uncertainty quantification for networks with power ...asa-qprc.org/2013/€¦ · uncertainty quanti...

Uncertainty Quantification for Networks with PowerDistribution Applications

Scott Vander Wiel* Russell Bent†

Earl Lawrence* Emily Casleton*

*Statistical Sciences Group†Energy & Infrastructure Analysis Group

Los Alamos National Laboratory

June 6, 2013

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Overview

Robust and reliable electricpower grid is essential forsociety to function

The network is complex,unpredictable, and only partiallyobservable

Grid operators determine theleast cost power generationbased on topology and demands

We estimate topologyprobabilities and optimize powergeneration accounting foruncertainty

Image Credit: U.S. DOE (2006).

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Conventional State Estimation

Estimate topology, voltages andflows given system measurements

Iterate nonlinear least squareswith greedy topology search

– estimate state x by NLS– large residuals indicate a

topology error– modify topology and repeat

Use single best fit to decideoptimal generation

Works well in practice, with a fewnotable failures

– Incorrect estimate of networktopology contributed to 2003northeast blackout

Image Credit: Slobodan Pajic (2007)

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Power Flow Solver

Computational model used bystate estimator

– implements electrical laws– uses physical properties of

grid components

Inputs: Grid topology and nodequantities (complex valuedpower load or voltage)

Outputs: power flowing on thelines, a subset of which areobserved

SINGH et al.: RECURSIVE BAYESIAN APPROACH FOR IDENTIFICATION OF NETWORK CONFIGURATION CHANGES 1331

In (9), is a diagonal matrix whose diagonal elements rep-resent the inverse of the variances corresponding to the errorcomponents in the error vector. Large value of the elements of

can magnify the model errors and cause the accelerationof convergence to a single model. The error vector is givenby

(10)

where is the real measurement vector common to allmodels and is the estimated value of the same realmeasurement vector obtained from the estimated states of theth model in the th iteration.

The algorithm proceeds recursively from an equal initialprobability assigned to each model. In each iteration thenew probabilities are computed according to (9). These newprobabilities are the improvements in the probabilities of theprevious iteration.

If a large number of iterations is considered, the model bankasymptotically converges to a single model. Over the iterationsone model has asymptotic probability equal to one while othershave zero probabilities. However, in practice, for the identifi-cation of the correct network configuration, asymptotic conver-gence to a single model with unit probability is not required.The algorithm can be terminated in few iterations as long as oneof the models attains a significantly higher probability than therest.

The main advantage of the recursive Bayesian approach isthat the identification is naturally constrained so the cases inwhich the state estimation diverges are automatically rejected.The rejection of a poor model is exponential and thus very fast.Furthermore, the algorithm is computationally inexpensive andhence a large number of models can be handled efficiently.

IV. STUDY SYSTEMS

Two 11 kV distribution networks, shown in Fig. 2, are consid-ered. The networks are based on the U.K. Generic DistributionSystem (UKGDS) [9], which was modified in order to demon-strate the concept of the proposed approach.

Network 1 consists of 26 buses, 25 overhead lines, 13 loads,and one distributed generator (DG), and Network 2 has 13 buses,13 overhead lines, and eight loads. All loads are in the rangeof 10 to 140 kW, apart from the load at bus #18 of Network1 which is 930 kW. The generator is fixed at 700 kW at 0.95power factor. The two networks are connected via a normallyopen point which can be closed for maintenance or emergencynetwork reconfiguration. In addition, each network is equippedwith a protective device (recloser, sectionalizer, or fuse), a typ-ical means of improving reliability and service continuity inoverhead lines.

Network parameters are obtained from [9]. The buses arerenumbered so the networks are easier to observe.

A. Model Bank Description

Two types of major contingencies are considered for themodel banks of the two study systems: topological changes

Fig. 2. Test network models.

that are associated with the operation of protective devices andstatus change of normally open points, and injection changesthat are associated with loss of a DG or disconnection of amajor load.

It can be shown that disconnection of small loads has little im-pact on the state estimation function. On the other hand, discon-nection of large loads has significant effect on the accuracy ofthe estimated quantities. Thus, loss of large load has been con-sidered as critical change and hence included in the model bank.Also line outages have not been considered individually as, ineffect, they cause the protective devices to operate, resulting inone of the configurations included in the model banks.

