chapter-04 economic operation of power system

Prepared by Balaram Das, EE Dept., GIET, Gunupur

Chapter-04

Economic Operation of Power System

Introduction

Economic operation is very important for a power system to return a profit on the

capital invested. Two things put pressure on power companies to achieve maximum

possible efficiency.

(a) Rates fixed by regulatory bodies and

(b) the importance of conservation of fuel place

Maximum efficiency minimizes the cost of electrical energy consumed by the

consumers. Also it reduces the rising prices for fuel, labor, supplies and maintenance.

Operational economics involving power generation and delivery of the power. Delivery

can be subdivided into two parts.

(i) One dealing with minimum cost of power production called Economic dispatch.

(ii) Other dealing with minimum loss of the generated power delivery to the loads.

For any specified load condition, economic dispatch

(i) determines the power output of each plant.

(ii)Minimizes the overall cost of fuel needed to serve the system load.

The economic dispatch problem can be solved by means of the optimal power flow

(OPF) program.

We first study the most economic distribution of power within the plant. The method

that we develop applies to economic scheduling of plant output for a given loading of

the system without considering of transmission losses.

Next we express transmission loss as a function of the outputs of the various plants.

Then we determine how the output of each of the plants of a system is scheduled to

achieve the minimum cost of power delivered to the load.

Because the total load of the power system varies throughout the day, coordinated

control of the power plant outputs is necessary to ensure generation to load balance

so that the system frequency will remain as close as possible to the nominal operating

value, usually 50 or 60 Hz. Also because of the daily load variation, the utility has to

decide on the basis of economics which generator to start up, which generators to

shut down, and in what order. The computational procedure for making such

decisions, called unit commitment.


Distribution of load between units within a plant

Power plant consisting of several generating units which are constructed by investing

huge amount of money. Fuel cost, staff salary, interest and depreciation charges and

maintenance cost are some of the components of operating cost.

Fuel cost is the major portion of operating cost and it can be controlled. Therefore we

shall consider the fuel cost alone for further considerations. To get different output

power, we need to vary the fuel input.

Fuel input can be measured in tones/hr or millions of BTU(British Thermal Unit)/hr.

Knowing the cost of the fuel, in terms of Rs/tone or Rs/Millions of BTU, input to the

generating unit can be expressed as Rs/hr.

Let Ci Rs/h be the input cost to generate a power of Pi MW in unit i. Fig.1 shows a

typical input output curve of a generating unit. For each generating unit there shall be

a minimum and maximum power generated as Pi,min and Pi,max.

If the input-output curve of Unit ‘I’ is quadratic , we can write

Where, Ci = Input cost, Pi – Output power in MW, α, β and ϒ are cost coefficient.

A power plant may have several generating units. If the input-output characteristic of

different generator is identical, then the generating units can be equally loaded. But

generating units will generally have different input-output characteristic. This means

that for a particular input cost, the generator power Pi will be different for different

generating units in a plant.

Incremental cost curve

As we shall see the criterion for distribution of the load between any two units is based

on whether increasing the generation of one unit, and decreasing the generation of

other unit by the same amount results in an increase or decrease in total cost. This


can be obtained if we can calculate the change in input cost ΔCi for a small change in

power ΔPi.

Thus while deciding the optimal scheduling, we are concerned with dCi/dPi,

Incremental cost (IC) which is determined by the slopes of the input-output curves.

Thus the incremental cost curve is the plot of dCi/dPi versus Pi. The dimension of

dCi/dPi is Rs/MWh.

Plot of Ic versus power o/p is shown in fig.2.

The fig.2 shows that the incremental cost is quite linear with respect to power output

over an appreciable range. In analytical work, the curve is usually approximated by

one or two straight lines. The dashed line in the fig-2 is a good representation of the

curve.

Economical division of plant load between generating units in a plant

Let total load in a plant is supplied by two units and that the division of load between

these units is such that the incremental cost of one unit is higher than that of the

other unit.

Now suppose some of the load is transferred from the unit with higher incremental

cost to the unit with lower incremental cost. Reducing the load on the unit with higher

incremental cost will result in greater reduction of cost than the increase in cost for

adding the same amount of load to the unit with lower incremental cost.


