# Dispatching a 19-Unit Indian Utility System Using a Refined Differential Evolution Algorithm

Authors: J. Jasper; T. Aruldoss Albert Victoire
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper presents a differential evolution with neighborhood based mutation (DE-NM) technique to solve Dynamic Economic Dispatch (DED) problem with valve point effects and multiple fuel options. A new mutation scheme based on neighborhood topology is presented with an aim to achieve the cost reduction together satisfying the dynamic behavior of the generating units over the considered time period. The neighborhood based mutation (NM) balances the exploration and exploitation of the search effort of differential evolution (DE) technique. The NM method enhances the convergence speed and the performance of the DE technique. The performance of the DE-NM is tested on a 10-unit and a real public Indian utility system with 19 generating units. Both the test systems are illustrated under different load patterns. The dispatch results obtained using the proposed method for the Indian system have considerably reduced the operating cost and optimized its operation.


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## Body

## 1. Introduction

Dynamic economic dispatch (DED) is a vital task in the economic operation of modern power system [1, 2].  The DED is defined as the optimal allocation of the predicted load profile among the online generators for a specific period of time. The objective of DED is to minimize the total fuel cost over the whole scheduling period subjected to practical and technical constraints. In DED the generation schedule is greatly affected due to ramp rate limits [3]. This ramp rate constraint sets a limit on the rate of change of electrical power output for the safe operation of the generating units. Thus the DED problem is a practical formulation in real-time power system problem.Actual DED formulation mainly depends on the representation of the fuel cost curve of the generators. Several conventional methods [4–6] such as lambda iteration, base point participation factor, Newton method, gradient method and Dynamic programming (DP) [7] is used to solve the DED problem by assuming the fuel cost curve of the generators piecewise linear and non-monotonically increasing. But, in practical generation the fuel cost curve is non-linear and non-convex because of valve point effect [8], multiple fuel option [9] and operational constraints such as prohibited operating zones [10], spinning reserves [11]. Thus makes the above conventional methods fail to solve for quality solution.To overcome the drawback in the conventional methods, heuristic techniques were introduced. Recently, artificial intelligence techniques such as Artificial Neural Network (ANN) [12], genetic algorithm (GA) [8], evolutionary programming (EP) [13], simulated annealing (SA) [14], particle swarm optimization (PSO) [15], differential evolution (DE) [16], modified differential evolution (MDE) [17], and hybrid algorithm (HA) [18, 19] are widely used to solve the DED problem. All the techniques have yielded success to certain extent, but they suffer from premature convergence, local trapping of the problem, and setting the control parameter.Combining deterministic techniques with evolutionary algorithms (EAs) is a promising alternative to solve highly nonlinear and nonconvex cost functions. Probabilistic methods do not always guarantee discovering the global optimum solution in finite time since updating their candidate’s position in the solution space requires probabilistic rules. Therefore, fine tuning of the above techniques was applied for every improvement in the solution. Plenty of literature work has been done on the fine tuning of above methods. To be concise, a few are included in the reference. Hybridization of EP with SQP (EP-SQP) [20], PSO with SQP (PSO-SQP) [21], and chaotic differential evolution with SQP (DEC-SQP) [22] are few examples.Differential evolution proposed by Storn and Price is one of the EA widely used in solving power system optimization problems. DE has got many advantages such that it is simple and easy to understand, and it can handle integer and discrete optimization, ease of use, fast convergence and robustness. In addition, DE is good at exploring the search space and locating the region of global optimum. Like other EAs, DE performance decreases as search space dimensionality increases. Also, DE is sensitive to the choice of the control parameters and it is difficult to adjust them for different problems [23]. Such contradictions in adjusting the control parameters can be overcome by self-adaptive techniques [24]. Hybridization of DE with other heuristic or local different algorithms is considered as an alternate direction of improvement of classical DE.From the literature [22–25], it can be observed that the main modifications, improvements, and developments on DE focus on adjusting control parameters in self-adaptive manner and or hybridization with other local search techniques. Very few enhancements have been implemented to modify the standard mutation strategies or to propose new mutation rules so as to enhance the total search ability of DE and to overcome the problems of stagnation or premature convergence [25]. In addition, DE is good at exploring the search space but slow at exploitation of solution [26]. In this paper DE algorithm with a neighborhood based mutation (DE-NM) [27] is developed for solving the DED problem. The new mutation scheme utilizes a concept of neighborhood of each population member. The neighborhood based mutation (NM) balances both the exploration and exploitation process to enhance the search ability of DE.The significant contribution of this paper is optimum generation schedule of an Indian utility system for three different load patterns. The Indian utility system consists of 19 generating units with the fuel cost function taking into account the valve-point effect and multiple fuel option. The DED problem of the Indian utility system is solved conventionally by priority list method where the generating units are allowed to run at full load to meet the demand. Also, real time constraints, such as ramp rate limits, prohibited operating zones, and so forth, are not taken into account in the solution. In recent years pilot attempts are taken to implement various artificial techniques in the operation and control of Indian power system. One such attempt is presented in this paper to solve the DED problem with valve point loading, ramp rate limits, prohibited operating zones, multiple fuel options, and spinning reserve.This paper is organized as follows: DED problem is formulated in Section2. Sections 3 and 4 give a detailed description of the DE-NM. Section 5 describes the implementation of DE-NM to DED problem. Analysis of DE-NM method with two systems is provided in Section 6 and Section 7 outlines the conclusion.


