unit ii -...
TRANSCRIPT
UNIT –II
SYSTEM OPERATION
CONTENTS
TECHNICAL TERMS
2. INTRODUCTION
2.1. SYSTEM LOAD FORECASTING
2.1.1. Important Factors for Forecasts
2.1.2. Forecasting Methods
2.1.3. Medium- and long-term load forecasting methods
2.2. ECONOMIC DISPATCH
2.2.1. Economic operation of power systems
2.2.2. Performance curves
2.2.3. Solution Methods
2.2.4. Economic dispatch problem
2.2.5. Thermal system dispatching with network losses considered
2.2.6. lambda-iteration method
2.2.7. Base point and participation factors
2.2.8. Economic dispatch controller added to LFC:
2.3. UNIT COMMITMENT
2.3.1. Constraints in Unit Commitment
2.3.2. Spinning Reserve
2.3.3. Thermal Unit Constraints
2.3.4 Other Constraints
2.3.4.1 Hydro-Constraints
2.3.4.2 Must Run
2.3.4.3 Fuel Constraints
2.3.5. Unit commitment solution methods
2.3.5.1. Priority-List Methods
2.3.5.2. Dynamic-Programming Solution
2.3.5.3. Forward DP Approach
2.3.4. Lagrange Relaxation Solution
TECHNICAL TERMS
Btu (British thermal unit): A standard unit for measuring the quantity of heat energy
equal to the quantity of heat required to raise the temperature of 1 pound of water by 1
degree Fahrenheit.
Capacity: The amount of electric power delivered or required for which a generator,
turbine, transformer, transmission circuit, station, or system is rated by the
manufacturer.
Combined Cycle Unit:
An electric generating unit that consists of one or more combustion turbines and one
or more boilers with a portion of the required energy input to the boiler(s) provided by
the exhaust gas of the combustion turbine(s).
Demand: The rate at which energy is delivered to loads and scheduling points by
generation, transmission, and distribution facilities.
Demand (Electric): The rate at which electric energy is delivered to or by a system,
part of a system, or piece of equipment, at a given instant or averaged over any
designated period of time.
Demand-Side Management: The planning, implementation, and monitoring of
utility activities designed to encourage consumers to modify patterns of electricity
usage, including the timing and level of electricity demand. It refers only to energy
and load-shape modifying activities that are undertaken in response to utility-
administered programs. It does not refer to energy and load-shape changes arising
from the normal operation of the marketplace or from government-mandated energy-
efficiency standards. Demand-Side Management (DSM) covers the complete range of
load-shape objectives, including strategic conservation and load management, as well
as strategic load growth.
Deregulation: The elimination of regulation from a previously regulated industry or
sector of an industry.
Electric Service Provider: An entity that provides electric service to a retail or end-
use customer.
Energy: The capacity for doing work as measured by the capability of doing work
(potential energy) or the conversion of this capability to motion (kinetic energy).
Energy has several forms, some of which are easily convertible and can be changed to
another form useful for work. Most of the world's convertible energy comes from
fossil fuels that are burned to produce heat that is then used as a transfer medium to
mechanical or other means in order to accomplish tasks. Electrical energy is usually
measured in kilowatthours, while heat energy is usually measured in British thermal
units.
Energy Charge: That portion of the charge for electric service based upon the
electric energy (kWh) consumed or billed
Outage: The period during which a generating unit, transmission line, or other facility
is out of service.
Forced Outage: The shutdown of a generating unit, transmission line or other
facility, for emergency reasons or a condition in which the generating equipment is
unavailable for load due to unanticipated breakdown.
Fuel: Any substance that can be burned to produce heat; also, materials that can be
fissioned in a chain reaction to produce heat.
2. INTRODUCTION
Accurate models for electric power load forecasting are essential to the operation and
planning of a utility company. Load forecasting helps an electric utility to make important
decisions including decisions on purchasing and generating electric power, load switching,
and infrastructure development. Load forecasts are extremely important for energy
suppliers,ISOs, financial institutions, and other participants in electric energy generation,
transmission, distribution, and markets.
Load forecasts can be divided into three categories: short-term fore-casts which are
usually from one hour to one week, medium forecasts which are usually from a week to a
year, and long-term forecasts which are longer than a year.
The forecasts for different time horizons are important for different operations within
a utility company. The natures of these forecasts are different as well. For example, for a
particular region, it is possible to predict the next day load with an accuracy of approximately
1-3%. However, it is impossible to predict the next year peak load with the similar accuracy
since accurate long-term weather forecasts are not available. For the next year peak forecast,
it is possible to provide the probability distribution of the load based on historical weather
observations. It is also possible, according to the industry practice, to predict the so-called
weather normalized load, which would take place for average annual peak weather conditions
or worse than average peak weather conditions for a given area. Weather normalized load is
the load calculated for the so-called normal weather conditions which are the average of the
weather characteristics for the peak historical loads over a certain period of time. The
duration of this period varies from one utility to another. Most companies take the last 25-30
years of data. Load forecasting has always been important for planning and opera-tional
decision conducted by utility companies. However, with the deregulation of the energy
industries, load forecasting is even more important. With supply and demand fluctuating and
the changes of weather conditions and energy prices increasing by a factor of ten or more
during peak situations, load forecasting is vitally important for utilities. Short-term load
forecasting can help to estimate load flows and to make decisions that can prevent
overloading. Timely implementations of such deci-sions lead to the improvement of network
reliability and to the reduced occurrences of equipment failures and blackouts. Load
forecasting is also important for contract evaluations and evaluations of various so-
phisticated financial products on energy pricing offered by the market. In the deregulated
economy, decisions on capital expenditures based on long-term forecasting are also more
important than in a non-deregulated economy when rate increases could be justified by
capital expenditure projects.