The transfer of load from one to other can be continued with a reduction of total cost

until the incremental costs of the two units are equal. The same reason can be

extended to a plant with more than two generating units also. In this case, if any two

units have different incremental costs, then in order to decrease the total cost of

generation, decrease the output power in units having higher incremental cost and

increase the output power in units having lower incremental cost.

When this process is continued, a stage will reach where incremental costs of all the

units will be equal. Now the total cost of the generation will be minimum. Thus the

economical division of load between units within a plant is that all units must operate

at the same incremental cost.

Economy loading neglecting transmission losses

Consider a system having two generating units having costs C1 and C2.

C1 and C2 are the fuel costs of the two units in Rs/hr.

Total output PT is equal to active power demand and is constant. It is desired to find P1

and P2 so that CT is minimum.

From equation (5)

For minimum total cost CT,

Combining equation (6), (7) and (8) we have

So it is concluded that the loads should be so allocated that the two units operate at

equal incremental costs.

The above concept can be extended to a system with any number of units.


Thus optimum economy is achieved if every unit operates at the same cost.


Distribution of load between different plants

Since plants are generally long distance a parts in an integrated system, it becomes

essential to consider transmission losses in deciding the load allocation to different

plants. This leads to the dispatch of power in an economic way so as to make the

overall cost to be the minimum.

Let there be ‘m’ no of plants in a system integrated by transmission line.

PG1, PG2..PGm - Generation output of the plants in MW

PD -Total load in MW

PL - Losses in transmission lines.

Where, Cn – Fuel cost of nth plant in Rs/hr

PGn – output of nth plant in MW.


Our objective is to obtain a minimum cost for a fixed system load PD subject to the

power balance constraint of equation (3). We now present the procedure for solving

such minimization problems called the method of Lagrange multipliers.

The new cost function C is formed by combining the total fuel cost and the equality

constraint of equation(3) in the following manner.

The cost function C is often called the lagrangian, and we shall see that the parameter

λ which we now call Lagrange multiplier is the effective incremental fule cost of the

system when transmission line losses are taken into account.

C and λ are expressed in Rs/hr

For minimum cost we require the derivative of C with respect to each PGm to equal

zero,

Since PD is fixed and the fuel cost of any one unit varies only if the power output of

that unit is varied. Therefore equation (5) becomes

Because Cm depends on only PGn, the partial derivative of Cm can be replaced by the

full derivative and equation (6) then gives

where Ln is called the penalty factor for plant n and is given by

Equation (7) is called the exact coordinate equation.


From equation (7)

So minimum fuel cost is obtained when the incremental fuel cost od each plant

multiplied by its penalty factor is the same for all the plants.

Representation of transmission losses

The transmission losses depend on line currents and line resistances. It is possible to

represent these losses as a function of plant loadings.

Fig. showa a simple system having two sources. Derive an expression for the

transmission loss and express it as a function of plant loadings. Assume the currents

I1 I2 are in phase.

Let ra, rb, rc be the resistances of line a,b and c respectively. The transmission loss is

Since I1 and I2 are in same phase

Let P1 and P2 be the power outputs and V1 and V2 be the bus voltage and cos Φ1 and

cos Φ2 be the power factors of sources 1 and 2 respectively. Then

Substituting these values in transmission line equation we get

Where,


The transmission loss equation:

To include the effect of transmission losses in deciding the load allocation, it is

necessary to represent the loss as a function of plant loading. The general form of loss

equation is

Where, PL – transmission losses in pu

P-plant loading in pu

B-Loss coefficient

For a two generating source system

For a three generating source system

The matrix representation of above loss equation is

Where for a total of k sources

Where, PT is the transposition of P. The B coefficients depend on the system network

parameters, configuration, plant power factor and voltages etc.

Example:1

The incremental cost characteristic of the two units in a plant are

IC1 = 0.1 P1+8.0 Rs/MWh


IC2 = 0.15 P2 + 3.0 Rs/MWh

When the total load is 100 MW, what is the optimum sharing of load?

