast model based on time series extrapolates future values of a variable through analysis of a

set of past values of that variable or other descriptive variables. This is the approach followed in

ARIMA models or Box-Jenkins, among others, which has proven to be useful for forecasting some

industrial processes. In the context of wind power forecasting, they provide reasonably good

results for horizons up to 6 hours. These models have been applied in [Boland2008], mainly

because of the ability of the ARMA models to extract significant statistical properties, following the

Box-Jenkins methodology.

Several time series models of irradiance measurements are compared in [Reikard2009] to forecast

short-term PV production, ranging from few minutes to 6h. Simple auto-regressive (AR) models are

used in [Bacher2009] to directly forecast the PV production, comparing its performance with other

methods.



The main advantage of using ARMA models is their flexibility, since they may represent different

order time series. It has been shown, in addition, that ARMA models are able to robustly process

time series with an underlying linear correlation.

Previously discussed physical and statistical methods are computationally expensive and large

amounts of data from weather forecasts are needed in order to provide accurate production

forecasts. Some authors propose forecast techniques based on Artificial Intelligence (AI), for

example [Mellit2008] [Yona2008], to model and forecast the sun irradiance. Results show that AI

methods, such as genetic algorithms (GA), Fuzzy logic, expert systems or neural networks, do not

require any a priori knowledge of the internal parameters of the system, demand less

computational resources than traditional methods and are robust to multivariate problems. ANNs

have been applied with great success for the estimation of the solar irradiance in [Guarnieri2008],

where it is shown that ANNs reduce the average normalized quadratic error of global horizontal

irradiance (GHI) by 15% when compared with numerical weather predictions (NWP) forecasts,

within a 12-18 h horizon.

2.3.2 IDE4L Concept

In the IDE4L project the forecast of photovoltaic installations and small wind turbine generation can be

used in the following work packages:

WP5: State Estimation and Power Control

WP2: Planning Tools for Distribution Network Management

WP6: Distribution networks dynamics


Generation forecasting methods may be classified as physical, statistical or AI-based. The statistical models

provide better results for short-term horizons than physical models do and need less information about the

plant. Among statistical methods, autoregressive models present the advantage of flexibility, since they

may represent different order time series. Moreover, in general, adding exogenous inputs, such as weather

forecasts, improves forecast results.



3 Design Specifications

3.1 State Estimation The state estimation algorithm will be designed so that the same algorithm can be used for both medium

voltage network state estimation (MVSE) and low voltage network state estimation (LVSE). Furthermore,

since the state estimators and state forecaster have several similarities and common inputs, also the state

forecasting will be done largely with the same algorithm.

The state estimator will be based on a state estimator core developed in earlier INTEGRIS and Smart Domo

Grid projects where TUT and A2A co-operated to create a state estimator suitable for decentralized

monitoring and control of smart grids. In the decentralized monitoring and control concept the MV and LV

network monitoring applications (e.g. state estimation) and control applications (e.g. congestion

management and fault management) are run in different physical locations. LV network applications are

run at the secondary substation automation unit (SSAU) and MV network applications are run at the

primary substation automation unit (PSAU). This reduces the data transfer need between smart meters and

upper level control systems. Only necessary information and alarms from the low voltage network are sent

to upper level systems.

The state estimator core is based on a weighted least squares (WLS) state estimator that uses branch

currents as state variables. In the IDE4L project, the earlier developed state estimator will be improved by

making it more automatic, as well as able to adapt to changing network configurations and measurement

setups. The improvements will be done largely by adding new support functions around the state estimator

core. Functions that can read the network topology from a CIM compatible database, adjust the network

topology based on switch status information and forecast the future load and DG production will be added.

State estimator and state forecaster will receive load and production estimates and forecasts from low and

medium voltage network load and production forecasters (LVF & MVF), which are also described in this

document. Another important relation is the connection to databases. Almost all the inputs for the state

estimator, including network topology and real-time measurements, are read from local SQL databases. In

this project, these databases are referred to as Data eXchange Platforms (DXPs) and they are used for

storing data and sharing data between different functions. State estimator software accesses databases by

using Octave Database package. These databases contain, real-time measurements received from RTUs and

smart meters, and they are organized according to the IEC 61850 standard. The network topology contains

information about the structure of the grid, starting from the primary substation until the point of energy

delivery at the customer premises. Also, a formal definition of the measurements placed over the network

topology is provided. The topology data is based on a CIM model.

Table 3.1.1 contains step-by-step descriptions for an algorithm executing both LVSE and LVSF. Figure 3.1.1

clarifies further the connections between the different steps. The steps and flowcharts for MVSE and MVSF

are practically identical with the ones presented below and can be found from [IDE4L2014a]. The design

specification documents [IDE4L2014a] and [IDE4L2014b] include also detailed descriptions for all the

information that is either read from the database or written to the database. Later in chapters 4.1.3.1 and

4.1.3.2 the inputs and outputs for step 6 (state estimation) are described in detail.



Table 3.1.1. LVSE and LVSF algorithm steps.

1 Network topology import function requests network topology information from the secondary substation

database (LV DXP).

2 Switch status import function requests switch status information from the secondary substation database

(LV DXP).

3 Topology information processing function reshapes the network topology information into a format

understood by the state estimator.

4 Real-time measurement reading function reads the real-time measurements from the secondary substation

database (LV DXP).

5 Real-time measurement filtering function evaluates the real-time measurements read in step 4, removes

erroneous measurements and saves them into the secondary substation database (LV DXP).

6 State estimation function calculates the best possible estimates for the network states using the available

information gathered in steps 1– 5 and received from LVF.

7 Export function writes the state estimation results into the secondary substation database (LV DXP)

8 State forecasting function calculates forecasts for the network states using the available information gathered

in steps 1–3 and received from LVF.

9 Export function writes the state forecasting results into the secondary substation database (LV DXP)

The state estimation algorithm will be run on a fixed schedule (e.g. once every minute which is expected to

be the RTU measurement reading interval). If the status of the network switches or fuses changes, the

network topology is re-evaluated and state estimation is re-calculated immediately. Fault Location,

Isolation and Restoration algorithm (FLIR, WP4) and Network Reconfiguration Algorithms (NRA, WP5)

supply information about the switch positions and blown fuses.

The state estimates are always made for the present time. The state forecasts are made from t=1 to a

predefined n-steps ahead. The length of the forecasting horizon depends on the forecasting moment and

can be anything between 24 – 48 hours. The forecasting resolution can vary – a higher temporal resolution

is used for the immediate future and lower temporal resolution is used for time moments further away.

The state estimation and state forecasting algorithm will be implemented as an Octave program. Octave

can be run either on Windows or Linux machine. Windows will be used during the development phase and

Linux during the testing and implementation phases.



1. Network topology import

2. Switch status import

3. Topology information

processing

has the network

topology or switching

status changed?

YES

NO

Load and production estimates

from LVF (external input)

4. Real-time measurement

reading

5. Real-time measurement

filtering

6. State estimation

7. Export state estimation

results

8. State forecasting

Do the state

forecasts need

updating?

YES

NO

Has someone

modified network

topology information?

YES

NO

START

YES YESt<= forecasting

horizon

3. Topology information

processing

NO

9. Export state

forecasting results

NO

Are the scheduled

switch statuses

same as in t-1?

Load and production forecasts

from LVF (external input)

t=t+1

Figure 3.1.1. State estimation and state forecasting algorithm flow chart.

3.2 Load forecasting

3.2.1 Low Voltage Load Forecasting

In the LV load forecaster core there is a time series load forecaster that uses measurements (LV load

demand, weather data), weather forecasts (from a local weather station) and flexible demand schedule.

The main program running the load forecaster can be divided into 4 distinct steps:

1) Real-time measurement reading

2) Real-time measurement filtering



3) Load forecasting

4) Exporting of load forecasting results

Depending on the resolution and time horizon for forecasts needed in the IDE4L project, the LV load

forecaster will be able to provide both 1) very short-term forecasts (up to 30 minutes ahead with 10

minutes resolution) and 2) short-term forecasts (up to 48h with hourly resolution).

1) Short-term load forecasts: The algorithm will give load forecasts for the next 24-48h and it will run

on demand (as required in the IDE4L project) or by schedule, e.g. every day at 00:00 GMT.

- Inputs:

a. Day-ahead forecast (24h-48h) from a local weather station. Data required are: wind speed

and direction, solar irradiance, temperature, humidity and pressure with hourly time step.

b. Last year weather measurements from a local weather station. Data required are: Previous

year time series that includes wind speed and direction, solar irradiance, temperature,

humidity and pressure with hourly time step.

c. Historical load measurements for every LV customer. Minimal amount of data needed is

the last year with hourly time step.

- Outputs:

a. Load forecasts for the next 24-48h for every LV customer.

b. Time resolution: 1h

c. Forecast update step: on demand (if needed in the IDE4L project) and also daily at a fixed

schedule hour (e.g. 00:00 GMT).

d. Forecast horizon: from 1h up to 24h – 48h

2) Very short-term load forecasts: The algorithm will generate load forecasts for the next 10-30

minutes and, as required in the IDE4L project, it will be run on demand.

- Inputs:

a. Historical load for every LV customer. Minimal amount of data needed is the last 168 hours

with 10 minutes time step.

- Outputs:

a. Load forecasts for the next 10-30 minutes for every LV customer.

b. Time resolution: 10-minutes

c. Forecast update step: on demand (as required in the IDE4L project)

d. Forecast horizon: from 10 minutes up to 30 minutes.



3.2.2 Medium Voltage Load Forecasting

In the MV load forecaster core there is a time series load forecaster that uses measurements (MV load

demand, weather data), weather forecasts (from a local weather station) and flexible demand schedule.

The main program running the load forecaster can be divided into 4 distinct steps:



3) Load forecasting

4) Exporting of load forecasting results

Depending on the resolution and time horizon for forecasts needed in the IDE4L project, the MV load



1) Short-term load forecasts: The algorithm will give load forecasts for the next 24-48h and it will run

on demand (as required in the IDE4L project) or by schedule, e.g. every day at 00:00 GMT.

- Inputs:

a. Day-ahead forecast (24h-48h with hourly resolution) from a local weather station. Data

required are: wind speed and direction, solar irradiance, temperature, humidity and

pressure with hourly time step.




c. Historical load measurements for every MV customer; aggregated load demand at the

MV/LV substation. Minimal amount of data needed is the last year with hourly time step.

- Outputs:

a. Load forecasts for the next 24-48h with hourly resolution for every MV customer;

aggregated load demand at the MV/LV substation.





2) Very short-term load forecasts: The algorithm will generate load forecasts for the next 10-30

minutes and, as required in the IDE4L project, it will be run on demand.

- Inputs:



a. Historical load measurements for every MV customer; aggregated load demand at the

MV/LV substation. Minimal amount of data needed is the last 168 hours with 10 minutes

time step.

- Outputs:

a. Load forecasts for the next 10-30 minutes for every MV customer; aggregated load demand

at the MV/LV substation.




3.3 Production Forecasting

3.3.1 Low Voltage Production Forecasting

In the LV production forecaster core there is a time series production forecaster that uses measurements

(LV DG production, weather data), weather forecasts (from a local weather station) and flexible demand

schedule.

The main program running the production forecaster can be divided into 4 distinct steps:



3) Production forecasting

4) Exporting of production forecasting results

Depending on the resolution and time horizon for forecasts needed in the IDE4L project, the LV production



1) Short-term production forecasts: The algorithm will give production forecasts for the next 24-48h

and it will run on demand (as required in the IDE4L project) or by schedule, e.g. every day at 00:00

GMT.