Most forecasting methods use statistical techniques or artificial intelligence
algorithms such as regression, neural networks, fuzzy logic, and expert systems. Two of the
methods, so-called end-use and econometric approach are broadly used for medium- and
long-term forecasting. A variety of methods, which include the so-called similar day
approach, various regression models, time series, neural networks, statistical learning
algorithms, fuzzy logic, and expert systems, have been developed for short-term forecasting.
As we see, a large variety of mathematical methods and ideas have been used for
load forecasting. The development and improvements of appropriate mathematical tools will
lead to the development of more accurate load forecasting techniques. The accuracy of load
forecasting depends not only on the load forecasting techniques, but also on the accuracy of
forecasted weather scenarios. Weather forecasting is an important topic which is outside of
the scope of this chapter.
2.1. SYSTEM LOAD FORECASTING
2.1.1. Important Factors for Forecasts
For short-term load forecasting several factors should be considered, such as time factors,
weather data, and possible customers’ classes. The medium- and long-term forecasts take
into account the historical load and weather data, the number of customers in different
categories, the appliances in the area and their characteristics including age, the economic
and demographic data and their forecasts, the appliance sales data, and other factors.
The time factors include the time of the year, the day of the week, and the hour of the day.
There are important differences in load between weekdays and weekends. The load on
different weekdays also can behave differently. For example, Mondays and Fridays being
adjacent to weekends, may have structurally different loads than Tuesday through Thursday.
This is particularly true during the summer time. Holidays are more difficult to forecast than
non-holidays because of their relative infrequent occurrence.
Weather conditions influence the load. In fact, forecasted weather parameters are the most
important factors in short-term load forecasts. Various weather variables could be considered
for load forecasting. Tem-perature and humidity are the most commonly used load
predictors.
Among the weather variables listed above, two composite weather variable functions, the
THI (temperature-humidity index) and WCI (wind chill index), are broadly used by utility
companies. THI is a measure of summer heat discomfort and similarly WCI is cold stress in
winter.
Most electric utilities serve customers of different types such as residential, commercial,
and industrial. The electric usage pattern is different for customers that belong to different
classes but is somewhat alike for customers within each class. Therefore, most utilities
distinguish load behavior on a class-by-class basis.
2.1.2. Forecasting Methods
Over the last few decades a number of forecasting methods have been developed. Two of
the methods, so-called end-use and econometric ap-proach are broadly used for medium- and
long-term forecasting. A variety of methods, which include the so-called similar day
approach, various regression models, time series, neural networks, expert systems, fuzzy
logic, and statistical learning algorithms, are used for short-term forecasting. The
development, improvements, and investigation of the appropriate mathematical tools will
lead to the development of more accurate load forecasting techniques.
Statistical approaches usually require a mathematical model that rep-resents load as
function of different factors such as time, weather, and customer class. The two important
categories of such mathematical models are: additive models and multiplicative models. They
differ in whether the forecast load is the sum (additive) of a number of components or the
product (multiplicative) of a number of factors.
For example, presented an additive model that takes the form of predicting load as the
function of four components:
L = Ln + Lw + Ls + Lr, -------------------------------------------- (1)
where L is the total load, Ln represents the “normal” part of the load, which is a set of
standardized load shapes for each “type” of day that has been identified as occurring
throughout the year, Lw represents the weather sensitive part of the load, Ls is a special event
component that create a substantial deviation from the usual load pattern, and Lr is a
completely random term, the noise.
Naturally, price decreases/increases affect electricity consumption. Large cost sensitive
industrial and institutional loads can have a significant effect on loads..
A multiplicative model may be of the form
L = Ln+ Fw +Fs+ Fr, -------------------------------------------------- (2)
where Ln is the normal (base) load and the correction factors Fw , Fs, and Fr are positive
numbers that can increase or decrease the overall load. These corrections are based on current
weather (Fw ), special events (Fs), and random fluctuation (Fr ). Factors such as electricity
pricing (Fp) and load growth (Fg ) can also be included. Weather variables and the base load
associated with the weather measures were included in the model.
2.1.3. Medium- and long-term load forecasting methods
The end-use modeling, econometric modeling, and their combinations are the most often
used methods for medium- and long-term load fore-casting. Descriptions of appliances used
by customers, the sizes of the houses, the age of equipment, technology changes, customer
behavior, and population dynamics are usually included in the statistical and simulation
models based on the so-called end-use approach. In addition, economic factors such as per
capita incomes, employment levels, and electricity prices are included in econometric
models. These models are often used in combination with the end-use approach. Long-term
fore-casts include the forecasts on the population changes, economic development, industrial
construction, and technology development.
End-use models. The end-use approach directly estimates energy consumption by using
extensive information on end use and end users, such as appliances, the customer use, their
age, sizes of houses, and so on. Statistical information about customers along with dynamics
of change is the basis for the forecast.
End-use models focus on the various uses of electricity in the residential, commercial, and
industrial sector. These models are based on the principle that electricity demand is derived
from customer’s demand for light, cooling, heating, refrigeration, etc.
Ideally this approach is very accurate. However, it is sensitive to the amount and quality of
end-use data. For example, in this method the distribution of equipment age is important for
particular types of appliances. End-use forecast requires less historical data but more in-
formation about customers and their equipment.