Solution:

But P1 + P2 = 100 MW

Optimum sharing of load is

Example:2

A power system consisting of two generators of capacity 210MW each supplies a total load of

310 MW at a certain time. The respective incremental fuel cost of Generator-1 and Generator-

2 are:

Where, powers PG in MW and costs C in Rs/hr. Determine (i) the most economical division of

load between the generators and (ii) the saving in Rs/day thereby obtained compared to equal

load sharing between the machines.

Solution:

Case-1: the most economical division of load between the generators


Case-2: the saving in Rs/day thereby obtained compared to equal load sharing between the

machines.

Th

The negative sign indicates a decrease in cost as output is decreasing.

Therefore the net increase in cost = 858.619-788.972=69.647 per hr.

Unit Commitment

The total load in the power system varies throughout the day and reaches different

peak value from one day to another. Different combination of generators are to be

connected in the system to meet the varying load. When the load increases, the utility

has to decide in advance the sequence in which the generator units are to be brought

in. Similarly when the load decreases, the operating engineer need to know in advance

the sequence in which the generating units are to be shut down. The problem of

finding the order in which the units are to be shut down over a period of time (say one

day), so the total operating cost involved on that day is minimum, is known as Unit

Commitment(UC). Thus UC problem is economic dispatch over a day. The period

considered may be a week, a month or a year.

Constraints on UC problem

a. Spinning reserve: There may be sudden increase in load, more than

what was predicted. Further there may be a situation that one generating

unit may have to be shut down because of fault in generator or any of its

auxiliaries. Some system capacity has to be kept as spinning reserve i) to meet an unexpected increase in demand and


ii) to ensure power supply in the event of any generating unit suffering a forced outage.

b. Minimum up time: When a thermal unit is brought in, it cannot be turned off immediately. Once it is committed, it has to be in the system for a specified minimum up time.

c. Minimum down time: When a thermal unit is decommitted, it cannot be turned on immediately. It has to remain decommitted for a specified minimum down time.

d. Crew constraint: A plant always has two or more generating units. It may not be possible to turn on more than one generating unit at the same time due to non-availability of operating personnel.

e. Transition cost: Whenever the status of one unit is changed some transition cost is involved and this has to be taken into account.

f. Hydro constraints: Most of the systems have hydroelectric units also. The operation of hydro units, depend on the availability of water. Moreover, hydro-projects are multipurpose projects. Irrigation requirements also determine the operation of hydro plants.

g. Nuclear constraint: If a nuclear plant is part of the system, another constraint is added. A nuclear plant has to be operated as a base load plant only.

h. Must run unit: Sometime it is a must to run one or two units from the consideration of voltage support and system stability.

i. Fuel supply constraint: Some plants cannot be operated due to deficient fuel supply.

j. Transmission line limitation: Reserve must be spread around the power system to avoid transmission system limitation, often called bottling of reserves.

Solving the unit commitment problem

The objective of unit commitment is not economical to run all the units available all

the time. To determine the units of plants that should operate for a particular load is

the problem of unit commitment.

This problem is important for thermal power plant because the operating cost and

start up time are high and hence their on-off status is important.

A simple approach to the problem is to impose priority ordering, wherein the most

efficient unit is loaded first, and then followed by the less efficient units in order as the

load increases.

Finding the most economical combination of units that can supply this load demand is

to try all possible combination of units that can supply this load; to divide the load

optimally among the units of each combination by use of the co-ordination equation,

so as to find most economical operating cost of combination then to determine the

combination which has the least operating cost among all these. These combinations

can be solved by dynamic programming method.

Solution of unit commitment problem by Forward Dynamic Programming (FDP)


In FDP, we start with the first duration obtain the unit commitment schedule for this

duration, go to second duration till we reach the last duration.

In FDP, the initial conditions are easily specified, calculation can be carried out for

any desired length of time and the previous history of each unit can be calculated at

every stage.

The operating cost for any stage needs method of economic dispatch. This is due to

the fact that for any given combination of units, the operating cost is minimum if all

the units in this combination are operating at equal incremental cost.

Two other variables enter the strategy for UC by Forward dynamic programming. This

is because a no of possible combinations exist for every state. Let this be denoted by

K. Another variable is the no. of paths or strategies to save at every step. Let this

variable be denoted as L.

