77

CHAPTER 4

HYDROTHERMAL COORDINATION OF UNITS

CONSIDERING PROHIBITED OPERATING ZONES – A

HYBRID PSO(C)-SA-EP-TPSO APPROACH

4.1 INTRODUCTION

HTC constitutes the complete formulation of the hydrothermal electric

power generation in power system operation. Earlier in chapters 2 and 3 it

was assumed that part of power demand was already met using the available

hydro generation and only the remaining load demand has to be taken over

and supplied by thermal generators. Thus HTC paves a way to proper

distribution of the required power demand between the available hydro and

thermal generators in a given system. It is a common practice that the

available hydro power should be completely used to meet the maximum

power demand (ie. Base load) and the thermal generators are used to supply

the remaining power demand if needed. Thus minimizing the cost of thermal

generation by maximally utilizing the hydro generation is the core feature of

the HTC.

The Hydrothermal Coordination problem is a sub-problem of UCP and

DEDP solves both thermal unit commitment and hydro schedules. It is therefore,

a more complex problem formulation which considers the prohibited operating

zones as constraints in addition to other hydro and equivalent thermal units.

Since the source for hydropower is the natural water resources, the objective of

hydrothermal scheduling is to minimize the operation cost of thermal units in

78

given period of time while preserving all constraints.

The hydrothermal scheduling problem is basically formulated as a

nonlinear and nonconvex optimization problem involving non-linear objective

function and a mixture of linear and non-linear constraints. Since, the

conventional methods are not suitable to solve this HTC formulation

considering prohibited operating zones, it is found that the proposed hybrid

technique is most adaptive to solve this HTC.

The main objective of this chapter is

To develop a heuristic based solution technique for solving the

HTC.

To develop an algorithm to find the optimal scheduling of hydro

units considering hydro constraints.

To formulate HTC to include all the constraints that were

considered in the UCP and DEDP formulate to validate the

feasibility of the solution procedure.

The formulations of the hydrothermal coordination problem and

applicability of the proposed solution procedures are validated through

several numerical simulations and are illustrated in detail.

4.2 HYDRO THERMAL COORDINATION PROBLEM

FORMULATION

The hydro generating units do not incur the fuel cost when

compared to thermal units. The hydrothermal scheduling problem is aimed at

minimizing the total thermal fuel cost while making use of the maximum

availability of hydro resource as much as possible. The hydro and thermal

constraints include generation load power balance, real power generation

79

limit, generating unit ramp rate limits and prohibited operating zones. In

addition to that the operating capacity limits of the hydro units, water

discharge rate, upper and lower bounds of the reservoir volume, water

spillage and hydraulic continuity restrictions are fully taken into account.

The objective function and the associated constraints of the

problem are formulated as follows:

Objective function: The total fuel cost for running the thermal system to

meet the load demand in a schedule horizon is given by

Min TF = j

z

jj PTf

1 (4.1)

where jj PTf is the fuel cost of the equivalent generator and jPT is the

thermal output at time period ‘j’.

Equality constraints: The equality constraints are the power balance, total

water discharge and the reservoir volume constraints.

a) The power balance constraints are described as,

jPD jjj PLPHPT for j = 1, 2, 3…………z (4.2)

The electric loss between the hydro plant and load jPL is given by

2jj PHkPL (4.3)

In the present work constant head operation is assumed and the

water discharge rate, jq , is assumed to be a function of the hydro plant

generation, ,jPH is given as

80

jj PHgq (4.4)

b) The total water discharge constraint is given by

z

jjjT qnq

1 (4.5)

where jq is the water discharge rate in the thj interval.

c) In the case of a storage reservoir with an given initial and final

volume, the reservoir volume constraints is given as

jjjjjj sPHgrnVV 1 for j = 1,2,3,………….z (4.6)

where jV is the volume of water in the reservoir. jn is the span of the

interval, jr and js are the water inflow rate and the spillage rate respectively

in the interval j .

