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CHAPTER 4

Unit Commitment --- Problem Formulation and Single

Solution Approaches

4.1 Introduction

The Unit Commitment (UC) is an important research challenge and vital optimization

task in the daily operational planning of modern power systems due to its combinatorial nature.

Because the total load of the power system varies throughout the day and reaches a different

peak value from one day to another, the electric utility has to decide in advance which generators

to start up and when to connect them to the network and the sequence in which the operating

units should be shut down and for how long. The computational procedure for making such

decisions is called unit commitment,and a unit when scheduled for connection to the system is

said to be committed.In this work the commitment of fossil-fuel units has been considered which

have different production costs because of their dissimilar efficiencies, designs, and fuel types.

Unit commitment plans for the best set of units to be available to supply the predict forecastload

of the system over a future time period.

In general, the UC problem may be formulated as a non-linear, large scale, mixed-integer

combinatorial optimization problem with both binary (unit status variable) and continuous (unit

output power) variables. This chapter presents the characteristics of power generation unit, unit

commitment problem formulation, modeling aspects of single approaches to solve UCP for

convex and non convex fuel cost function. The remaining discussions in this chapter focus on

algorithm development and their implementation, and case studies.

4.2 Characteristics of Power Generation Units

In analyzing the problems associated with the operation of power system, there are many

possible parameters of interest. Fundamental is the basic cost data and set of input-output

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characteristics of generation units. Different types of fuel are used in thermal power plants.

Depending on the types of turbine such as single value or multi value, the characteristic differs.

Although the operating cost of these units consists of both fuel and maintenance costs,

only the fuel cost varies directly with the units and also with the level of generation. The fuel

cost is incurred during the running (either at no-load or at any load), start-up and sometimes

shutdown conditions of the steam units.

4.2.1 Units Input-Output characteristic (Heat or Cost)

Unit (Boiler, turbine and generator) input-output curve establishes the relationship

between energy input to the driving system and the net energy output from the generator. A

typical boiler-turbine-generator unit is represented in Figure 4.1.

Figure 4.1 Boiler -Turbine -Generators Unit

In this characteristic the gross input (Rs. / h or tons of coal/h or millions of cubic feet of

gas/h or any other unit) being measured in millions of B.T.U. per hour (MBTU/h) is plotted

against the output in MW of the unit. The input is taken along y-axis. The output is normally the

net electrical output of the unit and is taken along x-axis. Z-axis represent the time axis, on

which usually one hour is taken to convert the output power P in MW to energy in MWh in order

to evaluate the per unit cost of input i.e., Plant is loaded at P (MW) for one hour, then input is

measured in Rs. / h or MBTU/h. (Z-axis can be omitted as each point loading pertains one hour).

B T G

A/P

Boiler fuel inputSteam Turbine

Generator

(Gross)

(Net)

Auxiliary Power supply

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The input-output characteristics of a steam unit in idealized form are represented in Figure

4.2.

Figure 4.2 Input-output Curve of a Steam Turbine Generator

For a single value turbine the governing is done by throttling of steam and for such units,

the input-output curve is substantially a straight line within its operating range.

4.2.2 Non convex fuel cost characteristic due to valve point effect

Non convex characteristic results due to valve point effect, multiple fuels and prohibited

operating zones. The valve point effects produce a ripple, which is highly non-smooth and

discontinuous as represented in Figure 4.3.

Figure 4.3 Input-output curve of a multi valve steam turbine generator with four steamadmission valves

A= primary valve B= secondary valve C= Tertiary valve D=Quaternary valve

E= Quinary valve

4.2.3 Incremental heat or cost characteristic

The incremental heat rate characteristic is the derivative of the inputoutput characteristic

(H/P or F/P). This characteristic is widely used in economic dispatching of units. It is

converted to an incremental fuel cost characteristic by multiplying the incremental heat rate. The

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incremental heat rate characteristics for single and multi value units are represented in Figures

4.3 and 4.4 respectively. The incremental heat rate characteristic of multi valve steam turbine is

discontinuous type.

Figure 4.4 Incremental Heat Rate or Cost characteristic.

Figure 4.5 Incremental Heat Rate Characteristics of a steam turbine with four valves.

4.2.4 Unit Heat rate (HR) characteristic

The heat rate curve is obtained from the unit input-output curve by dividing the input by the

corresponding output (H/P) at any loading condition versus the megawatt output of the unit. This

characteristic is plotted between H/P versus P. While incremental heat rate is given by the ratio

of the change in input (H) to the corresponding change in output (P) at any operating point.

Heat rate (HR) =MWinput

MBTU/hininput

Out (4.1)

The units of Hear rate are KWhBtuorMWhorMBtuMW

hRs/,,/

/.

and

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Incremental heat rate = Input = H or FOutput P P (4.2)

This is an important characteristic and defines the average heat rate per KWH of output. The

incremental efficiency, which is the ratio of the change in output to the corresponding change in

input at any loading condition, is clearly the reciprocal of the incremental heat rate. Thus lower is

the incremental heat rate, higher is the incremental efficiency. Since H/P = 1/, therefore this

characteristic if the reciprocal of the usual efficiency characteristic developed for a machine.

Both these quantities have the same unit which is B.T.U. per KWh. The heat rate and

incremental heat rate curves are represented in Figure 4.6.

Figure 4.6 Heat rate and incremental heat rate curves for convex cost function

The heat rate and incremental heat rate can be converted into fuel cost function and incremental

fuel cost by multiplying them with the cost of the fuel (Dollars per million of B.T.U.).

Fi(Pi) = Hi(Pi) x fuel cost $/h (4.3)

Incremental cost = dP

dF

i

T = $/MWh (4.4)

4.3 Unit Commitment Problem (UCP)

UCP was defined as preparing on/off schedule of generating units in order to minimize the

total production cost of utility and constraints such as system power balance, system spinning

reserve, and units minimum up and down times are satisfied. Figure 4.7 represents the

configuration that represent UC problem with on/off switches. The unit commitment problem is

discussed as follows:

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Figure 4.7 Unit Commitment

4.3.1 Objective Function

The principal objective is to prepare on/off schedule of the generating units in every sub-

period (typically 1h) of the given planning period (typically 1 day or 1 week) in order to serve

the load demand and spinning reserve at minimum total production cost (fuel cost, start up cost,

shut down cost), while meeting all unit, and system constraints. The following costs are

considered.