Econometric models. The econometric approach combines economic theory and statistical
techniques for forecasting electricity demand. The approach estimates the relationships
between energy consumption (de-pendent variables) and factors influencing consumption.
The relation-ships are estimated by the least-squares method or time series methods.
One of the options in this framework is to aggregate the econometric approach, when
consumption in different sectors (residential, commercial, industrial, etc.) is calculated as a
function of weather, economic and other variables, and then estimates are assembled using
recent historical data. Integration of the econometric approach into the end-use approach
introduces behavioral components into the end-use equations.
Statistical model-based learning. The end-use and econometric methods require a large
amount of information relevant to appliances, customers, economics, etc. Their application is
complicated and requires human participation. In addition such information is often not
available regarding particular customers and a utility keeps and supports a pro-file of an
“average” customer or average customers for different type of customers. The problem arises
if the utility wants to conduct next-year forecasts for sub-areas, which are often called load
pockets. In this case, the amount of the work that should be performed increases proportion-
ally with the number of load pockets. In addition, end-use profiles and econometric data for
different load pockets are typically different. The characteristics for particular areas may be
different from the average characteristics for the utility and may not be available.
We compared several load models and came to the conclusion that the following
multiplicative model is the most accurate
L(t) = F (d(t), h(t)) · f (w(t)) + R(t),--------------------------------------(3)
where L(t) is the actual load at time t, d(t) is the day of the week, h(t) is the hour of the day, F
(d, h) is the daily and hourly component, w(t) is the weather data that include the temperature
and humidity, f (w) is the weather factor, and R(t) is a random error.
In fact, w (t) is a vector that consists of the current and lagged weather variables. This
reflects the fact that electric load depends not only on the current weather conditions but also
on the weather during the previous hours and days. In particular, the well-known effect of the
so-called heat waves is that the use of air conditioners increases when the hot weather
continues for several days.
2.2. ECONOMIC DISPATCH
2.2.1. Economic Operation of Power Systems
One of the earliest applications of on-line centralized control was to provide a central
facility, to operate economically, several generating plants supplying the loads of the system.
Modern integrated systems have different types of generating plants, such as coal fired
thermal plants, hydel plants, nuclear plants, oil and natural gas units etc. The capital
investment, operation and maintenance costs are different for different types of plants.
The operation economics can again be subdivided into two parts.
i) Problem of economic dispatch, which deals with determining the power output of each
plant to meet the specified load, such that the overall fuel cost is minimized.
ii) Problem of optimal power flow, which deals with minimum – loss delivery, where in the
power flow, is optimized to minimize losses in the system. In this chapter we consider the
problem of economic dispatch.
During operation of the plant, a generator may be in one of the following states:
i) Base supply without regulation: the output is a constant.
ii) Base supply with regulation: output power is regulated based on system load.
iii) Automatic non-economic regulation: output level changes around a base setting as area
control error changes.
iv) Automatic economic regulation: output level is adjusted, with the area load and area
control error, while tracking an economic setting.
Regardless of the units operating state, it has a contribution to the economic
operation,
even though its output is changed for different reasons. The factors influencing the cost of
generation are the generator efficiency, fuel cost and transmission losses. The most efficient
generator may not give minimum cost, since it may be located in a place where fuel cost is
high. Further, if the plant is located far from the load centers, transmission losses may be high
and running the plant may become uneconomical. The economic dispatch problem basically
determines the generation of different plants to minimize total operating cost.
Modern generating plants like nuclear plants, geo-thermal plants etc, may require
capital
Investment of millions of rupees. The economic dispatch is however determined in terms of
fuel cost per unit power generated and does not include capital investment, maintenance,
depreciation, start-up and shut down costs etc.
2.2.2. Performance Curves
Input-Output Curve This is the fundamental curve for a thermal plant and is a plot of the input in British thermal
units (Btu) per hour versus the power output of the plant in MW as shown in Fig.1
Figure 1: Input – output curve
Heat Rate Curve
The heat rate is the ratio of fuel input in Btu to energy output in KWh. It is the slope of the
input – output curve at any point. The reciprocal of heat – rate is called fuel –efficiency. The
heat rate curve is a plot of heat rate versus output in MW. A typical plot is shown in Fig.2.
Figure .2 Heat rate curve.
Incremental Fuel Rate Curve
The incremental fuel rate is equal to a small change in input divided by the corresponding
change in output.
Incremental fuel rate =∆Input/∆ Output
The unit is again Btu / KWh. A plot of incremental fuel rate versus the output is shown in
Figure 3: Incremental fuel rate curve
Incremental cost curve The incremental cost is the product of incremental fuel rate and fuel cost (Rs / Btu or $ /Btu).
The curve in shown in Fig. 4. The unit of the incremental fuel cost is Rs / MWh or $ /MWh.
Figure 4: Incremental cost curve
In general, the fuel cost Fi for a plant, is approximated as a quadratic function of the
generated output PGi.
Fi = ai + bi PGi + ci PG2 Rs / h --------------------------------- (4)
The incremental fuel cost is given by
Rs / MWh ------------------------------------ (5)
The incremental fuel cost is a measure of how costly it will be produce an increment of
power. The incremental production cost, is made up of incremental fuel cost plus the
incremental cost of labour, water, maintenance etc. which can be taken to be some percentage
of the incremental fuel cost, instead of resorting to a rigorous mathematical model. The cost
curve can be approximated by a linear curve. While there is negligible operating cost for a
hydel plant, there is a limitation on the power output possible. In any plant, all units normally
operate between PGmin, the minimum loading limit, below which it is technically infeasible
to operate a unit and PGmax, which is the maximum output limit.