## 2. Problem Formulation

The objective function of DED problem is to minimize the total production cost of power over a given dispatch period, while satisfying various constraints. The objective function is formulated as(1)Minimize,FT=∑h=1H∑i=1NFih(Pih).Generally, the generator cost function is usually expressed as a quadratic polynomial as(2)Fih(Pih)=aiPih2+biPih+ci.For accurate nonconvex models of the objective function, the DED problem with valve point effects has to be considered by superimposing a rectified sinusoid component in the traditional quadratic fuel cost function as formulated in(3)Fih(Pih)=aiPih2+biPih+ci+|eisin(fi(Pihmin-Pih))|.Similarly practical generating units are supplied with multiple fuel sources and the cost functions of these units are represented with few or several piecewise quadratic functions. Such a cost function is called as a hybrid cost function and each segment of the hybrid cost function gives some information about the fuel burnt. The hybrid cost function is formulated as(4)Fih(Pih)={ai,1Pih2+bi,1Pih+ci,1,fuel1,Pihmin≤Pih≤Pih,1ai,2Pih2+bi,2Pih+ci,2,fuel2,Pih,1<Pih≤Pih,2⋮ai,kPih2+bi,kPih+ci,k,fuelk,Pih,k-1<Pih≤Pihmax.For more accurate dispatch results, the valve point effect and the multiple fuel options are integrated into the basic quadratic cost function. Thus the basic quadratic cost function given in (2) with N generating units and  NF fuel options for each unit is formulated as
(5)Fih(Pih)=ai,kPih2+bi,kPih+ci,k+|ei,ksin(fi,k(Pih,kmin-Pih))|ifPih,kmin≤Pih≤Pih,kmax,fuel  optionk,k=1,2,…,NF.The objective function as given in (1) is subjected to the following equality and inequality constraints.The power output from all the generating units must satisfy the total demand and the transmission losses of the system. The equality constraint is formulated as(6)∑i=1NPih=PDh+PLoss,h.The transmission loss is expressed in a quadratic form as(7)PLoss,h=∑m=1N∑n=1NPmhBmnPnh.The real power output of each generating unit is limited by the maximum and minimum power limit of the units. It is formulated as(8)Pimin≤Pih≤Pimax.The operating range of the generating units is restricted by their ramp rate limits. This is formulated as(9)Pih-Pi(h-1)≤URiifgenerationincreases,Pi(h-1)-Pih≤DRiifgenerationdecreases.Thus (8) is modified as
(10)max(Pimin,Pi(h-1)-DRi)≤Pih≤min(Pimax,Pi(h-1)+URi).Physical limitations of power plant components restrict the operation of generating units in certain operating regions known as prohibited zones. The power generated by each unit should lie either above or below the prohibited zones. Thus the feasible operating zones for the generating units are formulated as(11)Pimin≤Pih≤Pi,1l,Pi,j-1u≤Pih≤Pi,jlj=2,3,…,n,Pi,nu≤Pih≤Pimax.The spinning reserve constraint considering the prohibited operating zones is formulated as(12)∑i=1NSih≥SR,Sih={min{(Pimax-Pih),Simax},forunitswithoutprohibitedzones0,otherwise.From (12) it is shown that a unit with prohibited operating zones does not contribute to the spinning reserve of the system. This is because the prohibited zones severely restrict the generating unit’s flexibility to regulate system load.


## 3. Neighborhood Based Mutation (NM)

The optimization process in DE involves three basic operations such as mutation, crossover, and selection. Mutation in DE is a special kind of differential operator which is used to generate mutant or donor vectors. Actually it is the process of mutation, which demarcates one DE scheme from another. The rate of convergence of DE as well as its accuracy can be improved largely by applying different mutation strategies. Storn and Price proposed many DE variants applicable to different problems [28]. In this paper a new DE variant, based on neighborhood topology of the parameter vectors, is presented to improve the convergence characteristics of DE [29]. This concept of neighborhood topology is borrowed from the PSO algorithm.


### 3.1. Neighborhood Model

A graph of interconnection of vectors is called as neighborhood structure. The vectors in the neighborhood structure are assumed to be arranged in a circular fashion. Consider a DE populationP=[X→1,X→2,…,X→NP] where each, X→i is a D-dimensional parameter vector. For every X→i a neighborhood of radius K is defined which consists of vectors X→i-K,…X→i,…,X→i+K. K is a nonzero integer from 0 to (NP-1)/2. A proper balance between exploration and exploitation is necessary for efficient and effective operation of DE. Two kinds of neighborhood mutation models are presented to control the exploration and the exploitation process. One is local mutation model which has a greater tendency to locate the optimal solution of the objective function but needs more iteration. Second is global mutation model which rapidly converges to the optimal solution of the objective function but suffers from premature convergence problem.


#### 3.1.1. Local Neighborhood Mutation Model

For each vectorX→i of the population, a local donor vector is created by employing the best vector in the neighborhood of that member. In this mutation model each vector is mutated using the best position found so far in the neighborhood of it and not in the entire population. The local donor vector L→i,G is created as
(13)L→i,G=X→i,G+α·(X→n_besti,G-X→i,G)+β·(X→p,G-X→q,G),
where the subscript n_besti  indicates the best vector in the neighborhood of X→i,G and X→p,G and X→q,G are two randomly chosen vectors from the same neighborhood, that is, p,q∈[i-k,i+k]  with p≠q≠i.


#### 3.1.2. Global Neighborhood Mutation Model

Similarly, the global donor vector is created by using the best vector of the entire population. The global donor vectorg→i,G is created as
(14)g→i,G=X→i,G+α·(X→g_best,G-X→i,G)+β·(X→r1,G-X→r2,G),
where the subscript g_best indicates the best vector in the entire population at generation G and X→r1,G and X→r2,G are two randomly chosen vectors from the population, that is, r1,r2∈[1,NP] with r1≠r2≠i.The local mutation model favours exploration since all the vectors of the population are biased by different individuals. Global mutation model favours exploitation since all the vectors of the population are biased by the same individual. The two mutation operators are then combined using a new parameter called weight factorw∈(0,1) to form the actual mutation model of the presented DE algorithm. 										(15)V→i,G=w·g→i,G+(1-w)·L→i,G.


## 3.1. Neighborhood Model

A graph of interconnection of vectors is called as neighborhood structure. The vectors in the neighborhood structure are assumed to be arranged in a circular fashion. Consider a DE populationP=[X→1,X→2,…,X→NP] where each, X→i is a D-dimensional parameter vector. For every X→i a neighborhood of radius K is defined which consists of vectors X→i-K,…X→i,…,X→i+K. K is a nonzero integer from 0 to (NP-1)/2. A proper balance between exploration and exploitation is necessary for efficient and effective operation of DE. Two kinds of neighborhood mutation models are presented to control the exploration and the exploitation process. One is local mutation model which has a greater tendency to locate the optimal solution of the objective function but needs more iteration. Second is global mutation model which rapidly converges to the optimal solution of the objective function but suffers from premature convergence problem.