- Inputs:

a. Day-ahead forecast (24h-48h) from a local weather station. Data required are: wind speed

and direction, solar irradiance, temperature, humidity and pressure with hourly time step.




c. Historical production measurements for every LV customer, LV DG production. Minimal

amount of data needed is the last year with hourly time step.



- Outputs:

a. Production forecasts for the next 24-48h for every LV customer, LV DG production.





2) Very short-term production forecasts: The algorithm will generate production forecasts for the next

10-30 minutes and, as required in the IDE4L project, it will be run on demand.

- Inputs:

a. Historical production measurements for every LV customer; LV DG production. Minimal

amount of data needed is the last 168 hours with 10 minutes time step.

- Outputs:

a. Production forecasts for the next 10-30 minutes for every LV customer; LV DG production.




3.3.2 Medium Voltage Production Forecasting

In the MV production forecaster core there is a time series production forecaster that uses measurements

(MV DG production, weather data), weather forecasts (from a local weather station) and flexible demand

schedule.

The main program running the production forecaster can be divided into 4 distinct steps:



3) Production forecasting

4) Exporting of production forecasting results

Depending on the resolution and time horizon for forecasts needed in the IDE4L project, the MV production



1) Short-term production forecasts: The algorithm will give production forecasts for the next 24-48h

and it will run on demand (as required in the IDE4L project) or by schedule, e.g. every day at 00:00

GMT.

- Inputs:

a. Day-ahead forecast (24h-48h with hourly resolution) from a local weather station. Data

required are: wind speed and direction, solar irradiance, temperature, humidity and

pressure with hourly time step.






c. Historical production measurements for every MV customer; aggregated production at the

MV/LV substation, MV DG production. Minimal amount of data needed is the last year with

hourly time step.

- Outputs:

a. Production forecasts for the next 24-48h with hourly resolution for every MV customer;

aggregated production at the MV/LV substation, MV DG production.





2) Very short-term production forecasts: The algorithm will generate production forecasts for the next

10-30 minutes and, as required in the IDE4L project, it will be run on demand.

- Inputs:

a. Historical production measurements for every MV customer; aggregated production at the

MV/LV substation, MV DG production. Minimal amount of data needed is the last 168

hours with 10 minutes time step.

- Outputs:

a. Production forecasts for the next 10-30 minutes for every MV customer; aggregated

production at the MV/LV substation, MV DG production.






4 Algorithms

4.1 State Estimation This chapter describes the state estimation algorithm used in IDE4L project. The same algorithm is used for

both MV and LV network state estimation. As is recommended in Chapter 2.1, a WLS estimator which uses

branch currents as state variables is selected for this task.

4.1.1 Formulation for Branch Current Based State Estimation

Branch current based state estimators use line branch currents as state variables. If the network topology,

line parameters and source bus voltage are all known, the network state can be fully defined with complex

branch currents. The complex branch currents can be expressed either in polar or in rectangular form. In

this work the polar expression is chosen and thereby the state variables are the branch current magnitudes

and angles. All other measurable network variables; node voltages, power injections and line power flows

can be calculated from these variables.

4.1.1.1 Basic WLS Formulas

In WLS estimation, the goal is to minimize the weighted differences between measured network variables

and their estimated values. The most likely state of the network can be calculated by solving equation

4.1.1.

min𝒙 𝐽(𝒙) = 𝑚𝑖𝑛𝑥 ∑[𝑧𝑖 − ℎ𝑖(𝒙)]2

𝜎𝑖2

𝑁𝑚

𝑖=1

, (4.1.1)

where 𝐽(𝒙) is the cost function to be minimized a.k.a the weighted least square equation

𝒙 is the state vector that contains all state variables

𝑧𝑖 is value of measurement i

ℎ𝑖(𝒙) is measured variable i as a function of state variables

𝜎𝑖2 is variance of measurement i

𝑁𝑚 is number of measurements

If measurements and measurement functions are presented in vector form and measurement variances are

presented in a matrix form, the equation 4.1.1 can be expressed in a simpler form as is done in equation

4.1.2 [Abur2004].

min𝒙 𝐽(𝒙) = [𝒛 − 𝒉(𝒙)]𝑇𝑹−1[𝒛 − 𝒉(𝒙)], (4.1.2)

where 𝒛 = [

𝑧1

𝑧2

⋮𝑧𝑁𝑚

] (measurement vector)

𝒉(𝒙) =

[

ℎ1(𝒙)

ℎ2(𝒙)⋮

ℎ𝑁𝑚(𝒙)]

(measurement functions)



𝑹 =

[ 𝜎1

2 0 ⋯ 0

0 𝜎22 ⋯ 0

⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜎𝑁𝑚

2]

(covariance matrix)

The minimum of cost function 𝐽(𝒙) can be found by differentiating it and searching for the zero point. The

cost function derivative in respect to state vector 𝒙 is equal to its gradient. Therefore, the state vector

minimizing the cost function, forces the gradient to zero. The gradient of 𝐽(𝒙) is given in equation 4.1.3.

∇𝐽(𝒙) = −2𝑯𝑇𝑹−1𝒛 + 2𝑯𝑇𝑹−1𝑯𝒙, (4.1.3)

where 𝑯 = [𝜕𝒉(𝒙)

𝜕𝒙] (Jacobian matrix)

When gradient is zero, we can solve 𝒙 from equation 4.1.4.

𝒙 = (𝑯𝑇𝑹−1𝑯)−1𝑯𝑇𝑹−1𝒛 (4.1.4)

Since equation 4.1.4 is non-linear, solving the state vector 𝒙 requires the use of iterative methods, such as

the Newton-Raphson method. On every iteration round, a linearized approximation of the state vector

change ∆𝒙, shown in equation 4.1.5, is added to the initial state vector value. The iteration is continued

until ∆𝒙 is small enough [Abur2004].

∆𝒙 = (𝑯𝑇𝑹−1𝑯)−1𝑯𝑇𝑹−1[𝒛 − 𝒉(𝒙)] (4.1.5)

4.1.1.2 Equality Constrained WLS Estimation

In WLS estimation, measurements are weighted according to their accuracies. Load models are used as pseudo measurements and they are given low weights. Real-time measurements are given high weights and zero-injection measurements are given very high weights (if there is no load or production connected to a certain node, power injection on that node is known to be zero).

The combination of high and low weights can cause the gain matrix (𝑯𝑇𝑹−1𝑯) to become ill-conditioned. Gain matrix ill-conditioning reduces state estimation accuracy and it can in the worst case it can prevent gain matrix inversion. In order to avoid these ill-conditioning problems we use equality constraints to force the zero-injection measurements to zero instead of giving them very high weights. The equality constrained WLS problem can be solved by using the method of Lagrange multipliers [Wu1988]. In the method of Lagrange multipliers the constrained minimization problem is solved by minimizing the Lagrangian function

𝐿(𝒙, 𝝀) =1

2[𝒛 − 𝒉(𝒙)]𝑇𝑹−1[𝒛 − 𝒉(𝒙)] + 𝝀𝑇𝒄(𝒙) (4.1.6)

where 𝒙 is the state vector

𝝀 is the Lagrange multiplier vector 𝒛 is the measurement vector 𝒉(𝒙) is the measurement function

𝑹 is the covariance matrix (𝑹 = diag[𝜎12 𝜎2

2 ⋯ 𝜎𝑁2] where 𝜎𝑖

2 is the variance of the measurement 𝑖)

𝒄(𝒙) is the zero-injection measurement function.



The minimization problem can be solved by differentiating 𝐿(𝒙, 𝝀) partially with respect to 𝒙 and 𝝀 and setting the differentials to zero. This yields the following equations:

𝜕𝐿(𝒙, 𝝀)

𝜕𝒙= −𝑯𝑇𝑹−1[𝒛 − 𝒉(𝒙)] + 𝑪𝝀 = 0 (4.1.7)

𝜕𝐿(𝒙, 𝝀)

𝜕𝝀= 𝒄(𝒙) = 0 (4.1.8)

where 𝑯 =𝝏𝒉

𝝏𝒙 and 𝑪 =

𝝏𝒄

𝝏𝒙 are the Jacobian matrices.

Equations 4.1.7 and 4.1.8 form a system of equations which can be solved iteratively by the Newton–Raphson method. At each iteration, the incremental change to the state vector (Δ𝒙) is calculated with equation

[Δ𝒙𝝀

] = [𝑯𝑇𝑹−1𝑯 𝑪𝑇

𝑪 0]−1

[𝑯𝑇𝑹−1[𝒛 − 𝒉(𝒙)]

−𝒄(𝒙)]. (4.1.9)

4.1.1.3 Jacobian Matrices and Measurement Equations

Active power flow, reactive power flow, current flow, node voltage, current injection, active power

injection and reactive power injection measurements can all be used in the developed branch current

based state estimator. Measurements, their symbols and measurement equations are shown in table 4.1.1.

Table 4.1.1. Measurement equations

Symbol Measurement description Measurement equation

𝑃𝑘𝑚 Active power flow between nodes k

and m 𝑃𝑘𝑚 = 𝑟𝑒𝑎𝑙(��𝑘(𝐼��𝑚)∗)

𝑄𝑘𝑚 Reactive power flow between nodes k

and m 𝑄𝑘𝑚 = 𝑖𝑚𝑎𝑔(��𝑘(𝐼��𝑚)∗)

𝐼𝑘𝑚 Current flow between nodes k and m 𝐼𝑘𝑚 = |𝐼��𝑚|

𝑉𝑘 Voltage at node k 𝑉𝑘 = |𝑉1 − ∑ 𝐼��𝑚��𝑘𝑚

𝑘𝑚∈𝐴

|,

𝑤ℎ𝑒𝑟𝑒 𝐴 𝑖𝑠 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑝 𝑜𝑓 𝑙𝑖𝑛𝑒𝑠 𝑙𝑜𝑐𝑎𝑡𝑒𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑛𝑜𝑑𝑒𝑠 1 𝑎𝑛𝑑 𝑘

𝐼𝑘 Current injection at node k

𝐼𝑘 = |∑𝐼��𝑘𝑖∈𝐵

− ∑𝐼𝑘𝑗

𝑗∈𝐷

|,

Where B is the group of upper side nodes connected to node k

And D is the group of lower side nodes connected to node k

𝑃𝑘 Active power injection at node k 𝑃𝑘 = 𝑟𝑒𝑎𝑙 (∑��𝑘(𝐼��𝑘)∗

𝑖∈𝐵

− ∑��𝑘(𝐼��𝑗)∗

𝑗∈𝐷

)

𝑄𝑘 Reactive power injection at node k 𝑄𝑘 = 𝑖𝑚𝑎𝑔 (∑��𝑘(𝐼��𝑘)∗

𝑖∈𝐵

− ∑��𝑘(𝐼��𝑗)∗

𝑗∈𝐷

)

The Jacobian matrices H and C contain partial derivates for these measurements in respect to the state

variables Ikm and ∝ as show in equations 4.1.10 and 4.1.11.