2.2.3. Solution Methods:
1. Lagrange Multiplier method
2. Lamda iteration method
3. Gradient method
4. Dynamic programming
5. Evolutionary Computation techniques
2.2.4. The Economic Dispatch Problem
Figure 2.5 shows the configuration that will be studied in this section.
This system consists of N thermal-generating units connected to a single bus-bar serving a
received electrical load Pload input to each unit, shown as FI,represents the cost rate of the
unit. The output of each unit, Pi, is the electrical power generated by that particular unit. The
total cost rate of this system is, of course, the sum of the costs of each of the individual units.
The essential constraint on the operation of this system is that the sum of the output powers
must equal the load demand.Mathematically speaking, the problem may be stated very
concisely. That is, an objective function, FT, is equal to the total cost for supplying the
indicated load. The problem is to minimize FT subject to the constraint that the sum of the
powers generated must equal the received load. Note that any transmission losses are
neglected and any operating limits are not explicitly stated when formulating this problem.
That is,
------------------------------------------ (6)
Figure2.5. N thermal units committed to serve a load of Pload.
This is a constrained optimization problem that may be attacked formally using advanced
calculus methods that involve the Lagrange function. In order to establish the necessary
conditions for an extreme value of the objective function, add the constraint function to the
objective function after the constraint function has been multiplied by an undetermined
multiplier. This is known as the Lagrange function and is shown in Eq(7)
------------------------------------------------ (7)
The necessary conditions for an extreme value of the objective function result when we
take the first derivative of the Lagrange function with respect to each of the independent
variables and set the derivatives equal to zero. In this case,there are N+1 variables, the N
values of power output, Pi, plus the undetermined Lagrange multiplier, λ. The derivative of
the Lagrange function with respect to the undetermined multiplier merely gives back the
constraint equation. On the other hand, the N equations that result when we take the
partial derivative of the Lagrange function with respect to the power output values one at a
time give the set of equations shown as Eq. 8.
----------------------------------------------- (8)
When we recognize the inequality constraints, then the necessary conditions may be
expanded slightly as shown in the set of equations making up Eq. 9
--------------------------------------------- (9)
Several of the examples in this chapter use the following three generator units.
EXAMPLE 2.1
Suppose that we wish to determine the economic operating point for these three units
when delivering a total of 850 MW. Before this problem can be solved,the fuel cost of each
unit must be specified. Let the following fuel costs are in effect.
Unit 1: Coal-fired steam unit: Max output = 600 MW Min output = 150 MW
Input-output curve:
Unit 2 Oil-fired steam unit: Max output = 400 MW
Min output = 100 MW
Input-output curve:
Unit 3: Oil-fired steam unit: Max output = 200 MW, Min output = 50 MW
Input-output curve:
Unit 1: fuel cost = 1.1 P/MBtu
Unit 2: fuel cost = 1.0 Jt/MBtu
Unit 3: fuel cost = 1.0 Jt/MBtu
Then
Using Eq. 3.5, the conditions for an optimum dispatch are
and then solving for, P1,P2,P3
P1 = 393.2 MW
P2 = 334.6 MW
P3 = 122.2 MW
Note that all constraints are met; that is, each unit is within its high and low limit and the total
output when summed over all three units meet the desired 850 MW total.
EXAMPLE 2.2
Suppose the price of coal decreased to 0.9 P/MBtu. The fuel cost function for unit 1 becomes
If one goes about the solution exactly as done here, the results are
This solution meets the constraint requiring total generation to equal 850 MW, but units 1 and
3 are not within limit. To solve for the most economic dispatch while meeting unit limits,
suppose unit 1 is set to its maximum output and unit 3 to its minimum output. The dispatch
becomes
we see that λ must equal the incremental cost of unit 2 since it
is not at either limit. Then
Next, calculate the incremental cost for units 1 and 3 to see if they meet the conditions.
Note that the incremental cost for unit 1 is less than λ, so unit 1 should be at its maximum.
However, the incremental cost for unit 3 is not greater than ,λ so unit 3 should not be forced
to its minimum. Thus, to find the optimal dispatch, allow the incremental cost at units 2 and 3
to equal λ as follows.
2.2.5. Thermal System Dispatching With Network Losses Considered
Figure 2.6. Shows symbolically an all-thermal power generation system connected to an
equivalent load bus through a transmission network. The economic dispatching problem
associated with this particular configuration is slightly more complicated to set up than the
previous case. This is because the constraint equation is now one that must include the
network losses. The objective function, FT, is the same as that defined for Eq.10
------------------ (10)
The same procedure is followed in the formal sense to establish the necessary conditions for a
minimum-cost operating solution, The Lagrange function is shown in Eq.11. In taking the
derivative of the Lagrange function with respect to each of the individual power outputs, Pi, it
must be recognized that the loss in the transmission network, Ploss is a function of the network
impedances and the currents flowing in the network. For our purposes, the currents will be
considered only as a function of the independent variables Pi and the load Pload taking the
derivative of the Lagrange function with respect to any one of the N values of Pi results in Eq.
11. collectively as the coordination equations.
----------------------------------- (11)
It is much more difficult to solve this set of equations than the previous set with no losses
since this second set involves the computation of the network loss in order to establish the
validity of the solution in satisfying the constraint equation. There have been two general
approaches to the solution of this problem. The first is the development of a mathematical
expression for the losses in the network solely as a function of the power output of each of
the units. This is the loss-formula method discussed at some length in Kirchmayer’s
Economic Operation of Power Systems. The other basic approach to the solution of this
problem is to incorporate the power flow equations as essential constraints in the formal
establishment of the optimization problem. This general approach is known as the optimal
power flow.