4.3.1.1 Fuel Cost

The quadratic approximation is the most widely used by the researchers, which is

basically a convex shaped function. The curve shown in Fig. 4.1 is the operating fuel cost

equation for unit i and is mathematically represented as:

Fi(Pih) = =

N

i 1

[ ai + biPih+ciPih

2] (4.5)

(Units without valve point effects)

To take the effects of valve points as shown in Fig. 4.6 a sinusoidal function is added to the

convex cost function and represented as:

Fi(Pih) = =

N

i 1

[ ai + biPih+ciPih

2+ | eisin fi( Pih min- Pih)|] (4.6)

(Units with valve point effects)

4.3.1.2 Start up cost

Start up cost is warmth-dependent. Mathematically it is represented as a step function:

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STih= h-costi: TidownXi

off(h) Tidown+ c-s-houri$/h (4.7)

c-costi: Xioff(h) > Ti

down+ c-s-houri $/h (4.8)

4.3.1.3 Shut down cost

The typical value of the shut down cost is zero in the standard systems. This cost is

considered as a fixed cost.

SDih= KPih $/h (4.9)

Where K is the incremental shut-down cost

4.4 Constraints

The UCP is subjected to many constraints that include:

The total power generated must meet the load demand. There must be enough spinning reserve to cover any shortfalls in generation. The loading of each unit must be within its minimum and maximum allowable rating

(limits).

The minimum up and down times of each unit must be observed. Unit availability constraint is either unit is available / not available, out aged/Must out,

Must run, and Fixed Output Power (F.O.P).

Unit initial status +/- either already up or already down.The constraints which are taken into consideration in this work may be classified into two

groups: system constraints and unit constraints.

4.4.1 System constraints or coupling constraints

Constraints that concern all the units of the system are called system or coupling constraints.

These constraints have two categories: the system power balance and system spinning reserve

constraints.

(i) System Power balance or load constraint:

The system power balance constraint is the most important constraint in the UCP. The

generated power from all committed units must be equal to the load demand. This is formulated

in the balance equation as:

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=

N

i 1

Uih(Pih) = Dh h = 1,2,.H (4.10)

(ii) System Spinning reserve requirements

In this work spinning reserve is computed as an amount which is a percentage of the

forecasted load demand and is represented as:

=

N

i 1

Uih(Pimax) Dh + SRh h = 1,2,H (4.11)

4.4.2 Unit constraints or local constraints

Constraints that concern individual units are called unit constraints or local constraints are

described as follow:

(i) Units minimum and maximum generation limits

The generation limits represent the minimum loading limit below which it is not economical

to load the unit, and the maximum loading limit above which the unit should not be loaded. Each

unit has generation range, which is represented as:

UihPimin Pih PimaxUih (4.12)

for i = 1,2,..N, h = 1,2,.,H

(ii) Minimum up and down time limits

Once the unit is running, it should not be turned off immediately. Once the unit is de-committed,there is a minimum time before it can be recommitted. These constraints can be represented as:

Xion(h) Tiup

Xioff

(h) Tidown (4.13)

For i = 1, 2, ., N. h = 1, 2, , H.

(iii) Unit availability constraints.

The availability constraint specifies the unit to be in one of the following different

situations: Available/ not available, out aged/Must out, Must run, Fixed Output (F.O.P).

Must run

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Some units are given a must run status during certain times of the year for reasons of voltage

support on the transmission network or for such purposes as supply of steam for uses outside the

steam plant itself, and to increase the reliability or stability of the system.

Must out units:

Units which are on forced outages and maintenance are unavailable for commitment and these

are the must out units.

Units on fixed generation: (F.O.P)

These are the units which have been prescheduled and have their generation specified for certain

time period. A unit of fixed generation is automatically a must run unit for the designated time

period.

(iv) Unit initial status

The initial status value, if it is positive indicates the number of hours the unit is already up,

and if it is negative indicates the number of hours the unit has been already down. The status of

the unit +/- before the first hour in the schedule is an important factor to determine whether its

new state violates the MUT/MDT constraints. The initial status also affects the start up cost

calculations.

(v) Unit derating constraint

During the life time of a unit its performance could be change due to aging and cause

derating of the unit. Therefore, the unit minimum and maximum limits are changed.

4.5 Unit Commitment mathematical formulation as an optimization problem

The objective function of the unit commitment problem is to minimize the total

production cost and is mathematically formulated as:

Objective function

Minimize TPC = =

H

h 1

=

N

i 1

[ Fi(Pih) + STih+ SDih] $/h (4.14)

Subject to:

The system constraints (4.10, 4.11) and unit constraints (4.12, 4.13).The UCP can be considered as two linked optimization problems: the Unit Scheduling Problem

(Allocation of Generators) and the Economic dispatch problem(Allocation of Generation) and

is represented in Figure 4.8. The unit schedule problem is the on/off or 0/1 combinatorial

optimization problem. A feasible unit schedule must satisfy the forecasted load demand, system

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spinning reserve requirements, and the constraints on the start up and shut down times during

each planning period. The economic dispatch problem is the constrained non-linear optimization

problem. The economic load dispatch is to allocate the generation requirement among the

available units so that the total cost of energy supplied to meet the load demand within

recognized constraints is minimized on minute to minute basis.

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Figure 4.8 Representation of Unit Commitment Problem (UCP)

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4.6 Generation of initial feasible unit commitment schedules

The generation of initial feasible solution is much important, for the Unit Commitment

problem.When initial feasible schedules (generation > load + spinning reserve) are generated

randomly, it is difficult to get feasible schedules for 24 hours loads considering MUT and MDT

constraints It takes a very long time. These randomly generated solutions are also far from the

optimal solution. The convergence is slow and likely to get trapped in the local minimum during

the exploration of the solution space.

4.6.1 Initial unit commitment scheduling by using priority List method and focusing on

peak and off-peak loads of the daily load curve.