𝑯 =

[ 𝜕𝑃𝑘𝑚,1

𝜕𝐼1⋯

𝜕𝑃𝑘𝑚,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝑃𝑘𝑚,𝐿

𝜕𝐼1⋯

𝜕𝑃𝑘𝑚,𝐿

𝜕𝐼𝑁

𝜕𝑃𝑘𝑚,1

𝜕𝛼1

⋯𝜕𝑃𝑘𝑚,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝑃𝑘𝑚,𝐿

𝜕𝛼1

⋯𝜕𝑃𝑘𝑚,𝐿

𝜕𝛼𝑁

𝜕𝑄𝑘𝑚,1

𝜕𝐼1⋯

𝜕𝑄𝑘𝑚,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝑄𝑘𝑚,𝐿

𝜕𝐼1⋯

𝜕𝑄𝑘𝑚,𝐿

𝜕𝐼𝑁

𝜕𝑄𝑘𝑚,1

𝜕𝛼1

⋯𝜕𝑄𝑘𝑚,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝑄𝑘𝑚,𝐿

𝜕𝛼1

⋯𝜕𝑄𝑘𝑚,𝐿

𝜕𝛼𝑁

𝜕𝐼𝑘𝑚,1

𝜕𝐼1⋯

𝜕𝐼𝑘𝑚,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝐼𝑘𝑚,𝐿

𝜕𝐼1⋯

𝜕𝐼𝑘𝑚,𝐿

𝜕𝐼𝑁

𝜕𝐼𝑘𝑚,1

𝜕𝛼1

⋯𝜕𝐼𝑘𝑚,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝐼𝑘𝑚,𝐿

𝜕𝛼1

⋯𝜕𝐼𝑘𝑚,𝐿

𝜕𝛼𝑁

𝜕𝑉𝑘,1

𝜕𝐼1 ⋯

𝜕𝑉𝑘,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝑉𝑘,𝐿

𝜕𝐼1⋯

𝜕𝑉𝑘,𝐿

𝜕𝐼𝑁

𝜕𝑉𝑘,1

𝜕𝛼1

⋯𝜕𝑉𝑘,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝑉𝑘,𝐿

𝜕𝛼1

⋯𝜕𝑉𝑘,𝐿

𝜕𝛼𝑁

𝜕𝐼𝑘,1

𝜕𝐼1⋯

𝜕𝐼𝑘,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝐼𝑘,𝐿

𝜕𝐼1⋯

𝜕𝐼𝑘,𝐿

𝜕𝐼𝑁

𝜕𝐼𝑘,1

𝜕𝛼1

⋯𝜕𝐼𝑘,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝐼𝑘,𝐿

𝜕𝛼1

⋯𝜕𝐼𝑘,𝐿

𝜕𝛼𝑁

𝜕𝑃𝑘,1

𝜕𝐼1⋯

𝜕𝑃𝑘,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝑃𝑘,𝐿

𝜕𝐼1⋯

𝜕𝑃𝑘,𝐿

𝜕𝐼𝑁

𝜕𝑃𝑘,1

𝜕𝛼1

⋯𝜕𝑃𝑘,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝑃𝑘,𝐿

𝜕𝛼1

⋯𝜕𝑃𝑘,𝐿

𝜕𝛼𝑁

𝜕𝑄𝑘,1

𝜕𝐼1⋯

𝜕𝑄𝑘,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝑄𝑘,𝐿

𝜕𝐼1⋯

𝜕𝑄𝑘,𝐿

𝜕𝐼𝑁

𝜕𝑄𝑘,1

𝜕𝛼1

⋯𝜕𝑄𝑘,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝑄𝑘,𝐿

𝜕𝛼1

⋯𝜕𝑄𝑘,𝐿

𝜕𝛼𝑁 ]

(4.1.10)

𝑪(𝒙) =

[ 𝜕𝑃𝑘,1

𝜕𝐼1⋯

𝜕𝑃𝑘,1

𝜕𝐼𝑁𝜕𝑄𝑘,1

𝜕𝐼1⋯

𝜕𝑄𝑘,1

𝜕𝐼𝑁⋮ ⋱ ⋮

𝜕𝑃𝑘,𝑀

𝜕𝐼1⋯

𝜕𝑃𝑘,𝑀

𝜕𝐼𝑁𝜕𝑄𝑘,𝑀

𝜕𝐼1⋯

𝜕𝑄𝑘,𝑀

𝜕𝐼𝑁

𝜕𝑃𝑘,1

𝜕𝛼1

⋯𝜕𝑃𝑘,1

𝜕𝛼𝑁

𝜕𝑄𝑘,1

𝜕𝛼1

⋯𝜕𝑄𝑘,1

𝜕𝛼𝑁

⋮ ⋱ ⋮𝜕𝑃𝑘,𝑀

𝜕𝛼1

⋯𝜕𝑃𝑘,𝑀

𝜕𝛼𝑁

𝜕𝑄𝑘,𝑀

𝜕𝛼1

⋯𝜕𝑄𝑘,𝑀

𝜕𝛼𝑁 ]

(4.1.11)

where 𝐿 is the number of each type of measurements

𝑀 is the number of zero-injection nodes

𝑁 is the number of line sections (and corresponding line current flows)



The partial derivatives shown in equations 4.1.10 and 4.1.11 are given below:

1) Power flow measurements: When branch power flow measurements are in the same line segments as

the state variable, the partial derivatives are:

𝜕𝑃𝑘𝑚

𝜕𝐼𝑘𝑚

= 𝑉𝑘 cos(𝛿𝑘 − 𝛼𝑘𝑚) = 𝑃𝑘𝑚

𝐼𝑘𝑚

(4.1.12)

𝜕𝑃𝑘𝑚

𝜕𝛼𝑘𝑚

= 𝑉𝑘 Ikmsin(𝛿𝑘 − 𝛼𝑘𝑚) = 𝑄𝑘𝑚 (4.1.13)

𝜕𝑄𝑘𝑚

𝜕𝐼𝑘𝑚

= 𝑉𝑘 sin(𝛿𝑘 − 𝛼𝑘𝑚) = 𝑄𝑘𝑚

𝐼𝑘𝑚

(4.1.14)

𝜕𝑄𝑘𝑚

𝜕𝛼𝑘𝑚

= −𝑉𝑘 Ikmcos(𝛿𝑘 − 𝛼𝑘𝑚) = −𝑃𝑘𝑚 (4.1.15)

where 𝑉𝑘 is the voltage at node k

𝛿𝑘 is the voltage angle at node k

𝛼𝑘𝑚 is the current angle at line km.

Otherwise, when the measurement and the state variable are not in the same line segment, all the

partial derivatives are zeros.

2) Current magnitude measurements: When the current magnitude measurement is in the same line

segments as the state variable, the partial derivative with respect to current magnitude is one. If the

current measurement and state variable are on different line section, the partial derivative is zero.

Partial derivatives with respect to current angles are always zeros.

𝜕𝐼𝑘𝑚

𝜕𝐼𝑠𝑡= {

1, 𝑖𝑓 𝑠𝑡 = 𝑘𝑚 0, 𝑖𝑓 𝑠𝑡 ≠ 𝑘𝑚

(4.1.16)

𝜕𝐼𝑘𝑚

𝜕𝛼𝑠𝑡

= 0 (4.1.17)

where 𝐼𝑠𝑡 is the state variable corresponding to current magnitude at line st

𝛼𝑠𝑡 is the state variable corresponding to current angle at line st.

3) Voltage magnitude measurements: When the source node voltage and line currents are known,

voltage at any point of the network can be calculated by subtracting the voltage losses that happen

between source node (node 1) and the studied node k from the source node voltage. In a radial feeder,

the voltage at node k can be calculated with equation 4.1.18 assuming the feeder nodes have been

numbered as in Figure 4.1.1.

��𝑘 = ��1 − ∑𝐼��−1,𝑖��𝑖−1,𝑖

𝑘

𝑖=2

(4.1.18)



1 2 3 k-1 k

nodes

Figure 4.1.1. Node numbering on a radial feeder.

The Jacobian matrix elements related to voltage magnitude measurements can be divided into two

groups. The first group contains elements that are between node 1 and measured node k. Then the

partial derivatives are:

𝜕𝑉𝑘

𝜕𝐼𝑖−1,𝑖

= −cos 𝛿𝑘 ∙ 𝑍𝑖−1,𝑖 cos(𝛼𝑖−1,𝑖 + 𝜃𝑖−1,𝑖)

− sin 𝛿𝑘 ∙ 𝑍𝑖−1,𝑖 sin(𝛼𝑖−1,𝑖 + 𝜃𝑖−1,𝑖) (4.1.19)

𝜕𝑉𝑘

𝜕𝛼𝑖−1,𝑖

= cos 𝛿𝑘 ∙ 𝐼𝑖−1,𝑖 𝑍𝑖−1,𝑖 sin(𝛼𝑖−1,𝑖 + 𝜃𝑖−1,𝑖)

− sin 𝛿𝑘 ∙ 𝐼𝑖−1,𝑖𝑍𝑖−1,𝑖 cos(𝛼𝑖−1,𝑖 + 𝜃𝑖−1,𝑖) (4.1.20)

where 𝐼𝑖−1,𝑖 is the current magnitude on line that goes from node i−1 to node i, where i

belongs to a group of nodes that are between nodes 1 and k (node k belongs

to this group, node 1 is excluded from this group)

𝛼𝑖−1,𝑖 is the current angle on the line that goes from node i−1 to node i

𝑍𝑖−1,𝑖 is the impedance on the line that goes from node i−1 to node i

𝜃𝑖−1,𝑖 is the impedance angle on the line that goes from node i−1 to node i

𝛿𝑘 is the voltage angle at node k.

The second group contains elements that are not between node 1 and measured node. All partial derivatives in this group are zeros.

4) Current injection measurements: When the state variable is connected to a line segment feeding the

measured node (t=k), the current injection measurement partial derivative with respect to branch

current magnitude is one. If the state variable is connected to a line segment below the measured node

(s=k), then the partial derivative is minus one. The partial derivative is zero always when the state

variable is connected to a line segment that does not connect to the measured node (𝑠 ≠ 𝑘 𝑎𝑛𝑑 𝑡 ≠

𝑘). Partial derivatives with respect to the current angle are always zero.

𝜕𝐼𝑘𝜕𝐼𝑠𝑡

= {

1, 𝑖𝑓 𝑡 = 𝑘 −1, 𝑖𝑓 𝑠 = 𝑘

0, 𝑖𝑓 𝑠 ≠ 𝑘 𝑎𝑛𝑑 𝑡 ≠ 𝑘 (4.1.21)

𝜕𝐼𝑘𝜕𝛼𝑠𝑡

= 0 (4.1.22)



5) Power injection measurements: The Jacobian matrix elements related to power injection measurements can be divided into three groups. When a line segment is connected to the measured node and is feeding it, the partial derivatives are:

𝜕𝑃𝑘

𝜕𝐼𝑠𝑘= 𝑉𝑘 cos(𝛿𝑘 − 𝛼𝑠𝑘) (4.1.23)

𝜕𝑃𝑘

𝜕𝛼𝑠𝑘

= 𝑉𝑘 Isksin(𝛿𝑘 − 𝛼𝑠𝑘) (4.1.24)

𝜕𝑄𝑘

𝜕𝐼𝑠𝑘= 𝑉𝑘 sin(𝛿𝑘 − 𝛼𝑠𝑘) (4.1.25)

𝜕𝑄𝑘

𝜕𝛼𝑠𝑘

= −𝑉𝑘 Iskcos(𝛿𝑘 − 𝛼𝑠𝑘) (4.1.26)

When a line segment is connected to the measured node and is below it, the partial derivatives are:

𝜕𝑃𝑘

𝜕𝐼𝑘𝑚

= −𝑉𝑘 cos(𝛿𝑘 − 𝛼𝑘𝑚) (4.1.27)

𝜕𝑃𝑘

𝜕𝛼𝑘𝑚

= −𝑉𝑘 Ikmsin(𝛿𝑘 − 𝛼𝑘𝑚) (4.1.28)

𝜕𝑄𝑘

𝜕𝐼𝑘𝑚

= −𝑉𝑘 sin(𝛿𝑘 − 𝛼𝑘𝑚) (4.1.29)

𝜕𝑄𝑘

𝜕𝛼𝑘𝑚

= 𝑉𝑘 Ikmcos(𝛿𝑘 − 𝛼𝑘𝑚) (4.1.30)

In equations 4.1.23 – 4.1.26, subscript s can be any node that is above the node k and is connected to it with a line. In equations 4.1.27 – 4.1.30, subscript m can be any node that is below the node k and is connected to it with a line. When a line segment is not connected to the measured node, all partial derivatives are zeros [Wang

2004].