Procedure

• The stage in load cycle is specified as i. We start with i=1 i.e. first stage.

• The operating cost for this stage is computed. This has to be repeated for all

possible combination K.

• We now go to (i+1) th stage.

• The no. of feasible paths in duration (i-1) is found and stored.

• The minimum total cost is calculated using the formula

• Tcost (i,n)=min[least total cost to reach state(i,n)+ operating cost for

state(i,n)+transition cost from state (i-1,m) to state(i,n)]

• This has to be repeated for all states in the ith interval.

• The lowest cost paths (number L) are saved in computer memory.

• Check whether the program has reached the last stage in load cycle. If yes

optimal schedule is printed. Otherwise i is uploaded to (i+1) and program is

repeated.

Automatic Generation Control (AGC)

Almost all generating companies have tie line interconnections to neighboring utilities.

Tie lines allow the sharing of generation resources in emergencies and economies of

power production under normal conditions of operation.

For purpose of control the entire interconnected system is subdivided into control

areas which usually confirm to the boundaries of one or more companies. The net


interchange of power over the tie lines of an area is the algebraic difference between

area generation and area load(plus losses).

Frequency changes occur because system loads varies randomly throughout the day

so that an exact forecast of real power demand cannot be assured. The imbalance

between real power generation and load demand (plus losses) throughout the daily

load cycle causes kinetic energy of rotation to be either added to or taken from the

online generating units, and frequency throughout the interconnected system varies

as a result. Each control area has a central facility called Energy control centre, which

monitors the system frequency and the actual power flows on its tie lines to

neighboring areas. The deviation between desired and actual system frequency is then

combined with the deviation from the scheduled net interchange to form a composite

measure called the area control error (ACE).

To remove area control error, the energy control center sends command signals to the

generating units at the power plants within its area to control the generator outputs so

as to restore the net interchange power to scheduled values and assist in restoring the

system frequency to its desired value. The monitoring, telemetering, processing and

control functions are coordinated within the individual area by the computer based

automatic generation control(AGC) system at the energy control centre. The governors

on units of the interconnected system tend to maintain load-generation balance rather

than a specific speed and the supplementary control of the AGC system within the

individual control area functions so as to:

Cause the area to absorb its own load changes,

Provide the prearranged net interchange with neighbors,

Ensure the desired economic dispatch output of each area plant,

Allow the area to do its share to maintain the desired system frequency.

The ACE is continuously recorded within the energy control centre to show how well

the individual area is accomplishing these tasks.

Short Questions

1. What is Automatic load dispatching?’ 2017

Economic load dispatching is the distribution of the load among the generating units

in such a manner so as to minimize the cost of supplying the minute to-minute

requirements of the system.

In a large interconnected system it is humanly impossible to calculate and adjust such

generations and hence the help of digital computer system along with analogue


devices is used. The whole process is carried out automatically; hence this process is

known as automatic load dispatching.

2. Define load curve? 2017

The curve drawn between the variations of load on the power station with reference to time is known as load curve. There are three types of load curve

• Daily load curve, • Monthly load curve, • Yearly load curve.

3. What is Load factor? 2017

The ratio of average load to the maximum demand during a given period is known as load factor.

4. What is AGC? 2017

In an electric power system, automatic generation control (AGC) is a system for

adjusting the power output of multiple generators at different power plants, in

response to changes in the load. Since a power grid requires that generation and load

closely balance moment by moment, frequent adjustments to the output of generators

are necessary. The balance can be judged by measuring the system frequency; if it is

increasing, more power is being generated than used, which causes all the machines

in the system to accelerate. If the system frequency is decreasing, more load is on the

system than the instantaneous generation can provide, which causes all generators to

slow down.

5. What is incremental cost criterion? 2017

The operating cost C of a generating unit is a function of its power output. dC/dP is

the incremental cost. The total load should be so allocated to different generating units

that they operate at equal incremental costs.(Unit: Rs/MWh)

Or

Incremental Cost Criteria of a generating unit is stated as “Allocate Generation so that

total Demand is Satisfied & All Incremental Costs are equal in Economic Dispatch

Problem”

6. Write the equality and inequality constraints considered in economic dispatch

problem’ 2017


7. What are the constraints of economic load dispatch problem?2016

Refer answer of question number 06.