Figure.2.6.N thermal units serving load through transmission network
2.2.6. The Lambda-Iteration Method
Figure 2.7.is a block diagram of the lambda-iteration method of solution for the all-thermal,
dispatching problem-neglecting losses. We can approach the solution to this problem by
considering a graphical technique for solving the problem and then extending this into the
area of computer algorithms. Suppose we have a three-machine system and wish to find the
optimum economic operating point. One approach would be to plot the incremental cost
characteristics for each of these three units on the same graph, such as sketched in Figure 3.4.
In order to establish the operating points of each of these three units such that we have
minimum cost and at the same time satisfy the specified demand, we could use this sketch
and a ruler to find the solution. That is, we could assume an incremental cost rate (λ) and find
the power outputs of each of the three units for this value of incremental cost. the three units
for this value of incremental cost. Of course, our first estimate will be incorrect. If we have
assumed the value of incremental cost such that the total power output is too low, we must
increase the 3. value and try another solution. With two solutions, we can extrapolate (or
interpolate) the two solutions to get closer to the desired value of total received power. By
keeping track of the total demand versus the incremental cost, we can rapidly find the desired
operating point. If we wished, we could manufacture a whole series of tables that would show
the total power supplied for different
incremental cost levels and combinations of units. That is, we will now establish a set of
logical rules that would enable us to accomplish the same objective as we have just done with
ruler and graph paper. The actual details of how the power output is established as a function
of the incremental cost rate are of very little importance.
Figure: 2.7. Lambda-iteration method
We could, for example, store tables of data within the computer and interpolate between the
stored power points to find exact power output for a specified value of incremental cost rate.
Another approach would be to develop an analytical function for the power output as a
function of the incremental cost rate, store this function (or its coefficients) in the computer,
and use this to establish the output of each of the individual units.
This procedure is an iterative type of computation, and we must establish stopping rules. Two
general forms of stopping rules seem appropriate for this application..The lambda-iteration
procedure converges very rapidly for this particular type of optimization problem. The actual
computational procedure is slightly more complex than that indicated in Figure 2.7 since it is
necessary to observe the operating limits on each of the units during the course of the
computation. The well-known Newton-Raphson method may be used to project the
incremental cost value to drive the error between the computed and desired generation to
zero.
Example: 2.3
Given the generator cost functions found in Example 2.1, solve for the economic
dispatch of generation with a total load of 800 MW.Using α = 100 and starting from P10= 300
MW, P20 = 200 MW, and P3
0=300 MW, we set the initial value of λ. equal to the average of
the incremental costs of the generators at their starting generation values. This value is
9.4484.The progress of the gradient search is shown in Table 3.2. The table shows that the
iterations have led to no solution at all. Attempts to use this formulation
will result in difficulty as the gradient cannot guarantee that the adjustment to the generators
will result in a schedule that meets the correct total load of 800 MW.A simple variation of
this technique is to realize that one of the generators is always a dependent variable and
remove it from the problem. In this case, we pick P3 and use the following:
Then the total cost, which is to be minimized, is:
Note that this function stands by itself as a function of two variables with no
load-generation balance constraint (and no λ). The cost can be minimized by
a gradient method and in this case the gradient is:
\
Note that this gradient goes to the zero vector when the incremental cost at generator 3 is
equal to that at generators 1 and 2. The gradient steps are performed in the same manner as
previously, where:
------------------------------------ (12)
Each time a gradient step is made, the generation at generator 3 is set to 800 minus the sum of
the generation at generators 1 and 2. This method is often called the “reduced gradient”
because of the smaller number of variables.
2.2.7. Base Point and Participation Factors
This method assumes that the economic dispatch problem has to be solved repeatedly by
moving the generators from one economically optimum schedule to another as the load
changes by a reasonably small amount. We start from a given schedule-the base point. Next,
the scheduler assumes a load change and investigates how much each generating unit needs
to be moved (i.e.,“participate” in the load change) in order that the new load be served at the
most economic operating point.Assume that both the first and second derivatives in the cost
versus power output function are available (Le., both F; and Fy exist). The incremental cost
curve of the ith
unit is given in Figure 3.7. As the unit load is changed by an amount, the
system incremental cost moves from λ0toλ
0 for a small change in power output on this single
unit,
----------------------------------------- (13)
This is true for each of the N units on the system, so that
---------------------------------------(14)
The total change in generation (=change in total system demand) is, of course, the sum of the
individual unit changes. Let Pd be the total demand on the generators (where Pload+Ploss&),
then
---------------------------------------- (15)
The earlier equation, 15, can be used to find the participation factor for each unit as follows
--------------------------------------- (16)
The computer implementation of such a scheme of economic dispatch is straightforward. It
might be done by provision of tables of the values of FY as a function of the load levels and
devising a simple scheme to take the existing load plus the projected increase to look up these
data and compute the factors. somewhat less elegant scheme to provide participation factors
would involve a repeat economic dispatch calculation at. The base-point economic generation
values are then subtracted from the new economic generation values and the difference
divided to provide the participation factors. This scheme works well in computer
implementations where the execution time for the economic dispatch is short and will always
give consistent answers when units reach limits, pass through break points on piecewise
linear incremental cost functions, or have nonconvex cost curves.
EXAMPLE 2.4
Starting from the optimal economic solution found in Example 2A; use the participation
factor method to calculate the dispatch for a total load of 900 MW.