4.1.1.4 Bad Data Detection

In IDE4L project, all the input data given to the state estimator has gone through a filter that filters out bad

measurements. However, to make absolutely sure that bad measurements are not used in state estimation,

bad data detection is added also to the state estimator. If undetected, bad data will corrupt the state

estimation results and can in some cases prevent the state estimator convergence.

Measurements may contain errors due to various reasons. Meters can have biases, drifts or wrong

connections. Telecommunication system failures can also lead to large deviations in recorded

measurements. Some measurement errors are easy to detect with simple logical rules. For example,

negative voltage and current magnitudes and measurements, which are several orders of magnitude larger

or smaller than expected, are easily recognized as bad data. In our state estimation algorithm, this kind of

rough bad data detection is done right in the beginning of the algorithm. Unfortunately, not all types of bad

data are detected that easily. However, in more indistinct cases, other detection methods can be utilized.



In WLS state estimation, bad data detection can be made by examining the measurement residuals. This

has to be done after the estimation process. The bad data detection is essentially based on the statistical

properties of the residuals. One of the most used bad data detection methods is the Largest Normalized

Residual 𝑟𝑚𝑎𝑥𝑁 -test. This test is composed of the following steps [Abur2004]:

1) Solve the WLS estimation and obtain the elements of the measurement residual vector (𝒓):

𝒓 = 𝒛 − 𝒉(𝒙) (4.1.31)

2) Compute the normalized residuals (𝒓𝑁):

𝒓𝑁 =|𝒓|

√𝛀𝐢𝐢

, (4.1.32)

where 𝛀𝐢𝐢 is 𝑑𝑖𝑎𝑔(𝛀)

𝛀 is 𝐶𝑜𝑣(𝒓).

3) Find the largest normalized residual (𝒓𝑚𝑎𝑥𝑵 ).

4) If 𝒓𝑚𝑎𝑥𝑁 > 𝑐, then the corresponding measurement is erroneous. Here, c is the chosen detection

threshold, usually 3.0 (3σ threshold, i.e. all data that is more than three standard deviations away

from the expected value is labelled as bad data).

5) If bad data is detected, eliminate the faulty measurement from the measurement set and go back

to step 1.

The faulty measurements are eliminated one by one. After each elimination, WLS state estimation

procedure is repeated.

The largest normalized residual test can detect bad data if the removal of the corresponding measurement

does not render the system unobservable. It is possible to identify all cases of single bad data where the

faulty measurements are not critical or belong to a critical pair or critical k-tuple. Critical measurements are

those measurements whose removal would cause the system to become unobservable. A critical pair and

k-tuple contain two or more measurements, respectively, whose simultaneous removal would make the

system unobservable.

In the case of multiple bad data, only part of the measurement errors can be identified. Faulty

measurements with weakly correlated measurement residuals can be identified. If the measurement

residuals are strongly correlated, the bad data can be identified only in the case of non-conforming bad

data. If the identification of faulty measurement fails, the largest normalized residual test can incorrectly

remove a faultless measurement.

Because our state estimator is based on equality constrained WLS estimation, the measurement residual

covariance matrix cannot be solved as shown in [Abur2004]. Solution for this problem can be found from

[Wu1988]. In equality constrained state estimation the measurement residual covariance matrix 𝛀 is equal

to

𝐶𝑜𝑣(𝒓) = 𝑹−𝟏 − 𝑯𝑬𝟏𝑯𝑻, (4.1.33)



where 𝑬𝟏 is the upper left corner of the inverse of 𝑭.

𝑭−1 = [𝑯𝑇𝑹−1𝑯 𝑪𝑇

𝑪 0]−1

= [𝑬𝟏 𝑬𝟐

𝑇

𝑬𝟐 𝑬𝟑], (4.1.34)

where 𝑪 is the Jacobian matrix of the equality constraint function.

The problem with measurement residual based bad data detection is that it requires a certain amount of

redundancy from the measurement configuration. In distribution networks, the number of measurements

and thus also the redundancy level is very limited. Considering traditional distribution network

measurement setup, real-time power flow measurements only in the beginning of the feeder and load

pseudo measurements, we cannot identify bad data unambiguously. We can only detect that bad data exist

and it is either in the feeder power flow measurement or in one of the load pseudo measurements. In this

situation we use the pseudo measurements to set plausible limits to the real-time measurements and if the

real-time measurement is outside these limits, then it is interpreted as bad data. When using this approach,

the bad data detection threshold should be raised from the commonly used 3σ threshold and accurate load

models should be used as pseudo measurements [Mutanen2011]. In IDE4L project the measurement

redundancy will be higher than in traditional distribution systems and bad data detection will be more

useful.

Sometimes bad data can cause non-convergence to the state estimation algorithm. If the algorithm does

not converge, the measurement residuals cannot be calculated. In this case the solution is to remove all

real-time measurements and calculate state estimation using only pseudo measurements. After the pseudo

measurement based state estimates have been calculated, the largest normalized residual test is used to

evaluate the real-time measurements. Measurement with the largest normalized residual is identified as

bad data and removed from the measurement set. Then the state estimation is run again. This procedure is

repeated until all erroneous measurements are removed and the state estimator converges.

4.1.2 Algorithm Steps

The state estimation algorithm description can be divided into two levels; general level and detailed level.

The algorithm has been described on general level shortly in chapter 3.1 and in more detail in design

specification documents [IDE4L2014a] and [IDE4L2014b]. Here we will focus on describing what happens

inside the state estimation block that is step 6 in general description (see Table 3.1.1 and Figure 3.1.1) and

contains the actual state estimator that does the WLS calculation.

The state estimation algorithm has been implemented as an Octave function. The algorithm flow chart has

been given in Figure 4.1.3 and the 13 distinct steps have been described below.

1) Input validity check: The state estimation algorithm inputs go through a simple validation process. The

purpose of input validation is to filter out coarse errors with simple logical rules. For example, negative

current magnitude measurements and node voltage measurements that are twice as large as the

nominal voltage are labelled as bad data and are removed.

2) Branch current calculation: Initial branch currents are calculated using the load pseudo measurements

provided as inputs. Backward sweep algorithm is used to calculate the branch currents. Backward

sweep starts from the end of the network and sums branch currents according to Kirchhoff’s current

law as it progresses upwards. This has been illustrated in Figure 4.1.2a.



1

S 2

34

5

6

7

S8

9

10

11

12

13

14

a) Backward sweep b) Forward sweep

Figure 4.1.2. Backward/forward sweep calculation steps.

The line charging capacitances are taken into account when calculating the branch currents. π-model is

used to model the lines. As can be seen from Figure 4.1.3, the branch current Ikm has three different

values depending weather it is measured from the beginning, middle or end of the line. In this work,

the current at the beginning of the line is chosen as state variable. The charging currents are calculated

with equations 4.1.35 and 4.1.36.

Figure 4.1.3. Line π-model.

𝐼𝐶,𝑘𝑚,𝑘 = 𝑗𝑉𝑘𝐵𝐶,𝑘𝑚

2= 𝑗𝑉𝑘𝜋𝑓𝐶𝑘𝑚 , (4.1.35)

𝐼𝐶,𝑘𝑚,𝑚 = 𝑗𝑉𝑚𝐵𝐶,𝑘𝑚

2= 𝑗𝑉𝑚𝜋𝑓𝐶𝑘𝑚 , (4.1.36)

where 𝑉𝑘 is the node 𝑘 voltage

𝑉𝑚 is the node 𝑚 voltage

𝐵𝐶,𝑘𝑚 is the capacitive susceptance on line section 𝑘–𝑚

𝑓 is the system frequency

𝐶𝑘𝑚 is the capacitance on line section 𝑘–𝑚

3) Node voltage calculation: Initial node voltages are calculated using the forward sweep method. Here

the previously calculated mid-branch currents are used to calculate the voltage losses on each line

section. The calculation starts from node number 1 and continues downwards as depicted in Figure

4.1.2b.

4) Covariance matrix formation: Measurement covariance matrix is formed from the input measurement

accuracies. The covariance matrix is a diagonal matrix and the diagonal elements correspond to the

accuracy of each measurement (pseudo measurements included).



5) Measurement vector formation: The provided measurements are collected into a measurement

vector. The order of measurements is same as in Table 4.1.1 and the same order is applied throughout

this algorithm.

6) State variable vector formation: The state variable vector is formed from the previously calculated

branch currents. The N first elements are branch current magnitudes and the elements from N+1 to 2N

are branch current angles.

7) Jacobian matrix calculation: Subfunction Jacobian.m calculates the Jacobian matrices H and C

described in equations 4.1.10 and 4.1.11. Also the measurement function values h(x) and equality

constraint function values c(x) are calculated inside this subfunction.

8) Calculation of ∆x: Corrections to the state variable vector are calculated using the equation 4.1.9.

9) State variable vector update: The state variable vector is updated by summing the previously

calculated corrections ∆x to it.

10) Mid-line current calculation: First, the state variable vector is converted into a corresponding branch

current vector. Then the currents at the middle of each line are calculated by adding appropriate

charging currents.

11) Node voltage calculation: The node voltages are recalculated using the forward sweep.

12) Bad data detection: Once the largest value in correction vector ∆x falls below the pre-set threshold ε,

the algorithm exits from the first loop and starts bad data detection. The bad data detection is done

using the Largest Normalized Residual 𝑟𝑚𝑎𝑥𝑁 –test as described in chapter 4.1.1.4. If bad data is

detected, it is removed and the algorithm returns to step 4.

13) Output calculation: The other required state estimation function outputs are calculated from the

branch currents and node voltages.



1. Check input validity

2. Calculate branch currents backward_sweep.m

3. Calculate node voltages forward_sweep.m

4. Form covariance matrix R covariance_matrix.m

5. Form measurement vector z

7. Calculate:

Jacobian matrices H and C

measurement values h

equality constaint values c

6.Form state variable vector

Jacobian.m

8. Calculate correction ∆x using

the Lagrange method

9. Add correction ∆x to state

variable vector

11. Calculate node voltages

Is correction ∆x<ε

forward_sweep.m

Ready

NO

YES

Main function

Subfunctions

measurement_vector.m

10. Calculate currents in the

middle of the line

12. Bad data detection bad_data_detection.m

Bad data detected

NO

YES

Remove bad data

13. Calculate outputs

Figure 4.1.3. State estimation algorithm flow chart.



4.1.3 Inputs and Outputs

This chapter describes the inputs and outputs for the state estimation algorithm core function

state_estimator.m that contains the functionalities described in chapter 4.1.1. Some additional support

functions are needed to format the information read from PSAU and SSAU databases into a format

understood by the state estimation core. Some of these support functions are described in Chapter 4.1.5.

Some are still under development and will be finalized during the algorithm testing phase.

This chapter describes inputs and outputs only for the state estimation block presented in chapter 4.1.2.2.