Inequality constraints:

d) The operational range of the thermal plants is bounded by their

capability limits:

maxmin PTPTPT j (4.7)

Where minPT and maxPT are the minimum and maximum operating limits of the

equivalent thermal generator.

e) The operational range of the hydro plants is bounded by their

capability limits:

maxmin PHPHPH j (4.8)

81

Where minPH and maxPH are the minimum and maximum operating limits of

the hydro generation in interval j in the schedule horizon.

f) Reservoir storage limit:

maxmin jjj VVV (4.9)

Where minjV and maxjV are the volume limits in interval j .

g) Hydraulic continuity equation

jjjjj rsqXX 1 (4.10)

h) Water discharge rate limits:

min j maxq q q (4.11)

i) Reservoir initial and final volume:

inijj VV ,0

finjj VVt , (4.12)

j) Generation limits considering Prohibited operating zones:

The literature has shown the input-output characteristics of thermal

unit considering POZ as inequality constraints. The feasible operating zones

of unit i can be described as follows:

max

,

,1,

1,min

iiU

i

Lkii

Uki

Liii

PPP

PPP

PPP

ni

ink ,...3,2 (4.13)

where:

i = unit index min

iP = unit minimum generation limit

82

Thermal plant

X

Water Inflow

Water discharge

Hydro plant

Pd

PGT PGH

maxiP = unit maximum generation limit

in = number of prohibited zones for unit i

k = index of prohibited zones of a unit

ULkiP ,

, = lower/upper bounds of the thk prohibited zones for unit i

Adjusting the generating units output Pi must avoid unit operation

in the prohibited zones. The dynamic economic dispatch of power generation

of committed units is to be done for the given load demand by satisfying the

above constraints.

4.2.1 Traditional method of hydro and thermal power

The following formulation is used for solving the hydrothermal

coordination problem through PSO technique to be discussed in the next

section. Arbitrarily select a time interval‘d’ in the schedule horizon. Let the

unknown thermal generation dPT in the dth time interval be the dependent

generation as discussed in the previous chapter.

Figure 4.1 Hydrothermal coordination systems

83

The dPT can be calculated by assuming that the thermal

generations of the independent intervals, i.e., jPT for j = 1, 2, …………..d-

1, d+1,……..Z are known. In order to obtain the dPT , the hydro generation in

the dependent interval dPH is required to be calculated.

Solution of dPH and dPT

Consider the case that there are j intervals in the schedule horizon.

Substituting the equations (4.3) and (4.2) as given in Hota et al., we get the

expression as

02 jjjj PTPDPHPHk (4.14)

From the equation (4.4) and (4.5), the water discharge rate in any

interval then becomes a function g jj PTPD and the total water

discharge is expressed as

z

jjjjT PTPDgnq

1 (4.15)

and the equation (4.15) can be rewritten as

z

jjjT PHgnq

1 (4.16)

Given the quantity of total water discharge Tq and load pattern,

from equation (4.15) the water discharge rate in the dependent interval is

obtained as

d

z

djjjjTj nPHgnqPHg

,1

(4.17)

Where nd is the dependent time interval.

84

After obtaining the hydro discharges, hydro generation can be

calculated from Eq (4.4) by simple algebraic calculation, as the discharge in

the present case is a function of hydro generation. After calculating the hydro

generations, by using Eq (4.14) and the given load demand, the thermal

generation in the dependent interval can be calculated as

dddd PHPHkPDPT 2 (4.18)

The hydrogeneration in the non–dependent intervals are then

obtained by solving the following expressions.

02 jjjj PTPHPDPHk

for j = 1, 2, ……,(d-1), (d+1),…………..Z (4.19)

The volume of water in the reservoir at the end of each interval is

then calculated using an Equation (4.6). All generation level and water

volumes must be checked against their limiting values according to the

Equations (4.7) (4.8) and (4.9). In determining optimal solution for the

hydrothermal coordination according to the above mentioned problem solving

formulation, the main objective is to determine thermal generation in the non-

dependent interval. In this research particle swarm optimization based

algorithm has been applied to determine the non-dependent thermal

generations and hence, a better optimal hydrothermal coordination schedule is

obtained.