2.2.8. Economic dispatch controller added to LFC:
Both the load frequency control and the economic dispatch issue commands to change
the power setting of each turbine-governor unit. At a first glance it may seem that these two
commands can be conflicting. This however is not true. A typical automatic generation
control strategy is shown in Fig. 5.5 in which both the objective are coordinated. First we
compute the area control error. A share of this ACE, proportional to αi , is allocated to each
of the turbine-generator unit of an area. Also the share of unit- i , γi X Σ( PDK - Pk ), for the
deviation of total generation from actual generation is computed. Also the error between the
economic power setting and actual power setting of unit- i is computed. All these signals are
then combined and passed through a proportional gain Ki to obtain the turbine-governor
control signal.
Figure: 2.8. Economic dispatch controller added to LFC
2.3. UNIT COMMITMENT:
The life style of a modern man follows regular habits and hence the present society
also follows regularly repeated cycles or pattern in daily life. Therefore, the consumption of
electrical energy also follows a predictable daily, weekly and seasonal pattern. There are
periods of high power consumption as well as low power consumption. It is therefore
possible to commit the generating units from the available capacity into service to meet the
demand. The previous discussions all deal with the computational aspects for allocating load
to a plant in the most economical manner. For a given combination of plants the
determination of optimal combination of plants for operation at any one time is also desired
for carrying out the aforesaid task. The plant commitment and unit ordering schedules extend
the period of optimization from a few minutes to several hours. From daily schedules weekly
patterns can be developed. Likewise, monthly, seasonal and annual schedules can be prepared
taking into consideration the repetitive nature of the load demand and seasonal variations.
Unit commitment schedules are thus required for economically committing the units in plants
to service with the time at which individual units should be taken out from or returned to
service.
2.3.1. Constraints in Unit Commitment
Many constraints can be placed on the unit commitment problem. The list presented
here is by no means exhaustive. Each individual power system, power pool, reliability
council, and so forth, may impose different rules on the scheduling of units, depending on the
generation makeup, load-curve characteristics,
and such.
2.3.2. Spinning Reserve
Spinning reserve is the term used to describe the total amount of generation available
from all units synchronized (i.e., spinning) on the system, minus the present load and losses
being supplied. Spinning reserve must be carried so that the loss of one or more units does
not cause too far a drop in system frequency. Quite simply, if one unit is lost, there must be
ample reserve on the other units to make up for the loss in a specified time period.Spinning
reserve must be allocated to obey certain rules, usually set by regional reliability councils (in
the United States) that specify how the reserve is to be allocated to various units. Typical
rules specify that reserve must be a given percentage of forecasted peak demand, or that
reserve must be capable of making up the loss of the most heavily loaded unit in a given
period of time.Others calculate reserve requirements as a function of the probability of not
having sufficient generation to meet the load.Not only must the reserve be sufficient to make
up for a generation-unit failure, but the reserves must be allocated among fast-responding
units and slow-responding units. This allows the automatic generation control system to
restore frequency and interchange quickly in the event of a generating-unit outage. Beyond
spinning reserve, the unit commitment problem may involve various classes of “scheduled
reserves” or “off-line” reserves. These include quick-start
diesel or gas-turbine units as well as most hydro-units and pumped-storage hydro-units that
can be brought on-line, synchronized, and brought up to full capacity quickly. As such, these
units can be “counted” in the overall reserve assessment, as long as their time to come up to
full capacity is taken into account. Reserves, finally, must be spread around the power system
to avoid transmission system limitations (often called “bottling” of reserves) and to allow
various parts of the system to run as “islands,” should they become electrically disconnected.
2.3.3. Thermal Unit Constraints
Thermal units usually require a crew to operate them, especially when turned on and turned
off. A thermal unit can undergo only gradual temperature changes, and this translates into a
time period of some hours required to bring the unit on-line. As a result of such restrictions in
the operation of a thermal plant, various constraints arise, such as:
1. Minimum up time: once the unit is running, it should not be turned off immediately
2. Minimum down time: once the unit is decommitted, there is a minimum time before it
can be recommitted.
3. Crew constraints: if a plant consists of two or more units, they cannot both be turned on
at the same time since there are not enough crew members to attend both units while starting
up. In addition, because the temperature and pressure of the thermal unit must be moved
slowly, a certain amount of energy must be expended to bring the unit on-line. This energy
does not result in any MW generation from the unit and is brought into the unit commitment
problem as a start-up cost. The start-up cost can vary from a maximum “cold-start” value to a
much smaller value if the unit was only turned off recently and is still relatively close to
operating temperature. There are two approaches to treating a thermal unit during its down
period. The first allows the unit’s boiler to cool down and then heat back up to operating
temperature in time for a scheduled turn on. The second (called banking) requires that
sufficient energy be input to the boiler to just maintain operating temperature. The costs for
the two can be compared so that, if possible, the best approach (cooling or banking) can be
chosen.
Start-up cost when cooling = Cc × F+Cf------------------------------------------------- (17)
Where
Cc = cold-start cost (MBtu)
F = fuel cost
Cf= fixed cost (includes crew expense, maintenance expenses) (in R)
α = thermal time constant for the unit
t = time (h) the unit was cooled
Start-up cost when banking = Ct x t x F+Cf
Where
Ct = cost (MBtu/h) of maintaining unit at operating temperature
Up to a certain number of hours, the cost of banking will be less than the cost of cooling, as is
illustrated in Figure 5.3.Finally, the capacity limits of thermal units may change frequently,
due to maintenance or unscheduled outages of various equipment in the plant; this must also
be taken 2.3.4 Other Constraints
2.3.4.1 Hydro-Constraints
Unit commitment cannot be completely separated from the scheduling of hydro-units. In this
text, we will assume that the hydrothermal scheduling (or “coordination”) problem can be
separated from the unit commitment problem. We, of course, cannot assert flatly that our
treatment in this fashion will always
result in an optimal solution.