Inputs and outputs for the whole state estimation procedure have been described in chapter 3.1 and in the

design specification documents [IDE4L2014a] and [IDE4L2014b].

4.1.3.1 Inputs

The state estimation main function state_estimator.m shown in Figure 4.1.3 requires the following inputs:

1) Bus matrix, containing the bus numbers and load and production estimates or measurements for

each bus.

2) Line matrix, containing resistance, impedance and capacitive susceptance for each line section.

3) Line active power flow measurement matrix

4) Line reactive power flow measurement matrix

5) Line current flow measurement matrix

6) Bus current injection measurement matrix

7) Bus voltage measurement matrix.

The bus matrix contains as many rows as there are nodes in the network. The columns of the bus matrix

contain the information shown in table 4.1.2. The state estimation function state_estimator.m calculates

the system state using per unit values. Therefore, the inputs in bus matrix columns 5–12 are given in per

unit.

Table 4.1.2. Bus matrix columns.

Column Column name Column description

1 Bus number The bus numbering has to start from 1 and the numbering grows with

increments of 1 towards the end of the line

2 Phase A connection 1 if phase A is connected, 0 if not connected

3 Phase B connection 1 if phase B is connected, 0 if not connected

4 Phase C connection 1 if phase C is connected, 0 if not connected

5 3-phase power Estimate or measurement for the 3-phase power injection in this node

6 3-phase power std Standard deviation for the estimated or measured 3-phase power injection

7 Phase A power Estimate or measurement for phase A power injection

8 Phase B power Estimate or measurement for phase B power injection

9 Phase C power Estimate or measurement for phase C power injection

10 Phase A power std Standard deviation for the estimated or measured power injection in phase A

11 Phase B power std Standard deviation for the estimated or measured power injection in phase B

12 Phase C power std Standard deviation for the estimated or measured power injection in phase C

The line matrix contains as many rows as there are line sections in the network. The columns of the line

matrix contain the information shown in table 4.1.3. The inputs in line matrix columns 3–5 are given in per

unit.



Table 4.1.3. Line matrix columns.


1 Start node Node from which the line section starts

2 End node Node to which the line section ends

3 Line resistance Line resistance in per unit

4 Line reactance Line reactance in per unit

5 Line capacitive susceptance Line capacitive susceptance in per unit

6 Phase A connection 1 if line for phase A exists, 0 if not

7 Phase A connection 1 if line for phase B exists, 0 if not

8 Phase A connection 1 if line for phase C exists, 0 if not

The line active power, reactive power and current flow matrices contain as many rows as there are

measurements. The columns of these matrices contain the information shown in table 4.1.4. The inputs in

columns 4–9 are given in per units.

Table 4.1.4. Flow measurement matrix columns.


1 Start node Node from which the measured line sections starts

2 End node Node to which the measured line sections ends

3 Measurement location Measurements location

If at the beginning of the line section, then Col 3 = Col 1

If at the end of the line section, then Col 3 = Col 2

4 Phase A measurement Phase A measurement value in per unit

5 Phase B measurement Phase B measurement value in per unit

6 Phase C measurement Phase C measurement value in per unit

7 Phase A measurement std Standard deviation for phase A measurement

8 Phase B measurement std Standard deviation for phase B measurement

9 Phase C measurement std Standard deviation for phase C measurement

The bus current injection and voltage measurement matrices contain as many rows as there are

measurements. The columns of these matrices contain the information shown in table 4.1.5. The inputs in

columns 2–7 are given in per unit.

Table 4.1.5. Bus current injection and voltage measurement matrix columns.


1 Bus Measurement location

2 Phase A measurement Phase A measurement value in per unit

3 Phase B measurement Phase B measurement value in per unit

4 Phase C measurement Phase C measurement value in per unit

5 Phase A measurement std Standard deviation for phase A measurement

6 Phase B measurement std Standard deviation for phase B measurement

7 Phase C measurement std Standard deviation for phase C measurement

The minimum requirement for the state estimation to run is that both bus and line matrices exists and bus

power injection values, either load/production estimates or measurements, have been written to the bus

matrix. The load and production estimates or measurements can be either 3-phase or single-phase. Single-

phase estimates or measurements are used primarily if they exist. If single-phase values do not exist, the



phase-wise values are estimated based on the 3-phase values. The standard deviation connected to each

measurement, reflects the accuracy of that measurement. During the WLS estimation, the measurements

are weighted according to their accuracies. The other measurement matrices are optional and contain data

only if measurements of that type exist.

4.1.3.2 Outputs

The state estimation function state_estimator.m gives out the following outputs:

1) Phase-wise bus voltages

2) Phase-wise line current flows in the beginning of each line section*

3) Phase-wise line current flows in the middle of each line section*

4) Phase-wise line current flows in the end of each line section*

5) Phase-wise line power flows on each line section

6) Phase-wise power losses on each line section

7) Phase-wise power injections on each bus

* Currents in the beginning, middle and end of the line are slightly different because the lines have been

modelled with a π-model (see Figure 4.1.3).

All these outputs are complex and it is possible to extract voltage and current magnitudes or angles or

active and reactive power components from these. The outputs are given in per unit and they are later

converted into corresponding volt, ampere and kilowatt values.

4.1.4 Coordination between MVSE and LVSE

MVSE and LVSE operate independently in PSAU and SSAU (primary and secondary substation automation

units). In normal operation mode, when a measurement setup corresponds to the IDE4L concept and

secondary substation voltage is measured with a RTU, no information is changed directly between the state

estimators.

LVSE outputs the phase-wise power flows on the distribution transformer secondary. The medium voltage

network load and production forecaster (MVF) needs information on primary side power flows. To fill this

gap, a distribution transformer model is fitted between these algorithms. The calculation block containing

the distribution transformer model takes into account the distribution transformer winding configuration

and other parameters and calculates the phase-wise primary side power flows from the phase-wise

secondary side power flows and voltages. Figure 4.1.4 illustrates this relationship.



LVSEMV/LV

transformer

model

Database

syncronization

SSAU database

MV/LV transformer

secondary side

power flows and

voltages

MV/LV transformer

primary side

power flowsMV/LV transformer

primary side

power flows

MVF

PSAU database

Historical

MV/LV transformer

primary side

power flows

MV/LV transformer

primary side

power flows

Figure 4.1.4. Data transfer between LVSE and MVSE in normal operation mode.

If the secondary substation voltage is not measured, the LVSE uses MV/LV network connection point

voltage estimates given by the MVSE. The state estimators become interdependent and their relationship

can be illustrated with Figure 4.1.5.

LVSEMV/LV

transformer

modelMVSE

SSAU database

MV/LV transformer

secondary side

voltages

MV/LV transformer

primary side

voltages

MV/LV transformer

primary side

voltages

MV/LV transformer

secondary side

power flows

Figure 4.1.5. Data transfer between MVSE and LVSE when secondary substation voltage measurement is

missing.

4.1.5 Optional (not implemented) Functionalities

The following functionalities would be possible to add to a WLS estimator.

4.1.5.1 State Estimate Uncertainties

If needed, a WLS estimator can output also the state estimate uncertainties. Uncertainties for the state

variables can be extracted easily from the gain matrix. The state variable variances are found from the

diagonal of the inverted gain matrix.

The calculation of other uncertainties requires some additional work. An additional Jacobian matrix

containing partial derivates (with respect to the state variables) for all those states for which we wish to

calculate uncertainties must be formed. Using equations 4.1.12 – 4.1.30, it is possible to calculate partial

derivates to any branch power flow, branch current flow, node voltage, node current injection or node

power injection. Once the new Jacobian (𝐊) matrix has been formed, the uncertainties (variances) can be

calculated as is shown in equation 4.1.37 [Cobelo2007].

𝑣𝑎𝑟(𝑋) = 𝑑𝑖𝑎𝑔(𝑲 ∙ 𝑮−𝟏 ∙ 𝑲𝑇), (4.1.37)

where 𝐊 is the Jacobian matrix

𝑮 is the gain matrix (𝑯𝑇𝑹−1𝑯).



4.1.5.2 Calculation of Meshed Networks

The branch current based WLS estimator can be modified to calculate also (weakly) meshed networks

[Baran1995], [Lin et al. 2001], [Pau2013]. Each mesh adds a constraint on branch currents because the

Kirchhoff’s voltage law must be satisfied in every loop. Kirchhoff’s voltage law states that the sum of

voltage losses around the loop must be zero.

∑ 𝝀𝑗𝒛𝑗𝒊𝑗𝑗∈Λ = 0, (4.1.38)

where Λ is the set of branches forming the loop backbone and 𝒛𝑗 and 𝒊𝑗 are the impedance and current

phasor of the jth branch. 𝝀𝑗 is +1 or −1 depending on the reference loop direction with respect to the

branch j direction. The loop reference direction can be chosen arbitrarily. The branch j direction is the

downward direction of the branch when the network is radial. Fig. 4.1.6 visualizes this.

Reference

direction

1

2

01

13 14

15

1

6

17 18

19

Loop break point

Figure 4.1.6. Example of λ values with respect to a chosen reference direction.

The above presented conditions can be added to the WLS problem either as virtual measurements or as

equality constraints. The output of equation 4.1.35 is complex and both real and imaginary part of the

output must be zero. From this we get two measurement equations corresponding real part (4.1.39) and

imaginary part (4.1.40). These are

𝒄(1) = ∑𝝀𝑗|𝒛𝑗||𝒊𝑗|

𝑗∈Λ

cos(𝜶𝑗 + 𝜽𝑗) (4.1.39)

𝒄(2) = ∑𝝀𝑗|𝒛𝑗||𝒊𝑗|

𝑗∈Λ

sin(𝜶𝑗 + 𝜽𝑗) (4.1.40)

where 𝒛𝑗 is the line j impedance and 𝒊𝑗 is the line j current. 𝛼 and 𝜃 are the line current and impedance

angles, respectively. The corresponding Jacobian entries (partial derivates with respect to the state

variables |𝒊| and 𝜶) for equation 4.1.39 are

{

𝜕𝑐(1)

𝜕𝑖= 𝝀𝑗|𝒛𝑗| cos(𝜶𝑗 + 𝜽𝑗), if 𝑗 ∈ Λ

𝜕𝑐(1)

𝜕𝑖= 0, if 𝑗 ∉ Λ

(4.1.41)



{

𝜕𝑐(1)

𝜕𝛼= −𝝀𝑗|𝒊𝑗||𝒛𝑗| sin(𝜶𝑗 + 𝜽𝑗), if 𝑗 ∈ Λ

𝜕𝑐(1)

𝜕𝛼= 0, if 𝑗 ∉ Λ

(4.1.42)

and the Jacobian entries corresponding equation 4.1.40 are

{

𝜕𝑐(2)

𝜕𝑖= 𝝀𝑗|𝒛𝑗| sin(𝜶𝑗 + 𝜽𝑗), if 𝑗 ∈ Λ

𝜕𝑐(2)

𝜕𝑖= 0, if 𝑗 ∉ Λ

(4.1.43)

{

𝜕𝑐(2)

𝜕𝛼= 𝝀𝑗|𝒊𝑗||𝒛𝑗| cos(𝜶𝑗 + 𝜽𝑗), if 𝑗 ∈ Λ

𝜕𝑐(2)

𝜕𝛼= 0, if 𝑗 ∉ Λ

(4.1.44)

When the network voltages are calculated, a temporary break point is added to each loop, so that radial

forward-sweep can be used to calculate the voltages.