4.3 PROPOSED SOLUTION METHODOLOGY

The minimization of the cost of operation in a hydrothermal power

system involves a proper thermal commitment as well as an appropriate

allocation of hydro generation in different time period. Hence, the solution

85

process consists of two sub processes. The first sub-process provides the

optimal dispatch of hydro units. The second sub-process optimizes the

thermal commitment provided that the outputs of hydro units are known.

4.3.1 Overview of PSO based hydrothermal scheduling

PSO is one of the modern heuristics algorithms. In 1995, Kennedy

and Eberhart first introduced the PSO technique, motivated by social behavior

of organism such as fish schooling and bird flocking. So, as a heuristic

optimization tool, it provides a population-based search procedure in which

individuals called particles change their position (states) with time. In a PSO

system, particles fly around in a multidimensional search space. During flight,

each particle adjusts its position according to its own experience and the

neighboring experiences of particles, making use of the best position

encountered by itself and its neighbors.

Let x and v denote a particle position and its corresponding velocity

in a search space respectively. The modified velocity and position of each

particle can be calculated using the current velocity and the distance from

pbest to gbest as shown in the following expressions:

)1( tidv = . )(t

idv + 1c * 1rand * ( idpbest - )(tidx ) + 2c * 2rand * ( dgbest - )(t

idx ) (4.20)

where,

)(tiv : Velocity of particle i at iteration t.

: Inertia weight factor

1c , 2c : Acceleration constant

1rand , 2rand : random number between 0 and 1.

)(tix : Current position of particle I at iteration t.

86

Pbestid : pbest of particle i.

gbestd : gbest of the group.

Similar to other evolutionary algorithms, PSO must also have a

fitness function that takes the particle’s position and assigns to it a fitness

value. For consistency, the fitness function is the same as for the other

algorithms. The position with minimum fitness value in the entire run is

called the global best (gbest). Also each particle keep tracking its minimum

fitness value, called as local best (l best or pbest). Each particle is initialized

with a random position and velocity. The velocity vjt of the jth particle, each

of n dimensions, accelerated towards the global best and its own personal

best.

PSO has a well balanced mechanism to enhance both global and

local exploration abilities. This is realized by inertia weight w and is

calculated by the following expression:

iteriter

max

minmaxmax

(4.21)

where, minmax , is the initial and final weight, maxiter is the maximum

iteration count, and iter is the current iteration number. From the above

equation certain velocity can be calculated, which gradually gets close to

pbest and gbest. The current position (searching point in the solution space)

can be modified by the following expression:

,)1()()1( vxx t

id

t

id

t

id

i = 1,2,3,………,n, and d =1,2,3,……..,m (4.22)

87

Craziness function:

The main drawbacks of the PSO are its premature convergence,

especially while handling problems with more local optima and heavily

constrained. To solve this, the concept of craziness, with the particles having

a predetermined probability of craziness, is incorporated, while in general

increases the probability of finding a better solution in the complex domain.

Thus, “crazy” agents are initiated, when they find a premature

convergence of the procedure. In this thesis, the probability of craziness

cr (identification of particles and randomizing its velocity) is expressed as a

function of inertia weight, to ensure the control of inertia weight during the

search.

maxmin exp

t

cr (4.23)

Thus

max crtj t

j

rand (0,v ), if rand (0,1)V

V , otherwise

(4.24)

where, t is the inertia weight at the tth iteration of the run, and rand(0,1) is

the random number between 0 and 1. It is obvious that, if the PSO procedure

gets stuck in the beginning of the run, a high value of cr will be used to

generate “crazy” particles. While the run progresses, a comparatively low

value of cr will be used to generate “Crazy” particles. Thus the significance

of control of inertia weight in the PSO algorithm is also retained.