In this work initial solution is generated using the priority list method. The priority list

method is very fast and efficient method but the solution obtained cannot fulfill all the

constraints particularly the MUT and MDT constraints. Graphical representation is given in

Figure 4.9. Generally more generators are started up at around the peak load, and few units are

started up at light loads based on full load average production cost. To satisfy minimum up time

constraint the units are set continuously ON.

The full load average production cost (FLAPC) is the Heat Rate (HR) multiplied by the fuel

cost Fi(P

i).

Mathematically it is represented as:

(FLAPC) = HRix Fi(Pi) = Fi(Pimax)/ (Pimax) (4.15)

(MBTU / MWh x $/ MBTU) = ($/MWh)

= ( ai + biPimax+ciPimax

2 ) / (Pimax) (4.16)

Start up of the base units

In power system some units have must run status. These are base units having large output

power. Two units located at the bottom of the priority list as base load units, the units have ON

fixed status.

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Unit10

Unit9

Unit8

Unit7 Unit6

Unit3 Unit4 Unit5 Unit2 Unit1

PriorityList

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Figure 4.9 Generation of initial solution by priority list method a graphical representation

4.6.2 Generation of trial solutions / neighbors

At first step in solving the combinatorial optimization problems is to have good

neighbors/trial solutions from an existing solution. More solutions were obtained taking upper

four units in the priority list at every time interval as shown in Figure 4.10. Introduction of these

feasible solutions makes the search closer to the optimum, leading to a faster convergence and

better results.

Unit10 Unit9 Unit8

Unit7

Unit6 Unit3 Unit4 Unit5 Unit2 Unit1

PriorityList

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

either on/off sate On state Off stateFigure 4.10 Generation of new schedule by taking upper 4 units

4.7 Minimum up and minimum down Time Constraint Handling

During the optimization process of unit commitment schedules the MUT and MDT

constraints may be violated. They will be checked and repaired if violation occurs.

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4.8 Minimum up and down time constraint repairing by using bit change operator

A bit change operator is used to modify the bit positions. This operator overcomes the

problems of the minimum up/down time constraint violation. The operator looks at the past

states, the future states and the present state itself of all units to decide whether or not the units

status for the present hour should be flipped or not. The units with small minimum up and down

times have more changes in their status, while the units with large minimum up and down times

will require less change. A simple way to achieve this is by the categorization of units as base

load, sub-base load, peaking units, must run, and can run. An example of repairing the minimum

up and down time is represented in Figures 4.11 and 4.12 respectively.

Units/Hr

t-1 t t+1i 0 0 1 1 0 0 0

j 0 0 1 1 1 0 0

Figure 4.11 Repairing of minimum up time

Units/Hr

t-1 t t+1

i 1 1 0 0 1 1 1

j 1 1 1 1 1 1 1

Figure 4.12 Repairing of minimum down time

4.9 Algorithm for the construction of initial unit commitment schedule and M.U.T and

M.D.T constraint handling

Step 1: Generate a feasible unit commitment schedule satisfying load demand and spinning

reserve using priority list method. Which is a matrix (H x N).

Step 2: Get the row values of the matrix and calculate the total generation in each scheduling

hour.

Step 3: Check that the generated power is greater than load demand plus spinning reserve?

Step 4: Get the column values of the matrix and calculate units start up and shut down times.

(Xionh, Xi

offh)

Step 5: Xionh Tiup and Xioffh Tidownif NO repair minimum up and down time violations

using bit change operator and modify start up cost else go to step 6.

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Step 6: Get the row values of the matrix and calculate power output of each unit, operating fuel

cost of each unit, total operating fuel cost of each row, start up cost and total production

cost of each row for each hour.

Step 7: The final UC solution is one having the lowest total production cost.

4.10 Unit commitment schedule and determination of number of units to be operated

Consider a system having a forecasted load as shown in Fig. 4.9 Assuming that 10 units

are available to carry the load.

.

Figure 4.13 Forecasted load curve

An initial unit commitment schedule may be as represented in Figure 4.14.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

units Priority10 Unit10 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

9 Unit9 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 08 Unit8 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 07 Unit7 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 06 Unit6 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 05 Unit3 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 04 Unit4 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 03 Unit5 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 02 Unit2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 Unit1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 on state 0 off state

Figure 4.14 Representation of initial unit commitment schedule

Withdrawal of the unit contributes to the saving of its running cost during the reduced load

period of the system, but either start up cost or shut down cost is incurred when the unit is

restored to service. If these costs are less than the spinning cost, the withdrawal is economically

justified. In fact, other schedules can also be considered, but before the shut down of any unit,

the cost of its continued operation should be weighed against the shut down or start up cost. The

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problem is thus based on the evaluation of the total production cost in supplying the forecasted

load with a specified spinning reserve with various schedules of commitment.

4.11 Economic dispatch Problem (EDP, Allocation of Generation)

The economic dispatch problem can be classified as convex and non convex. Most of the

researches used convex economic dispatch the non-convex economic dispatch may result due to

valve point effect.

4.11.1 Economic load dispatch (ELD) calculations

In the unit commitment problem, the economic dispatch calculations consume a large

amount of calculation time. In this work, the ELD calculation is performed only for feasible

solutions by gradient method, merit order method, load assigned by operational engineer and

genetic algorithm.

4.11.1.1 Equal Incremental Cost Criterion

Let there be N thermal generating units connected to a bus-bar supplying a load demand of

Pload. It is required to load the units so that the cost of operating fuel in minimum. This problem

may be solved by using Equal Incremental Cost Criterion. Working philosophy of their

criterion is as: When the incremental costs of all the machines are equal, and then cost of

generation would be minimum subject to equality constraints.

The economic dispatch problem mathematically may be defined as:

Minimize: ( )=

=N

1i

iiT PFF (4.17)

Subject to: equality constraint =

=N

1i

iD PP (4.18)

And inequality constraint( )

( )

0PP

0PP

imin.i

max.ii (4.19)

for i =1,2,,N.

Where

FT= F1+ F2+ FNis the total fuel input to the system

Fi= Fuel input to ithunit

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Pi = the real power generation of ithunit

However, in this analysis method of Lagrange multiplier will be used. The working

philosophy of this method is that constrained problem can be converted into an unconstrained

problem by forming the Lagrange, or augmented function. Optimum is obtained by using

necessary conditions.