### 3.1.1. Local Neighborhood Mutation Model

For each vectorX→i of the population, a local donor vector is created by employing the best vector in the neighborhood of that member. In this mutation model each vector is mutated using the best position found so far in the neighborhood of it and not in the entire population. The local donor vector L→i,G is created as
(13)L→i,G=X→i,G+α·(X→n_besti,G-X→i,G)+β·(X→p,G-X→q,G),
where the subscript n_besti  indicates the best vector in the neighborhood of X→i,G and X→p,G and X→q,G are two randomly chosen vectors from the same neighborhood, that is, p,q∈[i-k,i+k]  with p≠q≠i.


### 3.1.2. Global Neighborhood Mutation Model

Similarly, the global donor vector is created by using the best vector of the entire population. The global donor vectorg→i,G is created as
(14)g→i,G=X→i,G+α·(X→g_best,G-X→i,G)+β·(X→r1,G-X→r2,G),
where the subscript g_best indicates the best vector in the entire population at generation G and X→r1,G and X→r2,G are two randomly chosen vectors from the population, that is, r1,r2∈[1,NP] with r1≠r2≠i.The local mutation model favours exploration since all the vectors of the population are biased by different individuals. Global mutation model favours exploitation since all the vectors of the population are biased by the same individual. The two mutation operators are then combined using a new parameter called weight factorw∈(0,1) to form the actual mutation model of the presented DE algorithm. 										(15)V→i,G=w·g→i,G+(1-w)·L→i,G.


## 3.1.1. Local Neighborhood Mutation Model

For each vectorX→i of the population, a local donor vector is created by employing the best vector in the neighborhood of that member. In this mutation model each vector is mutated using the best position found so far in the neighborhood of it and not in the entire population. The local donor vector L→i,G is created as
(13)L→i,G=X→i,G+α·(X→n_besti,G-X→i,G)+β·(X→p,G-X→q,G),
where the subscript n_besti  indicates the best vector in the neighborhood of X→i,G and X→p,G and X→q,G are two randomly chosen vectors from the same neighborhood, that is, p,q∈[i-k,i+k]  with p≠q≠i.


## 3.1.2. Global Neighborhood Mutation Model

Similarly, the global donor vector is created by using the best vector of the entire population. The global donor vectorg→i,G is created as
(14)g→i,G=X→i,G+α·(X→g_best,G-X→i,G)+β·(X→r1,G-X→r2,G),
where the subscript g_best indicates the best vector in the entire population at generation G and X→r1,G and X→r2,G are two randomly chosen vectors from the population, that is, r1,r2∈[1,NP] with r1≠r2≠i.The local mutation model favours exploration since all the vectors of the population are biased by different individuals. Global mutation model favours exploitation since all the vectors of the population are biased by the same individual. The two mutation operators are then combined using a new parameter called weight factorw∈(0,1) to form the actual mutation model of the presented DE algorithm. 										(15)V→i,G=w·g→i,G+(1-w)·L→i,G.


## 4. DE with Neighborhood Based Mutation

DE begins with a population of NPD-dimensional parameter vectors representing the candidate solutions. The ith vector of the population at the current generation is given as
(16)X→i,G=[x1,i,G,x2,i,G,…,xD,i,G].The vectorX→i,G is known as target or parent vector. G represents the subsequent generation and it is given by G=1,2,…,Gmax. The initial population (i.e., G=0) is randomly created by covering the entire search space as much as possible within prescribed minimum and maximum bounds. The jth component of the ith target vector is created as
(17)xj,i,0=xj,min+rand(0,1)·(xj,max-xj,min),
where rand(0,1) is a uniformly distributed random number lying between 0 and 1 and is obtained independently for each component of the ith vector. Each target vector of the population is subjected to mutation, crossover, and selection which are explained in the following subsections.


### 4.1. Mutation

After initialization, DE creates a donor vectorV→i,G corresponding to each target vector X→i,G in the current generation through mutation. In this method the actual donor vector is created with the help of global and neighborhood mutation models.For each member a global and local donor vector is created using (13) and (14). After creating the global and local donor vector they are combined using a weight factor w using (15).


### 4.2. Crossover

In crossover operation few components of the donor vector are exchanged with target vector to form a trial vectorU→i,G=[u1,i,G,u2,i,G,…,uD,i,G]. Two kinds of crossover schemes—exponential and binomial crossovers—can be used in DE. In this paper, binomial crossover is performed on each of the D variables whenever a randomly picked number between 0 and 1 is less than or equal to the Cr value. The number of parameters inherited from the donor has a binomial distribution. The scheme is outlined as
(18)uj,i,G={vj,i,G,if(randi,j(0,1)≤Cr)xj,i,G,otherwise,
where randi,j(0,1)∈[0,1] is a uniformly distributed random number for each jth component of the ith parameter vector.


### 4.3. Selection

To keep the population size constant over subsequent generations, the next step of the algorithm calls for selection to determine whether the target or the trial vector survives to the next generation, that is, atG=G+1.The selection operation is given as(19)X→i,G+1={U→i,G,iff(U→i,G)≤f(X→i,G),X→i,G,iff(U→i,G)≥f(X→i,G),
where f(X→i) is the function to be minimized. So if the new trial vector yields an equal or lower value of the objective function than that of target vector, then it replaces the corresponding target vector in the next generation. Otherwise the target vector is retained in the next generation. Hence the population either gets better (with respect to the minimization of the objective function) or remains the same in fitness status but never deteriorates.The three operations are repeated until a stopping criterion is met which is usually the maximum generation, that is,Gmax.