The model banks selected for the two study networks are sum-marized in Table I. The model banks considered in this studyrepresent critical configuration changes that have a detrimentaleffect on the state estimation output. In general, these configu-rations are network specific and a detailed contingency study isrequired for their identification. In other words, the number ofmodels in a model bank can be prevented from being excessiveby utilizing the critical contingencies, operators’ experience anddetailed reliability and risk analysis. However, the speed of theproposed approach allows the use of models banks with large

Image Credit: Modified from Singh, Manitsas, Pal, Strbac (2010)Unclassified4

AC Power Flow Equations

Known Property:admittance Gi ,j , bi ,j

power flow Pi ,j ,Qi ,j

voltage|Vi |, θi

power loadPi ,Qi

node quantities2 given, 2 solved:|Vj |, θjPj ,Qj

node i line i , j node j

Lossless flow equations for real (Pi ,Pi ,j) and reactive (Qi ,Qi ,j) power:

Pi =N∑j=1

|Vi ||Vj |[Gi ,j cos(θi − θj) + bi ,j sin(θi − θj)] =N∑j=1

Pi ,j

Qi =N∑j=1

|Vi ||Vj |[Gi ,j sin(θi − θj)− bi ,j cos(θi − θj)] =N∑j=1

Qi ,j

(i = 1, . . . ,N)

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Model Bank Topology Estimation

Singh, et al. consider estimating thecorrect topology from a finitecollection of possibilities

a bank of models contains allimportant networkconfigurations

estimate probabilities for eachmodel in the bank bycombining

– system measurements– prior information about loads– power flow model

Did not implement Bayes rulecorrectly

SINGH et al.: RECURSIVE BAYESIAN APPROACH FOR IDENTIFICATION OF NETWORK CONFIGURATION CHANGES 1331

In (9), is a diagonal matrix whose diagonal elements rep-resent the inverse of the variances corresponding to the errorcomponents in the error vector. Large value of the elements of

can magnify the model errors and cause the accelerationof convergence to a single model. The error vector is givenby

(10)

where is the real measurement vector common to allmodels and is the estimated value of the same realmeasurement vector obtained from the estimated states of theth model in the th iteration.

The algorithm proceeds recursively from an equal initialprobability assigned to each model. In each iteration thenew probabilities are computed according to (9). These newprobabilities are the improvements in the probabilities of theprevious iteration.

If a large number of iterations is considered, the model bankasymptotically converges to a single model. Over the iterationsone model has asymptotic probability equal to one while othershave zero probabilities. However, in practice, for the identifi-cation of the correct network configuration, asymptotic conver-gence to a single model with unit probability is not required.The algorithm can be terminated in few iterations as long as oneof the models attains a significantly higher probability than therest.

The main advantage of the recursive Bayesian approach isthat the identification is naturally constrained so the cases inwhich the state estimation diverges are automatically rejected.The rejection of a poor model is exponential and thus very fast.Furthermore, the algorithm is computationally inexpensive andhence a large number of models can be handled efficiently.

IV. STUDY SYSTEMS

Two 11 kV distribution networks, shown in Fig. 2, are consid-ered. The networks are based on the U.K. Generic DistributionSystem (UKGDS) [9], which was modified in order to demon-strate the concept of the proposed approach.

Network 1 consists of 26 buses, 25 overhead lines, 13 loads,and one distributed generator (DG), and Network 2 has 13 buses,13 overhead lines, and eight loads. All loads are in the rangeof 10 to 140 kW, apart from the load at bus #18 of Network1 which is 930 kW. The generator is fixed at 700 kW at 0.95power factor. The two networks are connected via a normallyopen point which can be closed for maintenance or emergencynetwork reconfiguration. In addition, each network is equippedwith a protective device (recloser, sectionalizer, or fuse), a typ-ical means of improving reliability and service continuity inoverhead lines.

Network parameters are obtained from [9]. The buses arerenumbered so the networks are easier to observe.

A. Model Bank Description

Two types of major contingencies are considered for themodel banks of the two study systems: topological changes

Fig. 2. Test network models.

that are associated with the operation of protective devices andstatus change of normally open points, and injection changesthat are associated with loss of a DG or disconnection of amajor load.

It can be shown that disconnection of small loads has little im-pact on the state estimation function. On the other hand, discon-nection of large loads has significant effect on the accuracy ofthe estimated quantities. Thus, loss of large load has been con-sidered as critical change and hence included in the model bank.Also line outages have not been considered individually as, ineffect, they cause the protective devices to operate, resulting inone of the configurations included in the model banks.