Particle velocities on each dimension are clamped to maximum

allowable velocity vmax if the sum of accelerations exceeds the limit. Vmax is

88

an important parameter that determines the resolution with which regions

between the present position and the target positions are searched. If vmax is

too high, agents may fly past good regions. If it is low, agents may not

explore sufficiently beyond locally good regions. To enhance the performance

of the PSO, vmax is set to the value of the dynamic range of each control

variable in the problem.

4.3.2 Optimization scheme of PSO

i) Evaluation of Each Particle

Each Particle is evaluated using the fitness function of the problem

to minimize the fuel cost function given by (4.1). The best fitness value of

each particle up to the current iteration is set to that if the local best of that

particle jLbestp , .

ii) Modification of Each Searching Point:

Using the global best and the local best of each particle up to the

current iteration, the searching point of each agent has to be modified

according to the following expression:

)1()()( tj

ptj

vtj

p (4.25)

where ))1(,

(11

)1()()( tj

pjbestL

prandctj

vtwtj

v

))1((22

tj

pbestG

prandc (4.26)

where rand 1&2 are random numbers between 0 and 1, and C1, C2 are

acceleration factors. Similar to inertia weight, acceleration factors also

controls the exploration of the PSO. These are the stochastic acceleration

89

terms that pull each particle towards Pbest and Gbest positions. Thus improved

performance of PSO can be obtained carefully selecting suitable values for

inertia weight C1 and C2. Thus, new searching points were explored for the

next iteration to further exploit the search. The elements of the new searching

point matrix Pj (t) should be forced to satisfy the real power generation limits

given in Fan and Mcdonald (1994). Once the new searching points were

determined, inertia weight had to be modified using (4.21).

iii) Modification of the Global and the Local bests:

Each particle should be evaluated using the fitness function of the

dynamic economic dispatch, as was done in point (i) in this section. PGbest and

PLbest,j have to be modified according to the present fitness function values

evaluated using the new searching points of the particle. If the best fitness

value of all the fitness function values is better than the Oj,best (t-1), then

change PGbest to this value of the searching point of the corresponding particle

contribute for this best fitness value. Similarly, the local best of other particle

in the population should be changed accordingly if the present fitness function

value is better than the previous one.

iv) Termination Criteria:

Repeat from (i) until the maximum number of iterations is reached.

4.3.3 Implementation of HTC problem using PSO (C) algorithm

To apply a PSO algorithm for optimization problem, some essential

components need to be designed. The implementation of these PSO

components for solving the hydrothermal coordination problem as described

below.

90

i) Representation of trial solution vector:

According to the problem solution methodology, a dependent thermal generation dPT is randomly selected. The non-dependent thermal generation jPT

for j = 1, 2, …………Z, j d, are together taken as (Z – 1) dimensional trail

vector. Let zddi PTPTPTPTPTP ,........,,,......., )1()1(2,1 be the trial vector of the i

th component of a particle.

ii) Initialization of Particle of trial vectors:

Let the particle size be pN . Each initial parent particle trail

vector pi NiP ,....,2,1, , is selected randomly from a feasible range in each

dimension. This is done by setting the jth components of each parent particle as

maxmin , PTPTrandPT j , for j = 1, 2, …….,(d -1), (d + 1), …….Z (4.27)

where, maxmin , PTPTrand denotes a uniform random variable ranging over

maxmin , PTPT . The above procedure is delineated in the flowchart shown in

Figure 4.2.

4.4 NUMERICAL SIMULATION

In this section two different hydrothermal test systems are solved to validate the proposed technique. First system consists of one hydro and one thermal unit, second system comprises single hydro unit and ten thermal units. Also, the second system is solved for two different load demand patterns. The availability of maximum hydro units are taken into account and deducted from the total power demand. The balance load demand is met by the thermal units, such that the thermal units to be committed for generation is defined by the hybrid SA based technique. The PSO (C) has to be providing the optimal hydro schedule for 24 hrs based on the volume of the water reservoir and discharge rate. The thermal unit commitment schedule has been incorporated in the DEDP sub-problem and solved by the proposed hybrid SA-EP-TPSO technique, which minimizes the total generation cost.