4.1.6 Performance tests

First the validity of power flow equations used in the developed state estimator is verified by comparing it

to the power flow calculation algorithm found in the Power System Toolbox [PST2015]. The Power System

Toolbox was conceived and initially developed by Dr. Kwok W. Cheung and Prof. Joe Chow from Rensselaer

Polytechnic Institute in the early 1990s. From 1993 to 2009, it was marketed, and further developed, by

Graham Rogers (formerly Cherry Tree Scientific Software), and is in use by utilities, consultants and

universities worldwide. In this comparison the state estimator is used to calculate only the power flow by

omitting all measurements.

A modified IEEE 37-bus test feeder was used in this comparison. The modifications included:

Single-phase equivalent of the original 3 phase network was calculated

All unbalanced loads were changed into balanced loads

Voltage regulator between nodes 799 and 701 was removed

The unloaded transformer connected to node 709 and the unloaded node 775 were removed and

the network was reduced to 36 buses.

The one-line diagram of the modified test feeder is shown in Figure 4.1.7.

Figure 4.1.7. One-line diagram of the modified test feeder.



The node voltages calculated with both PST and state estimator (SE) are shown in Figure 4.1.8. The

differences in calculated voltages are so small that they cannot be seen from this figure. The average

difference in calculated node voltages was only 3.0410∙10-8 and the maximum error was 3.9154∙10-8. The

dedicated load flow algorithm was approximately 10 times faster in load flow calculation than the state

estimator. When repeated 100 times, the average calculation times with this 36-bus test feeder were 2.5

and 23 milliseconds, respectively.

Figure 4.1.8. Comparison between PST load flow and developed state estimator.

The same 36-bus test feeder was used to compare the state estimation accuracy of the developed method

to a reference method. The commercially used state estimation method described in chapter 2.1.2 of the

state of the art document [IDE4L2014c] was used as a reference. The loads were assumed to be normally

distributed and loads in area 1 assumed to have 50 % relative standard deviation and the loads in area 2

were assumed to have 20 % relative standard deviation. Monte Carlo simulation with 1000 repetitions was

calculated. Each simulation contained the following steps:

1) Draw random values for the normally distributed loads

2) Calculate load flow

3) Extract feeder active and reactive power flows (at the beginning of the feeder) from the load flow

results and use these as real-time measurements in state estimation

4) Add random measurement errors corresponding to ±1 % measurement accuracy to the power flow

measurements

5) Calculate state estimation with both studied methods

6) Calculate the difference between estimated node voltages and voltages calculated in step 2.

The comparison results are shown in Figure 4.1.9. The developed state estimator has 24 % smaller

estimation errors. With this measurement setup, also the reference method could achieve this same

estimation accuracy if the simple modifications proposed in chapter 2.1.3 of the state of the art document

[IDE4L2014c] were to be implemented. However, even then the reference method could not utilize

measurements as effectively as the developed WLS-based estimation method. The developed state

estimator can for example use voltage measurements at the end of the network to improve load and

0 5 10 15 20 25 30 35 400.98

0.99

1

1.01

1.02

1.03

1.04

1.05

Node

Voltage (

p.u

.)

PST

SE



voltage estimates also in other parts of the network. Figure 4.1.10a shows how voltage measurements at

nodes 11, 30 and 36 affect on the voltage estimation accuracy. The benefit of voltage measurements

depends heavily on the voltage measurement accuracy. Figure 4.1.10b shows that state estimation

accuracy improves also when the load pseudo measurement accuracy is improved. Therefore, it is

important that the load and production forecaster is able to provide accurate load and production

estimates.

Figure 4.1.9. Comparison between the reference method and the developed state estimation method.

Figure 4.1.10. Estimation accuracy a) with different voltage measurement accuracies and b) with improved pseudo measurement accuracies.

4.2 State Forecasting The same algorithm that was used for state estimation is also used for state forecasting. When used for

state forecasting, the algorithm inputs are load and productions forecasts instead of estimates and

measurements. The state estimator and state forecaster use the same algorithm core, but they have two

fundamental differences.

1) According to the IDE4L concept, only load and production forecasts will be made. This means that

the only inputs to the state forecaster are bus and line matrixes, where the bus matrix contains the

load and production forecasts and the line matrix contains the line parameters. The optional

measurement matrices are empty. The measurement redundancy is zero, and what the state

5 10 15 20 25 30 350.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Node

Avera

ge v

oltage e

stim

ation e

rror

(%)

Existing SE method

Proposed SE method

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Node

Ave

rag

e v

olta

ge

estim

atio

n e

rro

r (%

)

a) Voltage measurements

Base case

±2 % accuracy

±1 % accuracy

±0,5 % accuracy

±0,2 % accuracy

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Node

Ave

rag

e v

olta

ge

estim

atio

n e

rro

r (%

)

b) Pseudo measurements

Base case

-25 % RSD

-50 % RSD



forecaster actually calculates is just a simple load flow calculation. At this level, the state forecasts

could be calculated with any power flow algorithm, but by using the algorithm described in chapter

4.1 we reserve the possibility to improve the state forecasting accuracy with redundant

(overlapping) forecasts. Also, with this method, it is possible to extract uncertainties for the

forecasted states.

2) The state forecasting is done for N different future time spans in the forecasting horizon. Basically,

the state forecasting is repeated N times with different inputs, where each input forecasts the load

and production on that specific time span. See Figure 3.1.1 to observe how the state estimator and

forecaster are connected and how the N different state forecasts are calculated.

The state forecaster input structure is similar to the state estimator and also the outputs are the same that

were presented in chapter 4.1.3.2.

4.3 Load Forecasting For every node in the network at time t, the forecasting algorithm can provide a prediction of the load at

time t+1. In order to model the behaviour of the load over time, and thus be able to estimate the future

load three kinds of information are used: historical measurements, calendar information and

meteorological forecasts. This procedure is expanded to deliver h predictions in the future, with two

different resolutions, up to the forecasting horizon t+h.

The IDE4L project will follow two approaches for forecasting the load demand. Which one of the

approaches will be selected will depend on the quality of the recorded measurements of the consumer load

that is to be forecasted. In case the latest measurements are not available in the database, the future load

demand is estimated by querying a lookup table, containing a predefined load baseline. If every needed

measurement is stored in the data exchange platform, past measurements are used to infer the load

demand in the future. This load-forecasting algorithm has two separate parts. The training algorithm fits

the model based on historical data. This procedure is executed offline. The second part of the algorithm is

executed online, at every time step and for every node in the network. The forecaster is able to provide

several predictions in the future up to a forecasting horizon by recursively adding the new forecasts to the

feature vector.

Load forecasting for medium voltage and low voltage follow the same procedure, unless otherwise

indicated in the remainder of this section

4.3.1 Real Time Measurement Reading and Filtering

The implementation of this algorithm will account for errors in the data collection step. Real

implementations of smart meter networks are subject to errors in measurements due to communication

failures, corruption of the data or temporal unavailability of the meter.

New and historic measurements are read by the forecasting algorithm from the low voltage data exchange

platform. New measurements are filtered based on the statistical properties of the training data set. The

training data set must comprise a sufficient range of historic values with the appropriate resolution. Two

issues might arise when querying the database for the most recent measurement. The first one is the lack

of such a measurement due to communication failures, data corruption or meter malfunction. In this case

the measurement is substituted by the last forecasted value for the particular moment. The second issue is

erroneous measurements. Based on the mentioned training data set, any new measurement not contained

within five times the standard deviation of the time series is disregarded. In its place the last forecast for



that particular time is used instead. This procedure introduces uncertainties in the forecast and degrades its

behaviour. However, it allows for a forecast to be made in every situation. If the number of missing values

is above a predefined threshold the algorithm will automatically switch to the first scheme described in this

section, and future load demands will be estimated based on the predefined load baselines.

4.3.2 Load Baselines

Nodes with a similar load demand profile are grouped together, in order to create a load baseline that best

defines their behaviour. Similar nodes are clustered using the K-means algorithm.

The K-Means algorithm clusters data by trying to separate samples in n groups of equal variance,

minimizing a criterion known as within-cluster sum-of-squares, which is a measure of how internally

coherent clusters are.

∑min𝜇𝑗

(‖𝑥𝑗 − 𝜇𝑖‖2)

𝑛

𝑖=0

(4.3.1)

The k-means algorithm divides a set of N samples X into K disjoint clusters C, each described by the mean

𝜇𝑗 of the samples in the cluster, known as centroids.

The K-means algorithm used in this implementation randomly selects k samples from the training data set

as centroids. It then assigns each sample to the nearest of the k clusters. It then creates new centroids by

taking the mean value of all of the samples assigned to each previous centroid. The difference between the

old and the new centroids are computed and the algorithm repeats these last two steps until this value is

less than a threshold.

Once every node has been assigned to one of the k clusters, a lookup table is created by averaging all

measurements from all samples in the k cluster having the same time of the day and day of the week. The

length of this look-up table depends on the resolution of the measurements. As an example, a meter

recording measurements once every hour would produce a load baseline of length 168.



Figure 4.3.1 shows the average load demand over a week of the consumers in a network, clustered into 15

separate groups. Consumers with similar load demand patterns are group together in order to create a load

Figure 4.3.1. Average load grouped by hour of the week. For this plot, weeks begin on Monday and end on Sunday. The shadowed area represents two times the standard deviation. The Active energy is expressed in

kWh.



profile that can be used to estimate future load demands in the future, for any of the members of that

group. Figure 4.3.1 shows the load profiles of a network where 15 different groups have been identified.

Measurements in the historic time series are grouped based on the hour of the day (seconds since

midnight) and the day of the week. Future loads based on this load baseline can be estimated by querying

the resulting look-up table. The number of predictions to be queried would then depend on the forecasting

horizon. For the very short-term approach, three predictions are made, with 10-minute resolution up to a

forecasting horizon of 30 minutes. For the short-term approach, 24 or 48 predictions are made, up to a

horizon of 24h or 48h.

4.3.3 Short Term Model Training

The methodology used to predict the future load demand is based on training linear models with an array

of specially designed features. This vector comprises historic load measurements, calendar variables, such

as time of day and day of week, historic temperatures and forecasted temperatures.

The demand at time t is expressed as a non-parametric additive model

��𝑡 = 𝑐(𝑡) + 𝑙(𝑦𝑡) + 𝑚(𝑇𝑡) + 휀𝑡 (4.3.2)

Where:

��𝑡 is the predicted demand at time t.

𝑐(𝑡) is the contextual information at time t. This information contains calendar effects, such as the hour of the day, and the day of the week.

𝑙(𝑦𝑡) is a series of recent demand measurements, going backwards from t-1.

𝑚(𝑇𝑡) models the forecasted meteorological conditions at time t, as well as recent measurements, going backwards from t-1.

휀𝑡 is the model error at time t. The term 휀𝑡 integrates all errors, explaining the difference between predicted and observed values of the

time series. These differences are due to process fluctuations, measurement errors, and model

misspecifications.

An additional term 𝑥𝑑 may be added to take into account the flexible demand schedule. This new term is

the desired load. A separate model has to be trained with this expanded feature vector. The fit of this

model is only possible if there are enough samples to constitute a training data set.

Meteorological factors are widely considered to have an influence on the active load demand. The feature

vector contains a varying number of historic temperature measurements, as well as the forecasted

temperature as provided by a third party service. The historic temperature measurements that have been

considered are 𝑇𝑛, where 𝑇𝑛 is the temperature n steps behind. The meteorological section of the feature

vector is 𝑥𝑚 = 𝑇𝑛 = {𝑇1, … , 𝑇𝑛}. Only the most influential variable, temperature, has been included in

the results, as it has been found that no benefit was achieved by including the others in the feature vector.