Figure:2.9. Hydro-Constraints
2.3.4.2 Must Run
Some units are given a must-run status during certain times of the year for reason of voltage
support on the transmission network or for such purposes as supply of steam for uses outside
the steam plant itself.
2.3.4.3 Fuel Constraints
We will treat the “fuel scheduling” problem system in which some units have limited fuel, or
else have constraints that require them to burn a specified amount of fuel in a given time,
presents a most challenging unit commitment problem.
2.3.5. Unit Commitment Solution Methods
The commitment problem can be very difficult. As a theoretical exercise, let us postulate the
following situation.
1. We must establish a loading pattern for M periods.
2. We have N units to commit and dispatch.
3. The M load levels and operating limits on the N units are such that any one unit can
supply the individual loads and that any combination of units can also supply the
loads.
Next, assume we are going to establish the commitment by enumeration (brute force). The
total number of combinations we need to try each hour is,
C (N, 1) + C (N,2) + ... + C(N, N - 1) + C ( N , N ) = 2N – 1---------------------------------------
(18)
Where C (N, j) is the combination of N items taken j at a time. That is,
------------------------------------------
(19)
For the total period of M intervals, the maximum number of possible combinations is (2N -
l)M, which can become a horrid number to think about.
For example, take a 24-h period (e.g., 24 one-hour intervals) and consider systems with 5, 10,
20, and 40 units. The value of (2N - 1)24
becomes the following.
These very large numbers are the upper bounds for the number of enumerations required.
Fortunately, the constraints on the units and the load-capacity relationships of typical utility
systems are such that we do not approach these large numbers. Nevertheless, the real
practical barrier in the optimized unit commitment problem is the high dimensionality of the
possible solution space. The most talked-about techniques for the solution of the unit
commitment problem are:
1. Priority-list schemes,
2. Dynamic programming (DP),
3. Lagrange relation (LR).
2.3.5.1. Priority-List Methods
The simplest unit commitment solution method consists of creating a priority list of units. As
we saw in Example 5B, a simple shut-down rule or priority-list scheme could be obtained
after an exhaustive enumeration of all unit combinations at each load level. The priority list
of Example 5B could be obtained in
a much simpler manner by noting the full-load average production cost of each unit, where
the full-load average production cost is simply the net heat rate at full load multiplied by the
fuel cost.
Priority List Method:
Priority list method is the simplest unit commitment solution which consists of creating a
priority list of units.
Full load average production cost= Net heat rate at full load X Fuel cost
Assumptions:
N (2N - 1)24
5 6.2 ×1035
10 1.73×1072
20 3.12×10144
40 Too big
1. No load cost is zero
2. Unit input-output characteristics are linear between zero output and full load
3. Start up costs are a fixed amount
4. Ignore minimum up time and minimum down time
Steps to be followed
1. Determine the full load average production cost for each units
2. Form priority order based on average production cost
3. Commit number of units corresponding to the priority order
4. Alculate PG1, PG2 ………….PGN from economic dispatch problem for the feasible
combinations only
5. For the load curve shown
Assume load is dropping or decreasing, determine whether dropping the next unit will supply
generation & spinning reserve.
If not, continue as it is
If yes, go to the next step
6. Determine the number of hours H, before the unit will be needed again.
7. Check H< minimum shut down time.
If not, go to the last step
If yes, go to the next step
8. Calculate two costs
1. Sumof hourly production for the next H hours with the unit up
2. Recalculate the same for the unit down + start up cost for either cooling or banking
9. Repeat the procedure until the priority list
Merits:
1. No need to go for N combinations
2. Take only one constraint
3. Ignore the minimum up time & down time
4. Complication reduced
Demerits:
1. Start up cost are fixed amount
2. No load costs are not considered.
2.3.5.2. Dynamic-Programming Solution
Dynamic programming has many advantages over the enumeration scheme, the chief
advantage being a reduction in the dimensionality of the problem. Suppose we have found
units in a system and any combination of them could serve the (single) load. There would be
a maximum of 24 - 1 = 15 combinations to test. However, if a strict priority order is imposed,
there are only four combinations to try:
Priority 1 unit
Priority 1 unit + Priority 2 unit
Priority 1 unit + Priority 2 unit + Priority 3 unit
Priority 1 unit + Priority 2 unit + Priority 3 unit + Priority 4 unit
The imposition of a priority list arranged in order of the full-load averagecost rate would
result in a theoretically correct dispatch and commitment only if:
1. No load costs are zero.
2. Unit input-output characteristics are linear between zero output and full load.
3. There are no other restrictions.
4. Start-up costs are a fixed amount.
In the dynamic-programming approach that follows, we assume that:
1. A state consists of an array of units with specified units operating and
2. The start-up cost of a unit is independent of the time it has been off-line
3. There are no costs for shutting down a unit.
4. There is a strict priority order, and in each interval a specified minimum the rest off-line.
(i.e., it is a fixed amount).amount of capacity must be operating.
A feasible state is one in which the committed units can supply the required load and that
meets the minimum amount of capacity each period.