+=

+=

=

N

1i

iDT

T

PP.F

.FL

(4.20)

The necessary conditions for constrained local minima of L are the following:

0P

L

i

=

(4.21)

0L=

(4.22)

Condition-I

First condition gives

( ) 010.P

F

P

L

i

T

i

=+

=

or

P

F0

P

F

i

T

i

T =

=

Q N21T FFFF ++=

Then

dP

dF

P

F

i

T

i

T ==

And therefore the condition for optimum dispatchis

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dp

dF

i

T = (4.23)

or

P2cb iii =+ (4.24)

Where

2

iiiiiT PcPbaF ++=

Condition-II

Second condition results in

=

==

N

1i

iD 0PPL

or

=

=N

1i

Di PP (4.25)

Condition for economic operation

For most economical operation, all plants must operate at equal incremental production cost

while satisfying the equality constraint given by equation (4.25).

i

ii

2c

bP

= (4.26)

The relations given by equations (4.26) are known as the co-ordination equations. They are

function of . An analytical solution for is given by substituting the value of Pi in equation

(4.25), i.e.

D

N

1i i

i P2c

b=

=

(4.27)

=

=

+

= N

1i i

N

1i i

iD

2c

1

2c

bP

(4.28)

Optimal schedule of generation is obtained by substituting the value of from equation (4.28)

into equation (4.26).

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Iterative Method of economic dispatch

The equation (4.27) is a function of and can be expressed as: ( ) DPf = (4.29)

Expanding by Taylors series about an operating point (k) and neglecting higher order terms

results in

( )( ) ( ) ( )

D

k

k

kP

ddf

f =

+ (4.30)

( ) ( )( )

( ) ( )k

k

Dk

ddf

fP

=

( )( )

( ) ( )k

kk

d

df

P

=

( )( )

( )k

kk

ddPi

P

= (4.31)

or

( )k( )

=

i

k

2c

1

P (4.32)

and therefore

( ) ( ) ( )kk1k +=+ (4.33)

Where

( ) ( )=

=N

1i

k

iD

k PPP (4.34)

The process is continued until P(k)is less than a pre-specified accuracy.

4.11.1.2 Loading to most efficient load

Although the criterion of equal incremental production costs will result in the optimum

economic scheduling of generation. The above method is still in use by utilities. In this method

units are loaded in ascending order of their heat rates, to their most efficient loads, based on the

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forecasted load at that hour. The most economical unit is loaded first and the remaining load is

shifted on the next unit in the priority list. It the remaining load is less the minimum limit of that

unit than the unit is loaded up to its minimum power output and the remaining loaded is shifted

on the previous unit. This is very quick method and gives near optimal solution in very short

time.

4.11.1.3 Economic Dispatch using Genetic Algorithm (Real Power Search)

GA works better on non convex fuel cost function. The pseudo code for Real Power-

search method is shown in Figure 4.15.

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Figure 4.15 Pseudo Code for Genetic Algorithm Real Power-Search Method

4.12 Economic Dispatch versus Unit Commitment

The unit commitment assumes that there are N units available to meet the forecasted load

demand, satisfying spinning reserve and MUT and MDT constraints. The economic dispatch

load the units economically within their limits satisfying system and unit constraints

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4.13 Conventional/Classical Single Approaches for convex fuel cost function

The single approaches used to solve the UCP are, Complete Enumeration, Merit Order based

on Full Load Average Production Cost, and Merit order based on maximum power output of

each unit.

The algorithm of the classical single approaches is as follows:

Step 1: Read in system data ai, bi, ci, load demand (PD), Pminand Pmaxof each unit.

Step 2: Calculate the full load average production cost (FLAPC) cost of each unit.

Step 3-a: Generate all combinations (2N - 1). Check for the feasible combination

corresponding to the satisfaction of the equality constraint (Generation Loaddemand plus spinning reserve) for each hour of the forecasted load.

b. Generate initial schedule based on Full Load Average Production Cost (FLAPC).

c. Generate initial schedule based on Maximum Power Output of each unit (PMAX).

Step 4: Calculate systems lambda using equation 4.28.

Step 5: Calculate economic loading of machines for each feasible combination (based on

equal incremental cost criteria).

Step 6: Check that the Power output of each unit is within minimum and maximum

generating limits of machines.

Step 7: Is there any violation of minimum or maximum power limit (yes-clamp at

minimum or maximum limit), (No - go to step 9).

Step 8: Recalculate the system lambda and output power of each machine using equations

4.32, 4.33 and 4.34.

Step 9: Satisfying the power balance equation (Generation = Load).

Step 10: Print the unit commitment schedule, power output of each machine, operating fuel

cost of each machine, and total production cost and system lambda for each

feasible schedule.

Step 11: Print the best schedule.

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4.13.1 Single Approach I --- Complete Enumeration

This method takes all the combinations (2N 1) H and than calculated the economic

dispatch of each unit. Where N = number of units, H = number of hours (24). For a total period

of H interval, the maximum number of possible combinations is (2N

-1)H

.

For example, take a 24-h period (e.g. 24 one hour intervals) and consider with 5, 10, 20

and 40 units. The value of (2N

-1)H

becomes the following.

N (2N-1)H

5 6.25 x (10)35

10 1.73 x (10)72

20 3.12 x (10)144

40 Too Big

There very large numbers are the upper bounds for the numbers of enumerations

required. The constraints on the units and the loading capacity of the units limit the search space.

Never the less, the real practical barrier in the UCP is the high dimensionality of the possible

search space.

4.13.2 Single Approach II --- Conventional Priority ListThis approach is based on conventional priority list method. A priority order is created

based on the Full Load Average Production Cost (FLAPC). The UC schedule is based on

FLAPC and ED is based on Lambda Iteration Method.

In this method, units are committed to service by observing their heat rate values. Units with

the lowest heat rate are put into operation first. For shutting down the reserve order is followed,

i.e. the units with the highest heat rate is withdrawn first. The load dispatcher takes into account

the hourly forecasted load and spinning reserve requirement, and then schedules the units to

match the load and spinning reserve. The industry, however, still mostly used the simple "merit

order" method.