The model banks selected for the two study networks are sum-marized in Table I. The model banks considered in this studyrepresent critical configuration changes that have a detrimentaleffect on the state estimation output. In general, these configu-rations are network specific and a detailed contingency study isrequired for their identification. In other words, the number ofmodels in a model bank can be prevented from being excessiveby utilizing the critical contingencies, operators’ experience anddetailed reliability and risk analysis. However, the speed of theproposed approach allows the use of models banks with large

B

E

17

18

D

P1 Q1

P2 Q2

Image Credit: Modified from Singh, Manitsas, Pal, Strbac (2010)Unclassified6

Model Bank Topology Estimation

We compute probabilities ontopologies from a vector ofmeasured flows by

propagating priors on nodalloads through each possibletopology

approximating resultantdistributions of power flows

implementing Bayes rule byimportance sampling aroundthe measured flows

Resulting algorithm is fast enoughto be used in real time

Image shows topology estimates andrisk-optimal generation

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Statistical Model Set-Up

Model bank of m network topologies: Ω = ω1, . . . , ωmSystem measurements, Y, are normally distributed with some mean µand standard deviation proportional to the mean

Y ∼ MVN(µ,Σy ), Σy = diag(ρµ)2

The mean is the output from solving the power flow equations for a setof random loads, Z, and a topology, ω

µ = µ(Z, ω)

Loads are also normally distributed with known mean and proportionalerror

Z ∼ MVN(ν,Σz), Σz = diag(βν)2

Goal of analysis is to estimate Pr(ωk |Y = y), ∀k = 1, . . . ,m.

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Offline Precomputation

Large networks imply large dimension of Z and running the power flowsolver in real time becomes computationally prohibitive.

A distribution, ηk , is fit to random outputs from the power flow solverto avoid running the solver in real time:

µ(Z, ωk) ∼ ηk(µ) with Z ∼ MVN(ν,Σz)

Scheme:

1. Draw a large sample of loads Zi (i = 1, . . . ,M).2. Run the solver to compute µi,k = µ(Zi , ωk) (∀i , k).3. For each topology transform µ1,k . . . ,µM,k to approximate normality:

hk(µi,k) ∼ MVN (over i)

Transformations hk are determined through exploratory data analysis

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Example: IEEE Reliability Test System (RTS)

Designed as benchmark forcomparing reliabilitymethodologies

72 nodes

Power observed on the 5interconnects between subsystems

Model bank of 124 topologiesdefined by RTS Task Force

– Normal– All single lines down (except

those monitored)– Some two-line losses

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Example: Normal-Plots of Solver Output µ(Zi , ωk)

Each line assumes one of the 124 model bank topologies.

Real power flow, P Reactive power flow, Q

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Example: Transformation of Reactive Power Solver Output

Transform each measurement,`, to normality

hk,`(µ) = Φ−1[Fk,`(µ)]

where

– Fk,` is a fitted three–parameter Gamma CDF

– Φ is the standard normalCDF

Transformed reactive power, hk(Q)

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Importance Sampling

Reminder: Goal is to compute

Pr(ωk |Y = y) ∝ π(ωk)

∫f (y|µ(ωk ,Z)) πZ (Z)dZ

= π(ωk)

∫f (y|µ) ηk(µ)dµ

where f is the measurement density, MVN with proportional std. dev.

π is a prior on topologies

πZ is the MVN density of Z

ηk is the approximating density of µ(ωk ,Z)

No analytical solution

Brute force Monte Carlo is infeasible because µ ∼ ηk(µ) is rarely closeenough to y to contribute to the integral

⇒ Importance sample µ near y and re-weight to estimate the integral

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Online Real Time Monitoring

1. Obtain a new measurement, y

2. Draw an importance sample of µ near y.

3. Estimate

Pr(ωk |y) ∝ π(ωk)Eηk [f (y|µ)]

as an importance-weighted average for each scenario, ωk

4. Normalize to obtain model bank probabilities.

Result is a vector of probabilities, p(y), for each possible topology

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Simulation Study

For each topology:

1. Generate 1000 simulated observations, y

2. Compute p(y) for 124 toplogies

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Simulation Study

For each topology:

1. Generate 1000 simulated observations, y

2. Compute p(y) for 124 toplogies

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Simulation Study Results

Plot shows

EYk[Pr(ωk |Y)]

verses

max6=k

EYk[Pr(ω`|Y)]

True topology has highestprobability in all but onecase

Ambiguity is often large

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Pr(

true

sce

nario

)

Maximum Pr(incorrect scenario)

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Chance Constrained Optimal Power Flow (CCOPF)

OPF decides how much power to generate and where.

CCOPF makes these decisions with the added constraints that

P(line i , j overloaded) < ε

for all lines and over the uncertain topology

Chance constraints have been implemented for other classes ofuncertainty such as unit commitment and expansion planning toaccount for renewable energy generation

Define setsN = nodes

G = generators

G i = generators associated with node i , (G i ⊂ G )

E = edges (lines)

E i edges connected to node i , (E i ⊂ E )

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DC Power Flow Equations

Known Property:admittance Gi ,j , bi ,j

power flow Pi ,j ,Qi ,j

voltage|Vi |, θi

power loadPi ,Qi

node quantities2 given, 2 solved:|Vj |, θjPj ,Qj

node i line i , j node j

DC power flow equations for real, Pi power at node i ; i = 1, . . . ,N:

Pi =N∑j=1

bi ,j(θi − θj)

Results from multiple simplifying assumptions on the AC equations thatresults in a linear set of equations.