## 4.1. Mutation

After initialization, DE creates a donor vectorV→i,G corresponding to each target vector X→i,G in the current generation through mutation. In this method the actual donor vector is created with the help of global and neighborhood mutation models.For each member a global and local donor vector is created using (13) and (14). After creating the global and local donor vector they are combined using a weight factor w using (15).


## 4.2. Crossover

In crossover operation few components of the donor vector are exchanged with target vector to form a trial vectorU→i,G=[u1,i,G,u2,i,G,…,uD,i,G]. Two kinds of crossover schemes—exponential and binomial crossovers—can be used in DE. In this paper, binomial crossover is performed on each of the D variables whenever a randomly picked number between 0 and 1 is less than or equal to the Cr value. The number of parameters inherited from the donor has a binomial distribution. The scheme is outlined as
(18)uj,i,G={vj,i,G,if(randi,j(0,1)≤Cr)xj,i,G,otherwise,
where randi,j(0,1)∈[0,1] is a uniformly distributed random number for each jth component of the ith parameter vector.


## 4.3. Selection

To keep the population size constant over subsequent generations, the next step of the algorithm calls for selection to determine whether the target or the trial vector survives to the next generation, that is, atG=G+1.The selection operation is given as(19)X→i,G+1={U→i,G,iff(U→i,G)≤f(X→i,G),X→i,G,iff(U→i,G)≥f(X→i,G),
where f(X→i) is the function to be minimized. So if the new trial vector yields an equal or lower value of the objective function than that of target vector, then it replaces the corresponding target vector in the next generation. Otherwise the target vector is retained in the next generation. Hence the population either gets better (with respect to the minimization of the objective function) or remains the same in fitness status but never deteriorates.The three operations are repeated until a stopping criterion is met which is usually the maximum generation, that is,Gmax.


## 5. Implementation of DE-NM for DED Problem

Step 1 (initialization of the population).
For a population of sizeNP and dimension D, an initial vector (target vector) Xij,G is randomly generated. D represents the number of decision variables to be optimized. In DED problem D represents the number of generating units (i.e., N) to be considered. The vector Xij is the real power output (i.e., Pij) of jth unit of the ith population randomly generated within the operating limits using (8). The population is represented in a matrix form as
(20)Population=[X→11X→12X→13⋯X→1DX→21X→22X→23⋯X→2D⋮⋮⋮X→NP1X→NP2⋯X→NPD].
Each component of theith individual in the population is randomly generated such that the component is uniformly distributed within the minimum and maximum power limits of the generating units. The generation of the jth component of the ith individual is given by (17).Step 2 (handling the generation limit constraints).
In Step1 the power output from each generator is limited to be within minimum and maximum power limits. For the generator with ramp rate limits, the minimum and the maximum power limits are adjusted by using (10). Also, during the recombination and mutation operation, the power output of a generator can go above or below the maximum and minimum limits. Therefore, to restrict the Pi to be within the generating limit, a strategy is defined as follow:
(21)Pij={PiminifPij<Pimin,PimaxifPij>Pimax,Pijotherwise.Step 3 (mutation operation).
Consider the following.
Step  3.1. A neighborhood structure of radius K is created for each Xij vector.
Step  3.2 (local neighborhood model). In this model a donor vector is created using the best position found so far in a small neighborhood of it and not in entire population. The mutated vector L→i,G is known as local donor vector and it is obtained using (13).
Step  3.3 (global neighborhood model). In this model a globally best vector Xbest,G of the entire population is used for mutating a population member. The global donor vector g→iG is obtained using (14).
Both the local and global donor vectors are combined using (15) to obtain the actual donor vector Vi,G of the presented technique.Step 4 (recombination (crossover)).
Recombination is employed to generate a trial vectorUi by replacing certain components of Xi with corresponding components of donor vector Vi. The trial vector by crossover operation is obtained using (18).Step 5 (evaluation of fitness function).
The evaluation function is computed from (1) and the generator cost function Fi(Pi) used is the cost function considering the valve point effect and multiple fuel option as given in (5). The power balance and the prohibited operating zone constraints are included in the evaluation function by adding a penalty term to the objective function. The power balance constraint is considered without including the transmission losses for simplicity.
Consider(22)f(Pih)=∑i=1NFih(Pih)+λ·[∑i=1NPih-PDh]2+γ·[∑i=1nviR],
where λ is the penalty parameter for not satisfying the load demand and γ represents the penalty for a unit loading falling within a prohibited operating zone. viR is the violation of the prohibited zone constraint for the ith unit which is defined as
(23)viR={1,ifPiviolates  the  prohibited  zones,0.Step 6 (selection).
Members to constitute the population of next generation(G+1) are decided by (14). The new vector Xi,(G+1) is selected based on the comparison of fitness value of both Xi and Ui.Step 7 (verification of stopping criterion).
Set the generation countG=G+1. Go to Step 3 until stopping criterion is met which is usually maximum generation count Gmax.


### 5.1. Flowchart for DE-NM Approach for DED

See Figure1.Figure 1
Flowchart for DE-NM approach for DED.


## 5.1. Flowchart for DE-NM Approach for DED

See Figure1.Figure 1
Flowchart for DE-NM approach for DED.


## 6. Test Results and Analysis

### 6.1. Description of the Test Systems

In this section two case systems are studied using DE-NM method. The fuel cost functions of the two systems are nonconvex considering valve point effect and multiple fuel option in its fuel cost.Three different load patterns are applied for both these systems to demonstrate the robustness of the proposed solution technique as well as to show the effectiveness in scheduling for different load patterns (situations). The simulation horizon for all the three load patterns is taken as 24 hr with a dispatch interval of 1 hr. The coding is written using the MATLAB 7 programming language and executed in a personal computer machine.


### 6.2. Parameter Selection

DE-NM consists of four new parameters in addition toF and CR as in classical DE. They are α, β, w, and K.  The role of α and β same as that of F. Thus, the number of parameters reduced by considering α=β=F. The most important parameter in DE-NM is the weight factor w, which controls the balance between the exploration and exploitation capabilities. To obtain the balance, value of w is usually selected from the range [0,1]. It is adapted online during the execution of the algorithm by a linear increment scheme; that is, it is initialized with 0 and increased up to 1. Since w starts with 0, it favors the exploration process during the initial stage and on getting promoted to 1 it favors the exploitation process. The linear increment scheme is given by
(24)wG=GGmax.In this paper, the neighborhood size equal to 10% of the total population size of 100 is considered. The value for bothα and β is taken as 0.8 and crossover rate Cr=0.9. The DE-NM is applied to both the case studies for 30 independent trials (1000 iterations per trial) with the selected parameters.