91

it = it + 1

Is stopping rule satisfied?

Output the hydrothermal scheduling results

Input data on load pattern, fuel cost curve of equivalent thermal unit, operating limits on thermal and hydro

units, and reservoir water volume limits

Input control variables of PSO

Generate at random initial particle vectors

it = 1

Estimate the thermal generation and its respective generation cost using SA-EP-TPSO

method. As discussed in 3.7

Calculate total water discharge

Calculate dependent thermal generation

Are all generation levels and water volumes with

their limiting values?

Is it == Np?

Evaluate objective functions corresponding to each particle

Update the particles velocity & position

Update Inertia weight

No Yes

Yes

No

No

Yes

Figure 4.2. Flowchart for the HTC problem by PSO (C) technique

92

Test System 1:

In this section, the first system is tested to validate the proposed

hybrid PSO(C) technique. It comprises a hydro and an equalent thermal plant.

The schedule horizon is 3 days and there are six 12-hr intervals given in

Appendix A1.3. The test data are taken from the Wood and Woolenberg

(1984).This unit has got the prohibited operating zones as given in Table 4.1.

Table 4.1 Prohibited Operating Zones for the 1-unit DEDP system

Unit Zone 1 (MW)

Zone 2 (MW)

Zone 3 (MW)

Zone 4 (MW)

1

[870, 910]

[790, 810]

[750 775]

[1200 1230]

To validate the feasibility of the proposed algorithm, the test system

was also solved using the gradient search algorithm and the PSO algorithm

without including the crazy function. The simulation was conducted for 100

trial runs to study the robustness of the developed PSO (C) based HTC

algorithm.

The following simulation parameters are selected for the PSO

algorithms to solve the HTC problem. The selection procedure has been

adopted from Aruldoss and Ebenezer (2005). These simulation parameters are

found to be most suitable for the test case adopted to demonstrate the

feasibility of the proposed algorithms. Particle size = 100, Maximum inertia

weight = 1.3, Minimum inertia weight = 0.7, C1 = C2 = 2, maximum velocity

= iP , kmax = 30. For PSO the penalty parameters are taken as 10000.

Termination of the PSO algorithms is done when there is no improvement in

the solution for a pre-specified number of iterations.

93

Table 4.2: Best Hydrothermal Schedules obtained by proposed Technique

Technique

Interval

Thermal Generation

(MW)

Hydro Generation

(MW)

Volume

(acre – ft)

Discharge

(acre-ft h-1)

Cost

(Rs)

Proposed

1st day 0:00-12:00 12:00-0:00 2nd day 0:00-12:00

12:00-0:00

3rd day 0:00-12:00

12:00-0:00

968.212 870 870

870

810

775

231.78 630 230

930

140

525

106216.2 88683.0 95005.8

59580.6

71271.0

60000.0

1771.3 3461.1 1473.1

4952.1

1025.8

2939.25

710002.4

Table 4.3 Best Hydrothermal Schedules obtained by Gradient technique

Technique

Interval

Thermal Generation

(MW)

Hydro Generation

(MW)

Volume (acre – ft)

Discharge (acre-ft h-1)

Cost

(Rs)

Gradient search

1st day 0:00-12:00

12:00-0:00

2nd day 0:00-12:00

12:00-0:00

3rd day 0:00-12:00

12:00-0:00

1013.21

870

870

870

790

750

186.787

630

230

930

160

550

108900.0

91366.8

97689.6

62264.4

72762.0

60000.0

1258.33

3461.1

1473.1

4952.1

1125.2

3063.5

710422.76

Table 4.4: Best Hydrothermal Schedules obtained by PSO technique

Technique

Interval

Thermal Generation

(MW)