The DC equations are an approximation to the AC equations and areused in the CCOPF due to their simplicity.

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CCOPF formula

The problem in terms of objective function and constraints:

minimize∑k∈G

ckgk +∑i∈N

κiξi

subject to g−k ≤ gk ≤ g+k ∀k ∈ G

0 ≤ ξi ≤ `i ∀i ∈ N∑k∈G i

gk − `i + ξi +∑i,j∈E i

bωi,j(θωi − θωj ) = 0 ∀i ∈ N, ω ∈ Ω

ρωi,j ≥|bωi,j(θωi − θωj )| − qi,j

M∀i , j ∈ E , ω ∈ Ω∑

ω∈Ω

pωρωi,j ≤ ε ∀i , j ∈ E

ρωi,j ∈ 0, 1

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CCOPF: Objective function

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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:

Objective Function:∑k∈G

ckgk +∑i∈N

κiξi

Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation

Node i`i load demandξi amount shedκi cost to shed

Node j

qi ,j capacitybωi ,j(θ

ωi − θωj )flow on (i , j) in ω

ρωi ,j =

1 (i , j) overloaded in ω0 otherwise

CCOPF: Generation constraints

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ckgk +∑i∈N

κiξi

Constraint 1: g−k ≤ gk ≤ g+k ∀k ∈ G

Constrain the amount generated at each generator to be within plausiblelimits.



Node j



ρωi ,j =


CCOPF: Load shedding constraints

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ckgk +∑i∈N

κiξi

Constraint 2: 0 ≤ ξi ≤ `i ∀i ∈ N

Cannot shed more load at a particular node than is demanded.



Node j



ρωi ,j =


CCOPF: Flow Balance constraints

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ckgk +∑i∈N

κiξi

Constraint 3:∑k∈G i

gk − `i + ξi +∑i ,j∈E i

bωi ,j(θωi − θωj ) = 0 ∀i ∈ N, ω ∈ Ω

The amount of power generated and passed in must be either consumed,shed, or passed out.



Node j



ρωi ,j =


CCOPF: Overload detection

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ckgk +∑i∈N

κiξi

Constraint 4: ρωi ,j ≥|bωi ,j(θωi − θωj )| − qi ,j

M∀i , j ∈ E , ω ∈ Ω

Constraint 6: ρωi ,j ∈ 0, 1If a line is overloaded in scenario ω, set the variable ρωi ,j to 1. Thediscrete-ness of the ρωi ,j makes the problem computationally challenging.



Node j



ρωi ,j =


CCOPF: Chance constraints

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ckgk +∑i∈N

κiξi

Constraint 5:∑ω∈Ω

pωρωi ,j ≤ ε ∀i , j ∈ E

Ensures the probability that the flow of power on a line violates its thermallimits is smaller than ε.



Node j



ρωi ,j =


Algorithms

1. Commercial Mixed Integer Programming Solver

2. Branch and Bound on the Chance Constraints–Recursively branch onenforcing the capacity constraints on one violated chance constraintscenario

3. Constraint Injection on the Chance Constraints–Add chance constraintsincrementally to eliminate constraint violations

4. Disjunctive Programming–Cutting plane algorithm

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Empirical Study

Stressed network with model bank of 340 scenarios

Sample 100 observations from each scenario–perform CCOPF withvarious values of ε and OPF based on most likely topology

ε Violation Probability (%) Average Load Shed

CCOPF OPF CCOPF OPF0 0.00 33.38 143.42 11.77

0.001 0.82 33.38 93.14 11.770.01 2.29 33.38 59 11.770.1 13.53 33.38 21.66 11.77

0.25 26.85 33.38 12.73 11.770.5 38.44 33.38 7.42 11.77

0.75 44.12 33.38 4.83 11.771 100.00 33.38 0 11.77

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Summary

Fast algorithm for capturing uncertainty in topology.

Standard OPF does not recognize uncertainty

Simulations suggests there is more uncertainty than is realized

CCOPF algorithm incorporates uncertainty for robust operation

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Future Work

Fully quantify the uncertainty in state estimation by calculatingposterior load distributions for each topology

[Z | Y = y, ωk ]

– Early work has used Gaussian process emulation techniques– Larger problems will require more scalable algorithms.– Incorporate load uncertainty into OPF

Model bank as a good stepping stone.

– Can be solved for moderately large networks with large models banks.– Can apply the method to the complete set of possible topologies and find

metrics that vary smoothly with the Gaussian approximations.– Goal is to explore the topology space in a structured manner.

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