### 6.3. Case Study 1

In this test case, a 10-unit system is optimized to meet three different load patterns. The level of fluctuation of the three different load patterns is given in Figure2. The peak demand for load pattern 1, pattern 2, and pattern 3 is 3208 MW, 2460 MW, and 3210 MW, respectively. The system data and the related constraints for this case study are given in [30]. The complete generation schedule for load pattern 2 corresponding to the minimum generation cost obtained by DE-NM is presented in Table 1. Table 2 gives the fuel type chosen by the units for the corresponding best generation schedule obtained. The fuel chosen by the units for all three load patterns is given in Table 2. The table format represents the fuel type chosen by an unit for pattern 1 and it is followed by fuel type for the same unit for patterns 2 and 3, respectively.Table 1
Best power generation schedule obtained by DE-NM for case study 1.
Hour
Unit
1 (MW)
2 (MW)
3 (MW)
4 (MW)
5 (MW)
6 (MW)
7 (MW)
8 (MW)
9 (MW)
10 (MW)
1
168.862
177.000
250.000
250.000
240.000
166.000
250.000
180.570
243.000
250.000
2
163.886
177.000
250.000
250.000
234.691
163.000
250.000
173.854
253.000
250.000
3
165.705
177.000
250.000
250.000
235.047
160.543
250.000
178.591
253.000
250.000
4
165.826
177.000
250.000
250.000
240.000
163.000
250.000
181.576
249.509
250.000
5
172.019
177.000
250.000
250.000
457.000
150.375
250.000
183.538
252.999
247.372
6
192.564
177.000
269.658
265.000
451.287
196.324
250.000
249.687
302.656
254.658
7
181.538
177.000
253.465
253.465
446.999
180.580
250.000
246.700
218.376
250.000
8
172.014
177.000
250.000
250.000
457.000
163.000
250.000
176.040
249.509
250.000
9
171.935
177.000
250.000
250.000
240.000
153.000
250.000
183.521
253.000
250.000
10
170.438
177.000
250.000
250.000
240.000
163.000
250.000
176.607
253.000
246.927
11
179.939
177.000
253.528
253.528
457.000
174.322
250.000
191.455
273.573
250.000
12
165.807
177.000
250.000
250.000
234.616
163.000
250.000
180.585
251.231
247.169
13
178.362
177.000
251.413
251.413
457.000
161.753
250.000
190.465
270.278
250.000
14
205.211
177.000
250.000
250.000
457.000
163.000
282.078
254.494
345.626
264.559
15
205.220
177.000
310.646
265.000
457.000
224.225
277.774
254.494
406.509
273.310
16
194.144
177.000
274.350
265.000
457.000
199.127
262.903
250.463
304.055
246.630
17
178.354
177.000
261.510
261.510
457.000
162.985
250.000
247.775
266.450
246.924
18
167.277
177.000
250.000
250.000
233.617
163.000
250.000
179.579
253.000
246.382
19
163.876
177.000
250.000
250.000
451.287
163.000
250.000
177.557
248.712
250.000
20
171.953
177.000
250.000
250.000
235.047
160.543
250.000
184.512
253.000
250.000
21
168.889
177.000
250.000
250.000
234.691
163.000
250.000
181.563
253.000
250.000
22
178.361
177.000
252.512
252.512
240.000
171.466
250.000
192.440
263.232
246.630
23
165.864
177.000
250.000
250.000
240.000
156.074
250.000
183.479
253.000
246.925
24
176.904
177.000
250.000
250.000
457.000
158.553
250.000
181.986
251.996
250.000Table 2
Fuel switching of generators for case study 1.
Hour
Unit
1
2
3
4
5
6
7
8
9
10
1
2,1
,
1
1,1
,
1
1,1
,
1
3,3
,
2
3,3
,
1
3,1
,
1
2,1
,
1
3,2
,
2
3,1
,
1
1,1
,
1
2
2,1
,
1
1,1
,
1
1,1
,
1
3,3
,
2
3,3
,
1
3,1
,
1
1,1
,
1
3,3
,
2
3,1
,
1
1,1
,
1
3
1,1
,
1
1,1
,
1
1,1
,
1
2,3
,
2
1,3
,
1
1,1
,
1
1,1
,
1
2,2
,
2
1,1
,
1
1,1
,
1
4
1,1
,
1
1,1
,
1
1,1
,
1
2,2
,
2
1,3
,
1
1,1
,
1
1,1
,
1
2,2
,
2
1,1
,
1
1,1
,
1
5
1,1
,
1
1,1
,
1
1,1
,
1
3,2
,
2
1,3
,
3
1,1
,
1
1,1
,
1
3,3
,
2
1,1
,
1
1,1
,
1
6
1,1
,
2
1,1
,
1
1,1
,
1
3,3
,
3
1,3
,
3
1,3
,
1
1,1
,
1
2,3
,
3
1,3
,
1
1,1
,
1
7
1,1
,
2
1,1
,
1
1,3
,
1
3,3
,
3
3,3
,
3
1,3
,
1
1,3
,
1
2,3
,
3
1,3
,
1
1,1
,
1
8
1,1
,
2
1,1
,
1
1,2
,
1
2,3
,
2
3,3
,
3
1,3
,
1
1,3
,
1
2,3
,
2
1,3
,
1
1,2
,
1
9
2,1
,
2
1,1
,
1
1,2
,
1
3,3
,
2
1,3
,
1
3,3
,
1
1,3
,
1
3,3
,
2
3,3
,
1
1,2
,
1
10
2,1
,
2
1,1
,
1
1,2
,
1
3,3
,
2
3,3
,
1
3,3
,
1
1,3
,
1
3,3
,
2
1,3
,
1
1,3
,
1
11
2,1
,
2
1,1
,
1
1,1
,
1
3,3
,
1
3,3
,
3
3,3
,
1
1,3
,
1
3,3
,
2
1,3
,
1
1,1
,
1
12
2,1
,
2
1,1
,
1
1,1
,
1
3,3
,
2
3,3
,
1
3,3
,
1
1,1
,
1
3,3
,
2
1,3
,
1
1,1
,
1
13
2,1
,
2
1,1
,
1
1,1
,
1
3,3
,
3
3,3
,
3
3,3
,
1
1,1
,
1
3,3
,
2
3,3
,
1
1,1
,
1
14
2,2
,
1
1,1
,
1
1,1
,
1
3,2
,
3
3,3
,
3
3,1
,
1
1,1
,
1
3,2
,
3
3,1
,
1
1,1
,
1
15
2,2
,
1
1,1
,
1
1,1
,
1
3,3
,
3
3,1
,
3
3,3
,
3
1,1
,
1
3,2
,
3
3,1
,
1
1,1
,
1
16
2,1
,
1
1,1
,
1
2,1
,
1
3,2
,
3
3,3
,
3
3,1
,
1
2,1
,
1
3,3
,
3
3,1
,
1
1,1
,
1
17
2,1
,
1
1,1
,
1
2,1
,
1
3,3
,
2
3,3
,
3
3,3
,
1
3,1
,
1
3,3
,
3
3,1
,
1
1,1
,
1
18
2,1
,
2
1,1
,
1
2,1
,
1
3,3
,
2
3,3
,
1
3,3
,
1
3,1
,
1
3,3
,
2
3,1
,
1
2,1
,
1
19
2,1
,
2
11,1
1,3
,
1
2,3
,
2
3,3
,
3
3,3
,
1
2,3
,
1
3,3
,
2
3,3
,
1
1,1
,
1
20
2,1
,
2
1,1
,
1
1,1
,
1
2,3
,
2
3,3
,
1
3,3
,
1
1,1
,
1
3,3
,
2
3,3
,
1
1,1
,
1
21
2,1
,
2
1,1
,
1
1,1
,
1
2,3
,
2
3,3
,
1
3,3
,
1
1,1
,
1
3,3
,
2
3,3
,
1
1,1
,
1
22
1,1
,
2
1,1
,
1
1,1
,
1
2,3
,
3
3,3
,
1
1,3
,
1
1,1
,
1
3,3
,
2
1,3
,
1
1,1
,
1
23
1,1
,
1
1,1
,
1
1,1
,
1
2,3
,
2
1,3
,
1
1,3
,
1
1,1
,
1
3,3
,
2
1,1
,
1
1,1
,
1
24
1,1
,
1
1,1
,
1
1,1
,
1
2,3
,
2
1,3
,
3
1,1
,
1
1,1
,
1
2,3
,
2
1,1
,
1
1,1
,
1Figure 2
Load demand patterns for case study 1.The best cost obtained by the DE-NM method considering spinning reserve for pattern 1 is $10649.68, for pattern 2 is $8103.326, and for pattern 3 is $11054.83. The worst cost obtained by the DE-NM method for pattern 1 is $11952.07, pattern 2 is $9163.973, and pattern 3 is $12036.89. The average cost of pattern 1, pattern 2, and pattern 3 for all the 30 trials is $11451.82, $9711.448, and $11497.85.  Figures3(a), 3(b), and 3(c) shows the distribution of the cost obtained for load patterns 1, 2, and 3, respectively. A straight line passing through each distribution area denotes the average cost. The percentage of producing quality solutions by DE-NM is above 60% as seen from Figure 4.(a) Cost distribution obtained by DE-NM for case study 1 with pattern 1. (b) Cost distribution obtained by DE-NM for case study 1 with pattern 2. (c) Cost distribution obtained by DE-NM for case study 1 with pattern 3.
(a)
(b)
(c)Figure 4
Frequency of convergence for case study 1.