Hydro Generation

(MW)

Volume (acre – ft)

Discharge (acre-ft h-1)

Cost

(Rs)

PSO

1st day 0:00-12:00

12:00-0:00

2nd day 0:00-12:00

12:00-0:00

3rd day 0:00-12:00

12:00-0:00

890.23

910

870

910

810

775

309.77

590

230

890

140

525

101565

86417.4

92740.2

59580.6

71271.0

60000

1869.55

3262.3

1473.1

4753.3

1025.8

2939.25

710169.73

94

0 10 20 30 40 50 60 70 80 90 1007.1

7.105

7.11

7.115

7.12

7.125

7.13 x 10 5

PSO(C) PSO

Figure 4.3 Best solution obtained for 100 trial runs

Best results obtained using the proposed method and the gradient

search algorithms are shown in Table 4.2 and 4.3. This result was obtained for 71 trial runs. The average cost obtained by proposed PSO based HTC

algorithm is Rs. 710002.4, and even this cost is comparatively less than the best cost obtained using the gradient search algorithm and the PSO algorithm

without the crazy agents. During the simulation the crazy agents are generated

at an average of 12 times when the inertia weight reduces below an average of

0.83. The best solution obtained by the proposed hybrid method and PSO for 100 trail runs is plotted in Figure 4.3.

Test System 2: In this case, the second test system was analyzed

with two different load demands for the best hydro and thermal generators schedule for twenty-four hours. The optimal hydrothermal coordination

schedule for the proposed hybrid PSO(C)-SA-EP-TPSO method is given in

the table 4.5 and 4.7. In the second test system the reserve power also considered by satisfying the minimum and maximum reservoir volume with

other constraints. The numerical results are validated with optimal schedule of

the hydro and thermal generators. The test data and the units having prohibited operating zones are considered as same as in the chapter 2.

95

Table 4.5 Final optimal hydrothermal schedule using the proposed

method for the second system (LD1).

S.No Load Demand (MW)

Thermal Power Generation PT

(MW)

Hydro Power Generation PH (MW)

Reservoir Volume (acre ft)

Water Discharge

(acre ft h-1) 1. 1036 741.4 294.6 100205.5 1794.5 2. 1110 780 330 10023.4 1970.1 3. 1258 998 260 100613.2 1622.2 4. 1406 1026 380 100394.6 2218.6 5. 1480 880 600 99082.6 3312.0 6. 1628 828 800 96776.6 4306.0 7. 1702 1002 700 94967.6 3809.0 8. 1776 1026 750 92910.1 4057.5 9. 1924 1024 900 90107.1 4803.0 10. 2072 1222 850 87552.6 4554.5 11. 2146 1246 900 84749.6 4803.0 12. 2220 1270 950 81698.1 5051.5 13. 2072 1002 1070 78050.2 5647.9 14. 1924 1024 900 75247.2 4803.0 15. 1776 1006 770 74153.8 4156.9 16. 1554 1004 550 73090.3 3036.5 17. 1480 680 800 70784.3 4306.0 18. 1628 1028 600 69472.3 3312.0 19. 1776 1006 770 67315.4 4156.9 20. 2072 1502 570 66152.5 3162.9 21. 1924 1024 900 63349.5 4803.0 22. 1628 928 700 61540.5 3809.0 23. 1332 1000 332 61560.5 1980.0 24. 1184 534 650 60000.0 3560.5

Total Cost (Rs) = 333170

96

Table 4.6 Final commitment schedule using the Hybrid SA-EP-TPSO

technique for the second system (First load demand)

Hour Thermal Power (MW)

Unit Status

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

741.4 780 998 1026 880 828 1002 1026 1024 1222 1246 1270 1002 1024 1006 1004 680 1028 1006 1502 1024 928 1000 534

0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0

From the above tables, the generation cost obtained from the

second test system for the first load demand (LD1) using the proposed hybrid

technique is found to be superior to that of the cost obtained using gradient

search and other methods. After determining the hydroelectric power the final

committed units for the thermal generators using the hybrid SA-EP-TPSO

technique are scheduled and given in table 4.6.