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4.13.3 Single Approach III --- Proposed Single Approach

This approach is also based on priority list method. But the priority list in this case is

based on the Maximum Power (PMAX) limit of each unit. The UC schedule is based on PMAX

and ED is based on Lambda Iteration Method.

4.14 Case studies --- Convex cost function

Following two standard test systems have been selected for the validation of the single

approaches.

a. Test System I --- 3 units with 24 hours load.

b. Test System III --- 10 units system with 24 hours load

Input Data: The description and input data of test systems used for investigation in the case

studies are given in Tables A.1 and A.3 placed in Appendix-A.

Computer Implementation: All the three single approaches have been implemented in C++ on

P-IV Personal Computer.

Output Results: The summary and comparison of results is given in this chapter.

4.14.1 Numerical Results of test system I

The priority order based on conventional and proposed method is given in Table 4.1.

Table 4.1 Priority order based on single approach-II and III: test system-I

Unit

No.

PMAX FLAPC

(S/MWH)

Single approach-II

Conventional Priority

order

(FLAPC)

Single approach-III

Proposed Priority order

(PMAX)

1 600 9.7922 2 1

2 400 9.4010 1 2

3 200 11.1888 3 3

The output results of test system I are shown in the following tables:

Table 4.2 presents the comparison of operating fuel cost ($),

Table 4.3 presents the comparison of Summary of Unit Commitment Schedules for single

approaches,

Table 4.4 presents the Number of Units in Operation for single approaches,

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Table 4.5 presents the Comparison of single approaches with Genetic Algorithm, and Hopfield

Neural Network methods.

Tables 4.6 to 4.8 presents the best solutions obtained by the single approach-I, II and III.

The Salient features of the conventional approaches in the light of the observations from the

results are as follows:

1. Enumeration method gives good results. But the number of transitions is 10. In test

system III the number of transitions is 4.

2. The operating fuel cost obtained by all the three single approaches remains low compared

to genetic algorithm and Hopfield neural network methods.

3. Single approach-I gives $606.28 saving per day compared with GA and $ 691.28 saving

per day compared with HFNN.

4. Single approach-II gives $381.15 saving per day compared with GA and $ 466.15 saving

per day compared with HFNN.

5. Single approach-III gives $389.28 saving per day compared with GA and $ 474.28 saving

per day compared with HFNN.

All the three approaches are simple, fast and give fair amount of reduction in operating cost as

compared to GA and HFNN.

Table 4.2 Comparison of the operating fuel cost ($) for 3 Unit Systems :( Test System I)

200800201000201200201400201600201800202000202200202400202600

Enumeration

Conventional

prioritylist

G.A.

H.F.N.N

Proposed

approach

Series1

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Table 4.3 Summary of Unit Commitment Schedules for 3 Unit Systems (Test System I)

Single approach-

ISingle approach-

IIProposed Single

approach-III

Load(MW)

CompleteEnumeration

Merit Order (FLAPC)with ED

UC(2,1,3)

Merit Order (PMAX)with ED

UC(1,2,3)12001200115011001000900800600550500500500

50050060080085090095010001050110012001200

111111111111110110110100100011011011

011011100110110110110110111111111111

111111111111110110110110110110110110

110110110110110110110110111111111111

111111111111110110110100100100100100

100100100110110110110110111111111111

201415.789 202393.346 201632.089

No. of Transitions No. of Transitions No. of Transitions

10 2 4

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Table 4.4 Number of Units in Operation for 3 Unit Systems (Test System I)

Single approach-I Single approach-II Proposed Singleapproach-III

Complete

Enumeration

Merit Order (FLAPC)

with EDUC(2,1,3)

Merit Order (PMAX)

with EDUC(1,2,3)

Hr.Load(MW)

No. of units No. of units No. of units

1 1200 3 3 3

2 1200 3 3 3

3 1150 3 3 3

4 1100 3 3 3

5 1000 2 2 2

6 900 2 2 2

7 800 2 2 2

8 600 1 2 19 550 1 2 1

10 500 2 2 1

11 500 2 2 1

12 500 2 2 1

13 500 2 2 1

14 500 2 2 1

15 600 1 2 1

16 800 2 2 2

17 850 2 2 2

18 900 2 2 2

19 950 2 2 220 1000 2 2 2

21 1050 3 3 3

22 1100 3 3 3

23 1200 3 3 3

24 1200 3 3 3

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Table 4.5 Comparison of the Three Single approaches with Genetic Algorithm and HopfieldNeural Network Methods for 3 unit systems: Test system-I

Algorithm Daily

Operating

Fuel Cost($)

Amount

of Daily

Saving(compared

with GA)

% saving in fuel

cost compared

with GA

Amount of

Daily

Saving(compared

with

Hopfield

Neural

Network

% saving

in fuel

costcompared

with Hopfield

Neural

Network(HFNN)

GeneticAlgorithm[109]

202021.360 - - - -

HopfieldNeural

Network

Method[110]

202106.360 - - - -

Single

Approach-I

(Enumeration)

201415.089 606.28 0.300 691.28 0.342

Single

Approach-II

(Conventional

Priority List)

202393.346 -371.98 -0.184 -0.286.98 -0.142

Proposed

Single

Approach-III

201632.089 389.28 0.1926 474.28 0.2346

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Table 4.6 Unit Commitment Schedule and Power Sharing (MW) of the best solutionobtained from Single Approach-I (Enumeration): Test system-I

Power Output of each

unit(MW)

Fuel Cost of each unit($/h)Hour UC

Schedule

P-1 P-2 P-3

Load

(MW)

F-1 F-2 F-3

Operating

Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060

4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560

5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720

6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900

7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464

8 100 600.00 0.00 0 600 5875.320 0.000 0 5875.320

9 100 550.00 0.00 0 550 5389.505 0.000 0 5389.505

10 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.24011 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240

12 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240

13 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240

14 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240

15 100 600.00 0 0 600 5875.320 0.000 0 5875.320

16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464

17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290

18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900

19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905

20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720

21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.98022 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560

23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 201415.789

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Table 4.7 Unit Commitment Schedule and Power Sharing (MW) of the solution obtainedfrom the Single Approach-II (Conventional Priority List): Test system-I

Power Output of each

unit(MW)

Fuel Cost of each unit($/h)Hour UC

Schedule

P-1 P-2 P-3

Load

(MW)