### 6.4. A Short Overview of the Indian Utility System

Indian power system is a geographically dispersed network of generators. For ease of operation, the entire Indian power system has been divided into five regions, namely, Northern Region (NR), Western Region (WR), Southern Region (SR), Eastern Region (ER), and North Eastern Region (NER). While NR, WR, ER, and NER are interconnected as a single grid, the SR had not been interconnected with the national grid. This prevents the SR from receiving benefits from national power market. The SR covering 6, 51,000 Sq. Km of area comprises the states of Andhra Pradesh, Karnataka, Kerala, Pondicherry and Tamil Nadu. The Indian utility system considered in this paper belongs to the state of Tamil Nadu. The Electricity Board of Tamil Nadu state was recently restructured and the board is in the move of handing over transmission and distribution to the private operators.The South Indian network comprises of 19 thermal units, located at various parts of the state of Tamil Nadu. Since the system under consideration belongs to the state department, the technical data of the system cannot be published is available with the corresponding author, and will be provided on request. Also the data of system network is approximated neglecting several local feeder data this work neglects estimation of transmission losses for simplicity. Even then, as per the board’s advice, this work assumes a uniform 20% loss in the transmission and hence it is added to the 24-hour load demand data for scheduling.


### 6.5. Case Study 2

This system involves a standard Indian utility system with 19 units. All the 19 units include nonlinear characteristics such as valve point effect, ramp rate limit, multiple fuels, prohibited operating zones and spinning reserve constraint. This system is applied for three different load patterns as shown in Figure5. The demand for each hour includes approximately 20% of transmission losses. A generation schedule is obtained using the DE-NM method where each unit contributes to the demand and the transmission loss.  The maximum demand for pattern 1 is 4400 MW, for pattern 2 is 4186 MW, and for pattern 3 is 4173 MW. Table 3 gives the best generation schedule obtained by the DE-NM method for pattern 2. Few units in the system are provided with multiple fuel option. The fuel types include 1, 2, 3, 4, 5 and 6. Table 4 shows the fuel switching for units 3, 5, 7, 13, and 19 and all the remaining units utilize fuel 1 for every hour of time interval.Table 3
Best power generation schedule obtained by DE-NM for case study 2.
Hour
Power output (MW)
1
2
3
4
5
6
7
8
9
10
1
126.623
176.332
126.942
25.000
113.750
340.000
166.554
150.000
242.127
15.000
2
124.189
165.000
144.530
08.000
146.176
340.000
177.498
138.874
254.528
15.000
3
127.640
161.742
140.000
25.000
177.500
334.685
177.500
152.687
283.000
15.000
4
132.015
287.062
368.136
25.000
175.995
340.000
177.499
165.069
260.000
40.000
5
148.315
421.843
400.000
25.000
177.500
340.000
174.898
193.407
280.290
40.000
6
227.936
438.000
440.000
25.000
177.500
340.000
176.290
321.681
302.381
40.000
7
292.692
438.000
400.000
25.000
177.499
363.000
177.499
452.292
516.598
40.000
8
300.000
438.000
400.000
25.000
177.500
371.145
177.499
500.000
532.465
40.000
9
120.045
204.725
374.718
25.000
177.500
338.122
177.499
144.607
283.000
40.000
10
129.676
196.189
202.088
25.000
177.500
340.000
177.500
153.581
257.593
40.000
11
125.834
137.702
136.903
25.000
177.499
340.000
177.499
152.633
254.420
15.000
12
126.297
149.825
140.000
25.000
177.497
335.679
177.484
294.088
238.824
15.000
13
131.871
176.309
129.436
25.000
177.499
340.000
177.499
137.506
234.751
15.000
14
119.616
177.057
103.519
25.000
177.499
340.000
177.499
142.830
260.000
15.000
15
124.985
165.000
290.000
25.000
177.499
340.000
177.499
152.687
249.272
40.000
16
114.362
309.274
337.442
25.000
177.499
340.000
177.499
152.334
239.253
40.000
17
126.154
324.1193
377.603
25.000
177.499
340.000
177.499
158.274
254.610
40.000
18
126.332
308.015
275.423
25.000
166.567
340.000
177.499
152.908
258.483
40.000
19
125.370
187.353
288.980
25.000
177.499
340.000
177.499
123.743
260.000
15.000
20
129.077
185.934
140.000
22.658
177.499
338.663
177.499
129.939
260.000
15.000
21
103.540
154.576
301.215
25.000
167.760
332.659
177.500
149.915
267.546
15.000
22
128.134
151.202
140.000
08.000
177.499
340.000
157.499
151.103
250.829
15.000
23
118.221
165.000
140.000
08.000
177.499
331.385
168.524
147.996
291.043
15.000
24
131.233
165.000
140.000
25.000
176.952
340.000
177.493
142.536
250.797
15.000
Hour
Unit (MW)
11
12
13
14
15
16
17
18
19
1
050.036
75.000
177.499
56.908
032.620
80.000
80.000
079.895
460.000
2
050.000