Load follows a distinctive pattern throughout the day and also within the same week. Two calendar

features are included into the feature vector. The calendar section of the feature vector is 𝑥𝑐 = {𝑑, 𝑤},

where d is the hour of the day, encoded as seconds from midnight, and w is the day of the week, encoded

as a number between 0 and 6.



Arguably, the most important regressor of a time series is its historic evolution in the past. The feature

vector includes the historic active energy, expressed as kWh, as 𝑥𝑦(𝑙) = {𝑦1, … , 𝑦𝑙}. Where 𝑦𝑙 is the load l

steps in the past.

The final feature vector is 𝑥 = 𝑥𝑦(𝑙)||𝑥𝑐||𝑥𝑚(𝑛), where || designates the concatenation operator. This

feature vector has two autoregressive parts: the latest l measurements and the latest n temperature

values. The alternative feature vector, designed to take into account the demand schedule is 𝑥 =

𝑥𝑦(𝑙)||𝑥𝑐||𝑥𝑚(𝑛)||𝑥𝑑.

The feature vector is then standardized, so that individual features approximate a Gaussian with zero mean

and unit variance. The data is transformed by removing the mean value 𝜇𝑡𝑟(𝑥) = {𝜇𝑥(1), … , 𝜇𝑥(𝑝)} of

each feature, and then scale it by dividing non-constant features by their standard deviation 𝜎𝑡𝑟 =

{𝜎1, … , 𝜎𝑝}, where 𝜇𝑡𝑟 is the average value of each individual feature in the training set and 𝜎𝑡𝑟 is the

standard deviation of each individual feature in the training set.

The target variable 𝑦 follows the same standardization procedure. The data is transformed by removing the

mean value 𝜇𝑡𝑟(𝑦) of each target variable, and then scale it by dividing them by their standard deviation

𝜎𝑡𝑟(𝑦), where 𝜇𝑡𝑟(𝑦) is the average value of each individual feature in the training set and 𝜎𝑡𝑟(𝑦) is the


Assuming the future load can be expressed as a linear combination of the input values, the forecaster can be expressed as

��(𝑤, 𝑥) = 𝑤0 + 𝑤1𝑥1 + … + 𝑤𝑛𝑥𝑛 (4.3.3)

Where ��(𝑤, 𝑥) is the predicted value, 𝑋 = (𝑥1, … , 𝑥𝑛) is the feature vector, made from n inputs, and

𝑤 = (𝑤1, … , 𝑤𝑛) are its coefficients.

Ordinary Least Squares (OLS) methods fit a linear model with coefficients 𝑤 = (𝑤1, … , 𝑤𝑝) in order to

minimize the residual sum of squares between training set and the testing set samples, expressed as

min𝑤

|| ��(𝑤, 𝑥) − 𝑦||22 (4.3.4)

Where ��(𝑤, 𝑥) is the predicted value and y is the real load value. The linear approximation is then used to

predict future data.

The training procedure assumes that every feature vector is complete. That is, the training set of feature

vectors have been pruned of samples with missing values.

The training algorithm may be called at any time and will provide an updated model of the time series. How

often this update is necessary, if at all, will depend on the variability of the time series and on the quality of

the recorded measurements. This update is an offline process and should not disrupt the forecasting

algorithm, which would keep using the current model until a new one is available.

4.3.4 Very Short Term Model Training

The methodology used to predict the future load demand is based on training linear models with an array

of specially designed features. This vector comprises historic load measurements and calendar variables,

such as time of day and day of week.



The demand at time t is expressed as a non-parametric additive model

��𝑡 = 𝑐(𝑡) + 𝑙(𝑦𝑡) + 휀𝑡 (4.3.5)

Where:

��𝑡 is the predicted demand at time t.

𝑐(𝑡) is the contextual information at time t. This information contains calendar effects, such as the hour of the day, and the day of the week.

𝑙(𝑦𝑡) is a series of recent demand measurements, going backwards from t-1.



misspecifications.

An additional term 𝑥𝑑 may be added to take into account the flexible demand schedule. This new term is

the desired load. A separate model has to be trained with this expanded feature vector. The fit of this

model is only possible if there are enough samples to constitute a training data set.

Load follows a distinctive pattern throughout the day and also within the same week. Two calendar

features are included into the feature vector. The calendar section of the feature vector is 𝑥𝑐 = {𝑑, 𝑤},

where d is the hour of the day, encoded as a seconds from midnight, and w is the day of the week, encoded

as a number between 0 and 6.

Arguably, the most important regressor of a time series is its historic evolution in the past. The feature

vector includes the historic active energy, expressed as kWh, as 𝑥𝑦(𝑙) = {𝑦1, … , 𝑦𝑙}. Where 𝑦𝑙 is the load l

steps in the past.

The final feature vector is 𝑥 = 𝑥𝑦(𝑙)||𝑥𝑐, where || designates the concatenation operator. This feature

vector has one autoregressive part: the latest l measurements. The alternative feature vector, designed to

take into account the demand schedule is 𝑥 = 𝑥𝑦(𝑙)||𝑥𝑐||𝑥𝑑.










Assuming the future load can be expressed as a linear combination of the input values, the forecaster can be expressed as

��(𝑤, 𝑥) = 𝑤0 + 𝑤1𝑥1 + … + 𝑤𝑛𝑥𝑛 (4.3.6)





Ordinary Least Squares (OLS) methods fit a linear model with coefficients 𝑤 = (𝑤1, … , 𝑤𝑛) in order to minimize the residual sum of squares between training set and the testing set samples, expressed as

min𝑤

|| ��(𝑤, 𝑥) − 𝑦||22 (4.3.7)

Where ��(𝑤, 𝑥) is the predicted value and y is the real load value. The linear approximation is then used to

predict future data.


vectors have been pruned of samples with missing values.





4.3.5 Forecasting

The short term forecasting and the very short term forecasting follow the same produce. The only

difference is the kind of information used to create the feature vector. The short-term forecaster vector

comprises historic load measurements, calendar variables, such as time of day and day of week, historic

temperatures and forecasted temperatures. The very short-term forecaster vector comprises historic load

measurements and calendar variables, such as time of day and day of week. As an optional input to the

algorithm, the feature vector may be expanded with the flexible demand. If available, the forecasting

algorithm will use the alternative model in order to provide the estimations.

The online forecast algorithm retrieves the latest measurements from the database in other to create a

feature vector following the same steps as in the training algorithm. This new feature vector needs to be

scaled using the same parameters used in the training dataset. The feature vector is standardized, so that

individual features approximate a Gaussian with zero mean and unit variance. The data is transformed by

removing the mean value 𝜇𝑡𝑟(𝑥) = {𝜇𝑥(1), … , 𝜇𝑥(𝑝)} of each feature, and then scale it by dividing non-

constant features by their standard deviation 𝜎𝑡𝑟 = {𝜎1, … , 𝜎𝑝}, where 𝜇𝑡𝑟 is the average value of each

individual feature in the training set and 𝜎𝑡𝑟 is the standard deviation of each individual feature in the

training set.

The target variable 𝑦 follows the same standarization procedure. The data is transformed by removing the




The estimated load at time t is the linear combination of the input feature vector x weighted by the fitted coefficients w.

��(𝑤, 𝑥) = 𝑤0 + 𝑤1𝑥1 + … + 𝑤𝑛𝑥𝑛 (4.3.8)





If the forecasting horizon is larger than one, the algorithm is called recursively. Instead of using historic

values of the measurements, at least part of 𝑥𝑦 will be a forecast of the load at time t. For instance, the

load at time t+2 forecasted at time t will have an autoregressive part of its feature vector as 𝑥𝑦(𝑙) =

{��𝑡+1,𝑦𝑡 … , 𝑦𝑙−1} . For the very short-term approach, three predictions are made, with 10-minute

resolution up to a forecasting horizon of 30 minutes. For the short-term approach, 24 or 48 predictions are

made, up to a horizon of 24h or 48h.

The final forecast is then descaled by adding the mean value 𝜇𝑡𝑟(𝑦) of each target variable, and then

multiplying them by their standard deviation 𝜎𝑡𝑟(𝑦) , where 𝜇𝑡𝑟(𝑦) is the average value of each individual

feature in the training set and 𝜎𝑡𝑟(𝑦) is the standard deviation of each individual feature in the training set.

4.4 Production Forecasting For every node in the distributed production network at time t, the production forecasting algorithm can

provide a prediction of its production at time t+1. In order to model the behaviour of the production over

time, and thus be able to estimate the future production three kinds of information are used: historical

measurements, calendar information and meteorological information. This procedure is expanded to

deliver h predictions in the future, with two different resolutions, up to the forecasting horizon t+h.

The IDE4L project will follow two approaches for forecasting the distributed production. Which one of the

approaches will be selected will depend on the quality of the recorded measurements and the availability

of accurate meteorological forecasts. In case the latest measurements are not available in the database,

the future production is estimated by querying a lookup table, containing a predefined production baseline.

If every needed measurement is stored in the data exchange platform, past measurements are used to

infer the production in the future. This production-forecasting algorithm has two separate parts. The

training algorithm fits the model based on historical data. This procedure is executed offline. The second

part of the algorithm is executed online, at every time step and for every node in the network. The

forecaster is able to provide several predictions in the future up to a forecasting horizon by recursively

adding the new forecasts to the feature vector.

Distributed production forecasting for medium voltage and low voltage follow the same procedure, unless

otherwise indicated in the remainder of this section

4.4.1 Real Time Measurement Reading and Filtering

The implementation of this algorithm will account for errors in the data collection step. Real

implementations of smart meter networks are subject to errors in measurements due to communication

failures, corruption of the data or temporal unavailability of the meter.

New and historic measurements are read by the forecasting algorithm from the low voltage data exchange

platform. New measurements are filtered based on the statistical properties of the training dataset. The

training dataset must comprise a sufficient range of historic values with the appropriate resolution. Two

issues might arise when querying the database for the most recent measurement. The first one is the lack

of such a measurement due to communication failure, data corruption or meter malfunction. In this case

the measurement is substituted by the last forecasted value for the particular moment. The second issue is

erroneous measurements. Based on the mentioned training dataset, any new measurement not contained



within five times the standard deviation of the time series is disregarded. In its place the last forecast for

that particular time is used instead. This procedure introduces uncertainties in the forecast and degrades its

behaviour. However, it allows for a forecast to be made in every situation. If the number of missing values

is above a predefined threshold the algorithm will automatically switch to the first scheme described in this

section, and future distributed production will be estimated based on the predefined production baselines.

4.4.2 Production Baselines

Nodes with a similar production profile are grouped together, in order to create a production baseline that

best defines their behaviour. Similar nodes are clustered using the K-means algorithm.

The K-Means algorithm clusters data by trying to separate samples in n groups of equal variance,

minimizing a criterion known as within-cluster sum-of-squares, which is a measure of how internally

coherent clusters are.

∑min𝜇𝑗

(‖𝑥𝑗 − 𝜇𝑖‖2)

𝑛

𝑖=0

(4.4.1)

The k-means algorithm divides a set of N samples X into K disjoint clusters C, each described by the mean

𝜇𝑗 of the samples in the cluster, known as centroids.

The K-means algorithm used in this implementation randomly selects k samples from the training dataset

as centroids. It then assigns each sample to the nearest of the k clusters. It then creates new centroids by

taking the mean value of all of the samples assigned to each previous centroid. The difference between the

old and the new centroids are computed and the algorithm repeats these last two steps until this value is

less than a threshold.