8. What are the spinning reserve constraints in unit commitment problem’ 2017

There may be sudden increase in load, more than what was predicted. Further there

may be a situation that one generating unit may have to be shut down because of fault

in generator or any of its auxiliaries. Some system capacity has to be kept as spinning

reserve

i) to meet an unexpected increase in demand and

ii) to ensure power supply in the event of any generating unit suffering a forced

outage.

OR

Spinning reserve must be maintained so that the loss of one or more units does not

cause unacceptable decline in frequency i.e. there must be sufficient reserve such that

if one unit is lost , other unit can makeup for the loss in a specified time period.

9. What is meant by unit commitment?2016

The problem of finding the order in which the units are to be shut down over a period

of time (say one day), so the total operating cost involved on that day is minimum, is

known as Unit Commitment(UC).

10. What are typical conditions needed to be taken care of while distributing

loads among the plants of a system? 2015


To determine the economic distribution of a load amongst the different units of a

plant,

• The variable operating costs of each unit must be expressed in terms of its

power output.

• The fuel cost is the main cost in a thermal or nuclear unit. Then the fuel cost

must be expressed in terms of the power output.

• Other costs, such as the operation and maintenance costs, can also be

expressed in terms of the power output.

11. What is area control error?

The deviation between desired and actual system frequency is then combined with the

deviation from the scheduled net interchange to form a composite measure called the

area control error (ACE).

12. What is the purpose of economic dispatch?

The purpose of economic dispatch (or) optimal dispatch is to minimize the fuel costs

for the power system.

13. What is meant by unit commitment?

Unit commitment means optimum allocation of generators at each generating station

at various station load levels.

14. Name the methods of finding economic dispatch.

The two methods to find economic dispatch are:

(i) Load scheduling

(ii) Unit commitment

15. What is meant by total generator operating cost?

The total generator operating cost includes the cost of fuel, cost of transmission loss,

labor and maintenance costs.

16. What are the factors affecting the cost of generation? (or) List the various

constraints in the modern power systems.

The cost of generation depends on operating constraints or system constraints. They

are:

Equality constraint.

Inequality constraint

Generator constraints

Voltage constraints


Running space capacity constraints.

Transformers tap settings.

Transmission line constraints.

17. What are the advantages of using participation factor?

The advantages of using participation factor are:

(a) Computer implementation of economic dispatch is straight forward.

(b) Execution time for economic dispatch is short.

(c) It will always give consistent answers when units reach limits.

(d) It gives linear incremental cost functions or has non-convex cost curves.

18. What is the difference between load frequency controller and economic

dispatch controller?

The load frequency controller is a fast acting controller and the economic

dispatch controller is a slow acting control.

LFC adjusts the speed changer setting every minute in accordance with a

command signal generated by the central economic dispatch computer.

19. What is merit order scheduling?

This method ensures that the incremental cost of all the generators is constant over

the full range (or) over successive discrete portions within the range. This method of

scheduling is known as merit order scheduling.

20. What is Lagrangian multiplier?

The necessary condition for the existence of a minimum cost operating condition is

that the incremental cost rates of all the units be equal to some undetermined value(λ)

called Lagrangian multiplier.

21. What are the points to be noted for a economic load dispatch including

transmission losses?

1) The incremental cost of production of a plant is always positive; the incremental

transmission losses can be both positive and negative.

2) The individual generators will operate at different incremental cost of production.

3) The generation with highest positive incremental transmission loss will operate at

lowest incremental cost of production.

22. What are the assumptions for deriving loss coefficients?

1. The ratio X/R for all transmission line is same.

2. The phase angle of all the load currents is the same.


23. How is incremental operating cost related to economic dispatch?

Total system load has to be divided among all units that all the units operate at equal

incremental cost.

24. What are the criteria that should be satisfied for economic loading of

generating station?

Equal incremental cost criteria.

25. What is system incremental cost?

The incremental fuel cost of all the generating units must be the same. The common

value of incremental fuel cost λ is called the system incremental cost.

chapter-04 economic operation of power system

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