40.000
130.805
95.000
043.997
15.000
80.000
073.000
420.219
3
150.000
75.000
177.500
95.000
049.824
80.000
80.000
230.000
443.210
4
150.000
75.000
177.499
95.000
141.701
80.000
80.000
230.000
473.308
5
150.000
75.000
177.500
95.000
220.000
80.000
80.000
230.000
466.529
6
150.000
75.000
177.500
95.000
220.000
80.000
80.000
230.000
483.000
7
150.000
75.000
177.499
95.000
220.000
80.000
80.000
230.000
491.740
8
150.000
75.000
177.499
95.000
220.000
80.000
80.000
230.000
487.647
9
050.000
75.000
177.499
95.000
187.521
80.000
80.000
230.000
430.000
10
150.000
75.000
171.776
95.000
121.250
80.000
80.000
230.000
460.000
11
150.000
75.000
177.499
81.780
040.653
80.000
80.000
230.000
469.862
12
079.616
75.000
177.488
87.838
036.233
80.000
80.000
111.209
460.000
13
050.000
25.000
177.499
65.000
041.758
80.000
80.000
092.411
460.000
14
108.595
75.000
177.499
65.000
044.047
80.000
80.000
087.993
460.000
15
150.000
75.000
177.499
95.000
028.027
80.000
80.000
184.753
433.063
16
150.000
75.000
177.499
95.000
182.323
80.000
80.000
230.000
480.285
17
150.000
75.000
177.499
95.000
198.027
80.000
80.000
230.000
483.000
18
150.000
75.000
177.499
95.000
094.367
80.000
80.000
230.000
466.193
19
150.000
75.000
177.499
95.000
091.068
80.000
80.000
230.000
464.949
20
150.000
75.000
177.499
31.432
035.402
80.000
80.000
230.000
438.682
21
060.000
75.000
113.750
65.000
044.040
80.000
80.000
073.000
460.000
22
150.000
75.000
112.426
35.000
044.592
80.000
80.000
073.000
380.409
23
50.000
75.000
113.750
94.352
033.301
15.000
44.215
073.000
460.000
24
50.000
75.000
177.499
95.000
044.034
45.000
16.283
050.000
454.875Table 4
Fuel switching of generators for case study 2.
Hour
Unit (MW)
3
5
7
13
19
1
1,1
,
2
2,4
,
2
2,6
,
2
2,2
,
2
1,1
,
2
2
2,1
,
2
2,4
,
2
2,2
,
2
2,1
,
2
1,1
,
1
3
1,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
4
1,2
,
2
2,2
,
2
2,2
,
2
6,2
,
2
1,1
,
1
5
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
6
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
7
2,2
,
2
2,2
,
2
6,2
,
2
2,2
,
6
1,1
,
1
8
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
9
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
10
2,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
11
2,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
12
2,1
,
2
2,2
,
2
2,2
,
6
2,2
,
2
2,1
,
1
13
2,1
,
1
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
14
2,1
,
1
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
15
2,2
,
1
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
16
2,2
,
1
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
17
2,2
,
1
2,2
,
2
2,2
,
2
2,2
,
1
2,1
,
1
18
2,2
,
2
2,6
,
2
2,2
,
6
2,2
,
2
1,1
,
1
19
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
20
2,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
21
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
2
1,1
,
1
22
2,1
,
2
2,2
,
2
2,6
,
6
6,1
,
2
1,1
,
1
23
1,1
,
2
2,2
,
2
2,6
,
2
2,1
,
2
1,1
,
1
24
1,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1Figure 5
Load demand patterns for case study 2.The best cost obtained by the DE-NM method for pattern 1, 2, and 3 is $404122.623, $324962.343, and $372140.528, respectively. The worst cost for pattern 1 is $40565.659, pattern 2 is $341440.818, and pattern 3 is $374405.694. Figures6(a), 6(b), and 6(c) shows the cost distribution for the system with pattern 1, pattern 2, and pattern 3, respectively. The straight line running through the cost distribution area represents the average cost. The average cost for pattern1 is 405515.7524, pattern 2 is $334321.1071, and pattern 3 is $33109.9977. The percentage of producing quality solutions by DE-NM for the system is above 70% as seen from Figure 7. Table 5 clearly shows the computational time of the three patterns for both the test systems. Table 6 presents the relationship between k, computation time, and result quality for a 10-unit system.Table 5
Computation time for both systems for 1000 iterations with a population size of 100.
10-unit system
19-unit system
Load pattern
Sec
Load pattern
Sec
1
13.581
1
32.939
2
12.121
2
33.782
3
11.465
3
34.566Table 6
Relationship betweenk, computation time, and result quality for a 10-unit system.
k
Time in Sec
Cost range
Pattern 1
10%
13.581
11640–11500
20%
96.32
11583–11472
30%
1134.15
11953–11678
Pattern 2
10%
12.121
8600–8000
20%
89.56
8578–7926
30%
985.23
8927–8712
Pattern 3
10%
11.465
11600–11000
20%
73.42
11578–11215
30%
921.73
11827–11749(a) Cost distribution obtained by DE-NM for case study 2 with pattern 1. (b) Cost distribution obtained by DE-NM for case study 2 with pattern 2. (c) Cost distribution obtained by DE-NM for case study 2 with pattern 3.
(a)
(b)
(c)Figure 7
Frequency of convergence for case study 2.