2.3.5.3. Forward DP Approach
One could set up a dynamic-programming algorithm to run backward in time starting from
the final hour to be studied, back to the initial hour. Conversely, one could set up the
algorithm to run forward in time from the initial hour to the final hour. The forward approach
has distinct advantages in solving generator unit commitment. For example, if the start-up
cost of a unit is a function of the time it has been off-line (i.e., its temperature), then a
forward dynamic-program approach is more suitable since the previous history of the
unit can be computed at each stage. There are other practical reasons for going forward. The
initial conditions are easily specified and the computations can go forward in time as long as
required. A forward dynamic-programming algorithm is shown by the flowchart in Figure
2.11 The recursive algorithm to compute the minimum cost in hour K with combinati
Fcost(K,I)= min[Pcost(K,I)+Scost(K-1,L:K,I)+Fcost(K-1,L)] ----------------------------------(20)
Where
Fcost(K, I ) = least total cost to arrive at state ( K , I )
Pcost(KI, ) = production cost for state ( K ,I )
Scost(K - 1, L: K , I)= transition cost from state (K - 1, L) to state ( K , I )
State (K, 1) is the Zth combination in hour K. For the forward dynamic programming
approach, we define a strategy as the transition, or path, from one state at a given hour to a
state at the next hour.
Note that two new variables, X and N, have been introduced in Figure 2.11
X = number of states to search each period
N = number of strategies, or paths, to save at each step
These variables allow control of the computational effort (see below Figure).For complete
enumeration, the maximum number of the value of X or N is 2n – 1
Figure: 2.10. Compute the minimum cost
Figure: 2.11. Forward DP Approach
2.3.5.4. Lagrange Relaxation Solution
The dynamic-programming method of solution of the unit commitment problem has many
disadvantages for large power systems with many generating units. This is because of the
necessity of forcing the dynamic-programming solution to search over a small number of
commitment states to reduce the number of combinations that must be tested in each time
period.
We start by defining the variable as
We shall now define several constraints and the objective function of the unit commitment
problem:
1. Loading constraints:
----------------------- (21)
2. Unit limits:
3. Unit minimum up- and down-time constraints. Note that other constraints can easily be
formulated and added to the unit commitment problem. These include transmission
security constraints (see Chapter 1 l), generator fuel limit constraints, and system air
quality constraints in the form of limits on emissions from fossil-fired plants, spinning
reserve constraints,etc.
4. The objective function is:
------------------------- (22)
We can then form the Lagrange function similar to the way we did in the economic dispatch
problem:
---------------------------- (23)
Example:2.5
The costs of two units at the busses connected through a transmission line are (with P1and P2
in MW): IC1=15+0.125 P1; IC2=20+0.05 P2, If 125 MW is transmitted from unit-1 to the
load at bus-2, at which the unit-2 is present, a line loss of 15.625 MW is incurred. Find the
required generation for each of the units and the power received by the load when the system
lambda is Rs.24.0 per MWHr. Use Penalty Factor method.
Solution:
With unit-2 not contributing to the line loss, it is due to the unit-1 alone, and hence,
DPL/dP2 = ITL2 =0; where, PL=B11P12; i.e., B11= PL/ P1
2 = 15.625/1252 = 10
-3 MW
-1
Thus, PL=10-3
P12 so that dPL/dP1= ITL1 = 2(10
-3) P1 MW
Hence we have,
IC1 = 15+0.125 P1 = λ (1-ITL1) = 24 {1 - 2(10-3) P1} and
IC2 = 20+0.05 P2 = λ (1-ITL2) = λ = 24
Solving, we get, P1=52 MW and P2= 80 MW.
Total loss = Total Generation – Total Load = (P1+P2) – PLoad
QUESTION BANK
PART- A
1. What is meant by incremental cost curve?
2. Write the constraints in Unit Commitment.
3. What is the purpose of economic dispatch? What is meant by unit
commitment?
4. List the various constraints in modern power systems.
5. What is meant by unit commitment?
6. What is system load forecasting?
7. Compare unit commitment and economic dispatch.
8. What are the advantages of using participation factor?
9. Explain penalty factor.
10. Define Participation factor.
11. What is priority list method?
PART-B
1. Explain briefly the constraints on unit commitment problem.
2. Explain priority list method using two schemes.
3. Explain with a neat flow chart the procedure for finding the solution for unit
commitment problems using forward DP (FDP) method.
4. The input-output curve characteristics of three units are F1=940+5.46PG1+.0016P2
G1,
F2=820+5.35PG2+.0019P2G2, F3=99+5.65 PG3+.0032P
2G3.Total load is 600MW.use the
Participation factor method to calculate the economic dispatch for load is reduced to 550MW?
5. Consider three units,
a. C1=561+7.92P1+0.00156P12
b. C2=310+7.85P2+0.00194P22
c. C3=780+7.97P3+0.00482P32
6. Unit Minimum Maximum
1 150 600
2 100 400
3 50 200
Find the priority list method using full load average production cost which the units
are committed and de committed in unit commitment problem.
7. The input –output curve characteristics of three units are:
F1=750+6.49 PG1+0.0035P2G1, F2=870+5.75 PG2+0.0015P2G2
F3=620+8.56 PG3+0.001P2G3, the fuel cost of unit 1, 2, 3 is 1.0 Rs / Mbtu.Total load
is 800 MW. Use Participation factor method to calculate the dispatch for a load is
increased to 880MW?
8. Draw the flow chart for obtaining the optimum dispatch strategy of N-bus system
with and without transmission loss.
9. Explain the reserve requirement & types of reserve.
10. Explain the classifications of load forecasting and need for load forecasting.
11. Give λ-iteration algorithm for solving economic scheduling problem, without
transmission loss.
12. Explain the economic dispatch controller added to LFC control.