F-1 F-2 F-3

Operating

Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060

4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560

5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720

6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900

7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464

8 110 322.39 277.61 0 600 3276.676 2638.749 0 5915.425

9 110 294.69 255.31 0 550 3030.592 2440.639 0 5471.231

10 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.36411 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364

12 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364

13 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364

14 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364

15 110 322.39 277.61 0 600 3276.676 2638.749 0 5915.425

16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464

17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290

18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900

19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905

20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720

21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.98022 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560

23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 202393.346

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Table 4.8 Unit Commitment Schedule and Power Sharing (MW) of the solution obtainedfrom the Proposed Single approach-III (PMAX): Test system-I

Power Output of each

unit(MW)

Fuel Cost of each unit($/h)Hour UC

Schedule

P-1 P-2 P-3

Load

(MW)

F-1 F-2 F-3

Operating

Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060

4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560

5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720

6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900

7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464

8 100 600.00 0 0 600 5875.320 0 0 5875.320

9 100 550.00 0 0 550 5389.505 0 0 5389.505

10 100 500.00 0 0 500 4911.500 0 0 4911.50011 100 500.00 0 0 500 4911.500 0 0 4911.500

12 100 500.00 0 0 500 4911.500 0 0 4911.500

13 100 500.00 0 0 500 4911.500 0 0 4911.500

14 100 500.00 0 0 500 4911.500 0 0 4911.500

15 100 600.00 0 0 600 5875.320 0 0 5875.320

16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464

17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290

18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900

19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905

20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720

21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.98022 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560

23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 201632.089

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4.14.2 Numerical results: Test system-III --- (10 unit system)

The case studies are conducted based on the following considerations:

UC schedules without considering spinning reserve and without considering MUT andMDT constraints.

Without s. r. but with considering MUT and MDT constraint 10% s. r. without MUT and MDT constraint With 10 % s. r. without ED considering MUT and MDT constraint With 10 % s. r. with ED considering MUT and MDT constraint without transition cost. With 10% spinning reserve, minimum up/down time constraints, transition cost and

Economic Dispatch.

The priority order for ten unit system is given in Table 4.9.

Table 4.9 Priority order based on single approach-II and proposed single approach III:Test system-III --- ten unit system

Unit

No.

PMAX FLAPC

(S/MWH)

Single approach-II

Conventional

Priority order

(FLAPC)

Proposed Single

approach-III

(PMAX)

Proposed hybrid

approach

(PMAX-FLAPC)

1 455 18.6062 1 1 1

2 455 19.5329 2 2 2

3 130 22.2446 4 5 5

4 130 22.0051 3 3 4

5 162 23.1225 5 4 3

6 80 27.4546 6 7 7

7 85 33.4542 7 6 6

8 55 38.1472 8 8 8

9 55 39.4830 9 9 9

10 55 40.0670 10 10 10

The output results of test systemIII are shown in the following tables:

Table 4.10 presents the comparison of the results for single approaches based upon above

considerations.

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Table 4.11 presents the comparison of operating fuel cost ($) considering 10% spinning reserve

and minimum up/down time constraints with transition cost.

Table 4.12 presents the comparison of the results with Genetic Algorithm.

Table 4.13 presents the summary of Unit Commitment schedules for Single approaches-I, II and

III.

Tables 4.14-4.19 presents the Unit Commitment Schedules and Power Sharing (MW) , operating

fuel cost, start up cost and total production cost of the best solutions obtained from Single

approaches-I, II, and III, with MUT and MDT constraints with 10 % spinning reserve.

The Salient features of the conventional single approach-III in the light of the observations

from the results are as follows:

1. Proposed Single approach III give better results than GA. The proposed approach givescost saving of $949.39 per day equivalent to 0.167% compared with GA.

2. Proposed single approach III give better results the conventional priority list. Theproposed approach gives a cost saving of $1247.37 per day equivalent to 0.220%

compared with conventional priority list.

Table 4.10 Comparison of the operating fuel cost ($) for 10 unit systems considering 10%spinning reserve and minimum up/down time constraints without transition cost: Test System III

560200560400560600560800561000561200561400561600561800

Enumeration

Conventional

prioritylist

Proposed

approach

Series1

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Table 4.11 Comparison of the operating fuel cost ($) for 10 unit systems considering 10%spinning reserve and minimum up/down time constraints with transition cost:

Test System III

Single

Approaches

Single

approach - I(Enumeration)With ED, 10 % s. r

MUT, MDT

Single

approach - II(ConventionalPriority List FLAPC)With ED, 10 % s. r

MUT, MDT

Proposed Single

approach - III(Priority List PMAX)With ED, 10 % s. r.

MUT, MDT

UCSchedule

UC12543768910 UC12435678910 UC12534768910

Operatingfuel cost ($)

560744.47 561682.98 560775.6

Transitioncost ($)

4090 4440 4100

TotalProduction

Cost ($)

564834.47 566122.98 564875.61

Table 4.12 Comparison of the results of the proposed single approach-III with GeneticAlgorithm and conventional priority list

Approach/Model TotalProduction

Cost ($)

Amount ofDaily

Savingcompared

with GA($)

% CostSaving

comparedwith GA

Amountof dailysaving

compared

with CPL($)

% CostSaving

comparedwith CPL

Genetic Algorithm 565825.00 - - - -

Single Approach-I(Complete Enumeration)

564834.47 990.53 0.175 - -

Single Approach-II(Conventional Priority list,CPL)

566122.98 -297.98 -0.052 - -

Proposed Single

Approach-III

564875.61 949.39 0.167 1247.37 0.220

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Table 4.13 Summary of Unit Commitment schedules for 10 unit systems:Single approach-I, II and III: Test System III

Single

approach-I

Single

approach-IIProposed Single

approach-III

Hour Load(MW) Enumerationwith 10 % s.rwith ED, MUT

and MDT

constraint

12543768910

(FLAPC)with 10% s.r.

with ED, MUT

and MDT

constraint

12435678910

(PMAX) with10% s.r. with

ED, MUT and

MDT constraint

12534768910

UC Schedule UC Schedule UC Schedule

1234

56789101112131415

161718192021222324

700750850950

10001100115012001300140014501500140013001200

1050100011001200140013001100900800

1100000000110000000011001000001100100000

11011000001111100000111110000011111000001111111000111111110011111111101111111111111111110011111110001111100000