Once every node has been assigned to one of the k clusters, a lookup table is created by averaging all

measurements from all samples in the k cluster having the same time of the day and day of the week. The

length of this look-up table depends on the resolution of the measurements. As an example, a meter

recording measurements once every hour would produce a production baseline of length 168.

Measurements in the historic time series are grouped based on the hour of the day (seconds since

midnight) and the day of the week. Future production based on this production baseline can be estimated

by querying the resulting look-up table. For the very short-term approach, three predictions are made, with

10-minute resolution up to a forecasting horizon of 30 minutes. For the short-term approach, 24 or 48

predictions are made, up to a horizon of 24h or 48h.

4.4.3 Short Term Model Training

The methodology used to predict the future distributed production is based on training linear models with

an array of specially designed features. This vector comprises historic production measurements, calendar

variables, such as time of day and day of week, historic temperatures and forecasted temperatures.

The production at time t is expressed as a non-parametric additive model

��𝑡 = 𝑐(𝑡) + 𝑙(𝑦𝑡) + 𝑚(𝑆𝑡) + 휀𝑡 (4.4.2)

Where:



��𝑡 is the predicted production at time t.

𝑐(𝑡) is the contextual information at time t. This information contains calendar effects, such as the hour of the day.

𝑙(𝑦𝑡) is a series of recent production measurements, going backwards from t-1.

𝑚(𝑆𝑡) models the forecasted meteorological conditions at time t, as well as recent measurements, going backwards from t-1.



misspecifications.

Meteorological factors have a great influence on the distributed production, especially in photovoltaic

production and micro wind turbines. The feature vector contains a varying number of historic

measurements of solar radiation, wind speed and wind direction, as well as the forecasted solar radiation,

wind speed and wind direction as provided by a third party service. The historic meteorological

measurements that have been considered are 𝑆𝑛, where 𝑆𝑛 are the meteorological variables n steps

backwards. The meteorological section of the feature vector is 𝑥𝑚 = 𝑆𝑛 = {𝑆1, … , 𝑆𝑛}.

Production follows a distinctive pattern throughout the day. One calendar feature is included into the

feature vector. The calendar section of the feature vector is 𝑥𝑐 = {𝑑}, where d is the hour of the day,

encoded as seconds from midnight.

An important feature of a time series is its historic evolution in the past. The feature vector includes the

historic production, expressed in kWh, as 𝑥𝑦(𝑙) = {𝑦1, … , 𝑦𝑙}. Where 𝑦𝑙 is the production l steps in the

past.

The final feature vector is 𝑥 = 𝑥𝑦(𝑙)||𝑥𝑐||𝑥𝑚(𝑛), where || designates the concatenation operator. This

feature vector has two autoregressive parts: the latest l measurements and the latest n meteorological

values.










Assuming the future production can be expressed as a linear combination of the input values, the forecaster can be expressed as

��(𝑤, 𝑥) = 𝑤0 + 𝑤1𝑥1 + … + 𝑤𝑛𝑥𝑛 (4.4.3)





Ordinary Least Squares (OLS) methods fit a linear model with coefficients 𝑤 = (𝑤1, … , 𝑤𝑛) in order to minimize the residual sum of squares between the predictions and the actual samples, expressed as

min𝑤

|| ��(𝑤, 𝑥) − 𝑦||22 (4.4.4)

Where ��(𝑤, 𝑥) is the predicted value and y is the real production value. The linear approximation is then

used to predict future data.


vectors has been pruned of samples with missing values.





4.4.4 Very Short Term Model Training

The methodology used to predict the future distributed production is based on training linear models with

an array of specially designed features. This vector comprises historic production measurements and

calendar variables, such as time of the day.

The production at time t is expressed as a non-parametric additive model

��𝑡 = 𝑐(𝑡) + 𝑙(𝑦𝑡) + 휀𝑡 (4.4.5)

Where:

��𝑡 is the predicted production at time t.

𝑐(𝑡) is the contextual information at time t. This information contains calendar effects, such as the hour of the day.

𝑙(𝑦𝑡) is a series of recent production measurements, going backwards from t-1.



misspecifications.

Distributed production follows a distinctive pattern throughout the day. One calendar feature is included

into the feature vector. The calendar section of the feature vector is 𝑥𝑐 = {𝑑}, where d is the hour of the

day, encoded as a seconds from midnight.

The feature vector includes the lagged active energy, expressed in kWh, as 𝑥𝑦(𝑙) = {𝑦1, … , 𝑦𝑙}. Where 𝑦𝑙

is the production measurement l steps in the past.

The final feature vector is 𝑥 = 𝑥𝑦(𝑙)||𝑥𝑐, where || designates the concatenation operator. This feature

vector has one autoregressive part: the latest 𝑙 measurements.












Assuming the future production can be expressed as a linear combination of the input values, the forecaster can be expressed as

��(𝑤, 𝑥) = 𝑤0 + 𝑤1𝑥1 + … + 𝑤𝑝𝑥𝑝 (4.4.6)



Ordinary Least Squares (OLS) methods fit a linear model with coefficients 𝑤 = (𝑤1, … , 𝑤𝑝) in order to

minimize the residual sum of squares between training set and the testing set samples, expressed as

min𝑤

|| ��(𝑤, 𝑥) − 𝑦||22 (4.4.7)

Where ��(𝑤, 𝑥) is the predicted value and y is the real production value. The linear approximation is then

used to predict future data.


vectors has been pruned of samples with missing values.





4.4.5 Forecasting

The short term forecasting and the very short term forecasting follow the same produce. The only

difference is the kind of information used to create the feature vector. The short-term forecaster vector

comprises historic production measurements, calendar variables, such as time of day, historic

measurements of solar radiation, wind speed and wind direction, as well as forecasted values of solar

radiation, wind speed and wind direction. The very short-term forecaster vector comprises historic

production measurements and calendar variables, such as time of day.

The online forecast algorithm retrieves the latest measurement from the database in other to create a

feature vector following the same steps as in the training algorithm. This new feature vector needs to be

scaled using the same parameters used in the training dataset. The feature vector is standardized, so that

individual features approximate a Gaussian with zero mean and unit variance. The data is transformed by

removing the mean value 𝜇𝑡𝑟(𝑥) = {𝜇𝑥(1), … , 𝜇𝑥(𝑝)} of each feature, and then scale it by dividing non-

constant features by their standard deviation 𝜎𝑡𝑟 = {𝜎1, … , 𝜎𝑝}, where 𝜇𝑡𝑟 is the average value of each



individual feature in the training set and 𝜎𝑡𝑟 is the standard deviation of each individual feature in the

training set.

The target variable 𝑦 follows the same standarization procedure. The data is transformed by removing the




The estimated production at time t is the linear combination of the input feature vector x weighted by the fitted coefficients w.

��(𝑤, 𝑥) = 𝑤0 + 𝑤1𝑥1 + … + 𝑤𝑝𝑥𝑝 (4.4.8)



If the forecasting horizon is larger than one, the algorithm is called recursively. Instead of using historic

values of the measurements, at least part of 𝑥𝑦 will be a forecast of the production at time t. For instance,

the production at time t+1, forecasted at time t will have an autoregressive part of its feature vector as

𝑥𝑦(𝑙) = {��𝑡+1,𝑦𝑡 … , 𝑦𝑙−1} . For the very short-term approach, three predictions are made, with 10-minute

resolution up to a forecasting horizon of 30 minutes. For the short-term approach, 24 or 48 predictions are

made, up to a horizon of 24h or 48h.

The final forecast is then descaled by adding the mean value 𝜇𝑡𝑟(𝑦) of each target variable, and then

multiplying them by their standard deviation 𝜎𝑡𝑟(𝑥) , where 𝜇𝑡𝑟(𝑦) is the average value of each individual

feature in the training set and 𝜎𝑡𝑟(𝑦) is the standard deviation of each individual feature in the training set.

4.5 Interfaces

4.5.1 Medium Voltage Network Load and Production Forecaster

Almost all the inputs for medium voltage load and production forecasting are read from a SQL database

located at the primary substation computer. In the database, real-time measurement data contains values

received from RTU and smart meters, and they are organized according to an IEC 61850 data model.

Input Data exchanged Source Local / Remote Update schedule Format

MV Customer

information

- Contractual

power or fuse size

MV DXP Local Once a day - Contractual

power in

kilowatts [kW] or

fuse size in amps

[A]

- Power demand

[kW],

DG information - Nominal power

- Type (PV, wind

MV DXP Local Once a day Table with

integer and

floating point



etc.) numbers

Smart meter

measurement

time series 10

min interval

- Load and

production time

series from the

last 168 hours

with 10 minutes

time step

- Time stamp for

each

measurement

MV DXP Local Once a day - Power demand

[kW]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]

Smart meter

measurement

time series

hourly resolution

- Load and

production time

series from the

last year with

hourly time step

- Time stamp for

each

measurement

MV DXP Local Once a day - Power demand

[kW]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]

Weather

measurements

time series

- Outdoor

temperature,

solar irradiance

and wind speed

and direction

measurements

(hourly time step)

from the last year

Local weather

station (via web

service)

Remote Once a day - Temperature

[°C]

- Solar irradiance

[W/m2]

- Wind speed and

direction [m/s ;

degrees]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]

Weather

forecasts

- Outdoor

temperature,

solar irradiance

and wind speed

and direction

forecasts for the

next 24 – 48 hours

Hourly time step

Local weather

station (via web

service)

Remote On a fixed

schedule and “on

demand”

- Temperature

[°C]

- Solar irradiance

[W/m2]

- Wind speed

[m/s]

- Wind direction

[°]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]



4.5.2 Low Voltage Network Load and Production Forecaster

Almost all the inputs for low voltage load and production forecasting are read from a SQL database located

at the secondary substation computer. In the database, real-time measurement data contains values

received from RTU and smart meters, and they are organized according to an IEC 61850 data model.

Input Data exchanged Source Local / Remote Update schedule Format

LV Customer

information

- Contractual

power or fuse size

LV DXP Local Once a day - Contractual

power in

kilowatts [kW] or

fuse size in amps

[A]

- Power demand

[kW],

DG information - Nominal power

- Type (PV, wind

etc.)

LV DXP Local Once a day Table with

integer and

floating point

numbers

Smart meter

measurement

time series 10

min interval

- Load and

production time

series from the

last 168 hours

with 10 minutes

time step

- Time stamp for

each

measurement

LV DXP Local Once a day - Power demand

[kW]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]

Smart meter

measurement

time series

hourly resolution

- Load and

production time

series from the

last year with

hourly time step

- Time stamp for

each

measurement

LV DXP Local Once a day - Power demand

[kW]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]

Weather

measurements

time series

- Outdoor

temperature,

solar irradiance

and wind speed

and direction

measurements

(hourly time step)

Local weather

station (web

service via DXP)

Remote Once a day - Temperature

[°C]

- Solar irradiance

[W/m2]

- Wind speed and



from the last year direction [m/s ;

degrees]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]

Weather

forecasts

- Outdoor

temperature,

solar irradiance

and wind speed

and direction

forecasts for the

next 24 – 48 hours

Hourly time step

Local weather

station (web

service via DXP)

Remote On a fixed

schedule and “on

demand”

- Temperature

[°C]

- Solar irradiance

[W/m2]

- Wind speed

[m/s]

- Wind direction

[°]

- Time stamp

[YYYY-mm-dd

HH:MM:SS]



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