97

Table 4.7 Final optimal hydrothermal schedule using the proposed

Hybrid method for the second system (LD2).

S.No Load Demand (MW)

Thermal Power

Generation PT (MW)

Hydro Power

Generation PH (MW)

Volume of the

Reservoir (acre ft)

Water Discharge

(acre ft h-1)

1. 1036 723 313 82472.5 1885.6 2. 1110 594 516 81577.9 2894.5 3. 1258 1187 071 82895.1 0682.8 4. 1406 319 1087 79162.7 5732.3 5. 1480 839 641 77646.9 3515.7 6. 1628 1162 466 77000.9 2646.0 7. 1702 1135 567 75852.9 3147.9 8. 1776 1409 367 75698.9 2153.9 9. 1924 1448 476 75003.2 2695.7 10. 2072 1823 249 75435.7 1567.5 11. 2146 1508 638 73934.8 3500.8 12. 2220 1384 836 71449.9 4484.9 13. 2072 1489 583 70222.4 3227.5 14. 1924 1219 705 68388.5 3833.8 15. 1776 1546 230 68915.4 1473.1 16. 1554 1136 418 68508.0 2407.4 17. 1480 618 862 65893.8 4614.1 18. 1628 879 749 63841.3 4052.5 19. 1776 1269 507 62991.5 2849.7 20. 2072 1447 625 65552.9 3436.2 21. 1924 1050 874 58881.5 4673.7 22. 1628 1563 065 60228.4 0653.0 23. 1332 669 663 58603.3 3625.1 24. 1184 1129 055 60000.0 0603.3

Total Cost (Rs) = 397570

The total cost of power generation obtained by using the proposed hybrid

SA-EP-TPSO is given in table 4.7. In this system some of the units considered

POZ as constraints and the proposed technique could overcome the non-

convexity in the solution domain and could produce a quality solution.

98

The final generation cost obtained from the second test system for

the second load demand (LD2) is found to be superior in terms of the

performance of the heavily constrained hydrothermal coordination system

includes POZ in certain thermal units in addition to the security constraints.

The final commitment schedule for the 10 unit thermal system is given in

table 4.8. To validate the proposed hybrid algorithm several standard and

IEEE-39 bus test systems are taken and solved.

Table 4.8 Final commitment schedule using the hybrid SA-EP-TPSO

technique for the second system (second load demand)

Hour Thermal Power (MW)

Unit Status

1 2 3 4 5 6 7 8 9 10

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

795 653

1306 351 923

1278 1249 1550 1593 2005 1659 1522 1638 1341 1701 1250 680 967

1396 1592 1155 1719 736

1242

1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0

99

4.5 CONCLUSION

A hybrid PSO(C)-SA-EP-TPSO based algorithm to solve the HTC

problem is proposed. In this problem the prohibited operating zone as an

inequality constraint, considered in the thermal generating units. All the

thermal generating units are represented as single equivalent thermal plant. In

the second case a 10-unit thermal system is solved to schedule the committed

units. The POZ complicates the solution domain and makes the solution

algorithm to easily trap into a local minimum. To solve this heavily

constrained HTC problem, the PSO algorithm is improved by including the

craziness function. The UCP and DEDP part were solved using SA-EP-TPSO

techniques. The efficiency of this newly proposed algorithm is illustrated

using a standard test system. Quite impressive results have been obtained by

the proposed technique in terms of fuel cost savings. Also, Numerical results

demonstrated that the proposed technique provides a feasible schedule

compared with the gradient search technique in terms of quality and

reliability. The simulation time can be significantly reduced by implementing

the proposed algorithm in parallel processing machines. (Titus and Ebenezer

Jeyakumar 2007).