111110000011111000001111100000111110000011111111001111111000111111100011001000001100000000

1100000000110000000011010000001111000000

11110000001111100000111110000011111000001111111000111111110011111111101111111111111111110011111110001111100000

111110000011111000001111100000111110000011111111001111111000111111100011010000001100000000

1100000000110000000011001000001100100000

11101000001111100000111110000011111000001111111000111111110011111111101111111111111111110011111110001111100000

111110000011111000001111100000111110000011111111001111111000111111100011001000001100000000

Operatingfuel cost

($)

564834.47 566122.98 564875.61

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Table 4.14 Unit Commitment Schedule and Power Sharing (MW) of the best solutionobtained from Single approach-I (Enumeration) considering MUT and MDT constraints with 10

% spinning reserve: Test system-III

Power output of each unit (MW)Hour UC schedule

12543768910

Load

(MW)

1 2 3 4 5 6 7 8 9 10

1 1100000000 700 455 245 0 0 0 0 0 0 0 0

2 1100000000 750 455 295 0 0 0 0 0 0 0 0

3 1100100000 850 455 370 0 0 25 0 0 0 0 0

4 1100100000 950 455 455 0 0 40 0 0 0 0 0

5 1101100000 1000 455 390 0 130 25 0 0 0 0 0

6 1111100000 1100 455 360 130 130 25 0 0 0 0 0

7 1111100000 1150 455 410 130 130 25 0 0 0 0 0

8 1111100000 1200 455 455 130 130 30 0 0 0 0 0

9 1111111000 1300 455 455 130 130 85 20 25 0 0 0

10 1111111100 1400 455 455 130 130 162 33 25 10 0 0

11 1111111110 1450 455 455 130 130 162 73 25 10 10 0

12 1111111111 1500 455 455 130 130 162 80 25 43 10 10

13 1111111100 1400 455 455 130 130 162 33 25 10 0 0

14 1111111000 1300 455 455 130 130 85 20 25 0 0 0

15 1111100000 1200 455 455 130 130 30 0 0 0 0 0

16 1111100000 1050 455 310 130 130 25 0 0 0 0 0

17 1111100000 1000 455 260 130 130 25 0 0 0 0 0

18 1111100000 1100 455 360 130 130 25 0 0 0 0 0

19 1111100000 1200 455 455 130 130 30 0 0 0 0 0

20 1111111100 1400 455 455 130 130 162 33 25 10 0 0

21 1111111000 1300 455 455 130 130 85 20 25 0 0 022 1111111000 1100 455 315 130 130 25 20 25 0 0 0

23 1100100000 900 455 420 0 0 25 0 0 0 0 0

24 1100000000 800 455 345 0 0 0 0 0 0 0 0

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Table 4.15 Unit Commitment Schedule, fuel cost, start up cost and total production cost of thebest solution obtained from Single approach-I (Enumeration) considering MUT and MDT

constraints with 10 % spinning reserve: Test system-III

Hour UC schedule

12543768910

Load

(MW)

Fuel cost

($/h)

Startup

Cost

($/h)

Total

Production

Cost ($)

1 1100000000 700 13683.13 0 13683.13

2 1100000000 750 14554.50 0 14554.50

3 1100100000 850 16809.45 900 17709.45

4 1100100000 950 18597.67 0 18597.67

5 1101100000 1000 20020.02 560 20580.02

6 1111100000 1100 22387.04 1100 23487.04

7 1111100000 1150 23261.98 0 23261.98

8 1111100000 1200 24150.34 0 24150.34

9 1111111000 1300 27251.06 860 28111.06

10 1111111100 1400 30057.55 60 30117.5511 1111111110 1450 31916.06 60 31976.06

12 1111111111 1500 33890.16 60 33950.16

13 1111111100 1400 30057.55 0 30057.55

14 1111111000 1300 27251.06 0 27251.06

15 1111100000 1200 24150.34 0 24150.34

16 1111100000 1050 21513.66 0 21513.66

17 1111100000 1000 20641.82 0 20641.82

18 1111100000 1100 22387.04 0 22387.04

19 1111100000 1200 24150.34 0 24150.34

20 1111111100 1400 30057.55 490 30547.5521 1111111000 1300 27251.06 0 27251.06

22 1111111000 1100 23592.97 0 23592.97

23 1100100000 900 17684.69 0 17684.69

24 1100000000 800 15427.42 0 15427.42

SUM 21700 560744.47 4090.00 564834.47

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Table 4.16 Unit Commitment Schedule and Power Sharing (MW) of the best solutionobtained from Single approach-II (FLAPC) considering MUT and MDT constraints with 10 %

spinning reserve: Test system-III

Power output of each unit (MW)Hour UC schedule

12435678910

Load

(MW)

1 2 3 4 5 6 7 8 9 10

1 1100000000 700 455 245 0 0 0 0 0 0 0 0

2 1100000000 750 455 295 0 0 0 0 0 0 0 0

3 1101000000 850 455 265 0 130 0 0 0 0 0 0

4 1111000000 950 455 235 130 130 0 0 0 0 0 0

5 1111000000 1000 455 285 130 130 0 0 0 0 0 0

6 1111100000 1100 455 360 130 130 25 0 0 0 0 0

7 1111100000 1150 455 410 130 130 25 0 0 0 0 0

8 1111100000 1200 455 455 130 130 30 0 0 0 0 0

9 1111111000 1300 455 455 130 130 85 20 25 0 0 0

10 1111111100 1400 455 455 130 130 162 33 25 10 0 0

11 1111111110 1450 455 455 130 130 162 73 25 10 10 0

12 1111111111 1500 455 455 130 130 162 80 25 43 10 10

13 1111111100 1400 455 455 130 130 162 33 25 10 0 0

14 1111111000 1300 455 455 130 130 85 20 25 0 0 0

15 1111100000 1200 455 455 130 130 30 0 0 0 0 0

16 1111100000 1050 455 310 130 130 25 0 0 0 0 0

17 1111100000 1000 455 260 130 130 25 0 0 0 0 0

18 1111100000 1100 455 360 130 130 25 0 0 0 0 0

19 1111100000 1200 455 455 130 130 30 0 0 0 0 0

20 1111111100 1400 455 455 130 130 162 33 25 10 0 0

21 1111111000 1300 455 455 130 130 85 20 25 0 0 022 1111111000 1100 455 315 130 130 25 20 25 0 0 0

23 1101000000 900 455 315 0 130 0 0 0 0 0 0

24 1100000000 800 455 345 0 0 0 0 0 0 0 0

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Table 4.17 Unit Commitment Schedule, fuel cost, startup cost and total production cost of thebest solution obtained from Single approach-II (FLAPC) considering MUT and MDT constraints

with 10 % spinning reserve: Test system-III

Hour UC schedule Load

(MW)

Fuel cost

($/h)

Startup

Cost

($/h)

Total

Production

Cost ($)

1 1100000000 700 13683.13 0.00 13683.13

2 1100000000 750 14554.50 0.00 14554.50

3 1101000000 850 16892.15 560.00 17452.15

4 1111000000 950 19261.50 550.00 19811.50

5 1111000000 1000 20132.56 0.00 20132.56

6 1111100000 1100 22387.04 1800.00 24187.04

7 1111100000 1150 23261.98 0.00 23261.98

8 1111100000 1200 24150.34 0.00 24150.34

9 1111111000 1300 27251.06 860.00 28111.06

10 1111111100 1400 30057.55 60.00 30117.5511 1111111110 1450 31916.06 60.00 31976.06

12 1111111111 1500 33890.16 60.00 33950.16

13 1111111100 1400 30057.55 0.00 30057.55

14 1111111000 1300 27251.06 0.00 27251.06

15 1111100000 1200 24150.34 0.00 24150.34

16 1111100000 1050 21513.66 0.00 21513.66

17 1111100000 1000 20641.82 0.00 20641.82

18 1111100000 1100 22387.04 0.00 22387.04

19 1111100000 1200 24150.34 0.00 24150.34

20 1111111100 1400 30057.55 490.00 30547.5521 1111111000 1300 27251.06 0.00 27251.06

22 1111111000 1100 23592.97 0.00 23592.97

23 1101000000 900 17764.14 0.00 17764.14

24 1100000000 800 15427.42 0.00 15427.42

SUM 561682.99 4440.00 566122.99

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Table 4.18 Unit Commitment Schedule and Power Sharing (MW) of the best solutionobtained from Proposed Single approach-III (PMAX) considering MUT and MDT constraints

with 10 % spinning reserve: Test system-III

Power output of each unit (MW)Hour UC schedule

12534768910

Load

(MW)

1 2 3 4 5 6 7 8 9 10

1 1100000000 700 455 245 0 0 0 0 0 0 0 0

2 1100000000 750 455 295 0 0 0 0 0 0 0 0

3 1100100000 850 455 370 0 0 25 0 0 0 0 0

4 1100100000 950 455 455 0 0 40 0 0 0 0 0

5 1110100000 1000 455 390 130 0 25 0 0 0 0 0

6 1111100000 1100 455 360 130 130 25 0 0 0 0 0

7 1111100000 1150 455 410 130 130 25 0 0 0 0 0

8 1111100000 1200 455 455 130 130 30 0 0 0 0 0

9 1111111000 1300 455 455 130 130 85 20 25 0 0 0

10 1111111100 1400 455 455 130 130 162 33 25 10 0 0

11 1111111110 1450 455 455 130 130 162 73 25 10 10 0

12 1111111111 1500 455 455 130 130 162 80 25 43 10 10

13 1111111100 1400 455 455 130 130 162 33 25 10 0 0

14 1111111000 1300 455 455 130 130 85 20 25 0 0 0

15 1111100000 1200 455 455 130 130 30 0 0 0 0 0

16 1111100000 1050 455 310 130 130 25 0 0 0 0 0

17 1111100000 1000 455 260 130 130 25 0 0 0 0 0

18 1111100000 1100 455 360 130 130 25 0 0 0 0 0

19 1111100000 1200 455 455 130 130 30 0 0 0 0 0

20 1111111100 1400 455 455 130 130 162 33 25 10 0 0

21 1111111000 1300 455 455 130 130 85 20 25 0 0 022 1111111000 1100 455 315 130 130 25 20 25 0 0 0

23 1100100000 900 455 420 0 0 25 0 0 0 0 0

24 1100000000 800 455 345 0 0 0 0 0 0 0 0

8/13/2019 910S-4 (1)

41/41

Table 4.19 Unit Commitment Schedule fuel cost, startup cost and total production cost of thebest solution obtained from Single approach-III (PMAX) considering MUT and MDT constraints

with 10 % spinning reserve: Test system-III

Hour UC schedule Load

(MW)

Fuel cost

($/h)

Startup

Cost

($/h)

Total

Production

Cost ($)

1 1100000000 700 13683.13 0 13683.13

2 1100000000 750 14554.50 0 14554.50

3 1100100000 850 16809.45 900 17709.45

4 1100100000 950 18597.67 0 18597.67

5 1110100000 1000 20051.16 550 20601.16

6 1111100000 1100 22387.04 1120 23507.04

7 1111100000 1150 23261.98 0 23261.98

8 1111100000 1200 24150.34 0 24150.34

9 1111111000 1300 27251.06 860 28111.06

10 1111111100 1400 30057.55 60 30117.5511 1111111110 1450 31916.06 60 31976.06

12 1111111111 1500 33890.16 60 33950.16

13 1111111100 1400 30057.55 0 30057.55

14 1111111000 1300 27251.06 0 27251.06

15 1111100000 1200 24150.34 0 24150.34

16 1111100000 1050 21513.66 0 21513.66

17 1111100000 1000 20641.82 0 20641.82

18 1111100000 1100 22387.04 0 22387.04

19 1111100000 1200 24150.34 0 24150.34

20 1111111100 1400 30057.55 490 30547.5521 1111111000 1300 27251.06 0 27251.06

22 1111111000 1100 23592.97 0 23592.97

23 1100100000 900 17684.69 0 17684.69

24 1100000000 800 15427.42 0 15427.42

SUM 21700 560775.61 4100 564875.61