## 6.1. Description of the Test Systems

In this section two case systems are studied using DE-NM method. The fuel cost functions of the two systems are nonconvex considering valve point effect and multiple fuel option in its fuel cost.Three different load patterns are applied for both these systems to demonstrate the robustness of the proposed solution technique as well as to show the effectiveness in scheduling for different load patterns (situations). The simulation horizon for all the three load patterns is taken as 24 hr with a dispatch interval of 1 hr. The coding is written using the MATLAB 7 programming language and executed in a personal computer machine.


## 6.2. Parameter Selection

DE-NM consists of four new parameters in addition toF and CR as in classical DE. They are α, β, w, and K.  The role of α and β same as that of F. Thus, the number of parameters reduced by considering α=β=F. The most important parameter in DE-NM is the weight factor w, which controls the balance between the exploration and exploitation capabilities. To obtain the balance, value of w is usually selected from the range [0,1]. It is adapted online during the execution of the algorithm by a linear increment scheme; that is, it is initialized with 0 and increased up to 1. Since w starts with 0, it favors the exploration process during the initial stage and on getting promoted to 1 it favors the exploitation process. The linear increment scheme is given by
(24)wG=GGmax.In this paper, the neighborhood size equal to 10% of the total population size of 100 is considered. The value for bothα and β is taken as 0.8 and crossover rate Cr=0.9. The DE-NM is applied to both the case studies for 30 independent trials (1000 iterations per trial) with the selected parameters.


## 6.3. Case Study 1

In this test case, a 10-unit system is optimized to meet three different load patterns. The level of fluctuation of the three different load patterns is given in Figure2. The peak demand for load pattern 1, pattern 2, and pattern 3 is 3208 MW, 2460 MW, and 3210 MW, respectively. The system data and the related constraints for this case study are given in [30]. The complete generation schedule for load pattern 2 corresponding to the minimum generation cost obtained by DE-NM is presented in Table 1. Table 2 gives the fuel type chosen by the units for the corresponding best generation schedule obtained. The fuel chosen by the units for all three load patterns is given in Table 2. The table format represents the fuel type chosen by an unit for pattern 1 and it is followed by fuel type for the same unit for patterns 2 and 3, respectively.Table 1
Best power generation schedule obtained by DE-NM for case study 1.
Hour
Unit
1 (MW)
2 (MW)
3 (MW)
4 (MW)
5 (MW)
6 (MW)
7 (MW)
8 (MW)
9 (MW)
10 (MW)
1
168.862
177.000
250.000
250.000
240.000
166.000
250.000
180.570
243.000
250.000
2
163.886
177.000
250.000
250.000
234.691
163.000
250.000
173.854
253.000
250.000
3
165.705
177.000
250.000
250.000
235.047
160.543
250.000
178.591
253.000
250.000
4
165.826
177.000
250.000
250.000
240.000
163.000
250.000
181.576
249.509
250.000
5
172.019
177.000
250.000
250.000
457.000
150.375
250.000
183.538
252.999
247.372
6
192.564
177.000
269.658
265.000
451.287
196.324
250.000
249.687
302.656
254.658
7
181.538
177.000
253.465
253.465
446.999
180.580
250.000
246.700
218.376
250.000
8
172.014
177.000
250.000
250.000
457.000
163.000
250.000
176.040
249.509
250.000
9
171.935
177.000
250.000
250.000
240.000
153.000
250.000
183.521
253.000
250.000
10
170.438
177.000
250.000
250.000
240.000
163.000
250.000
176.607
253.000
246.927
11
179.939
177.000
253.528
253.528
457.000
174.322
250.000
191.455
273.573
250.000
12
165.807
177.000
250.000
250.000
234.616
163.000
250.000
180.585
251.231
247.169
13
178.362
177.000
251.413
251.413
457.000
161.753
250.000
190.465
270.278
250.000
14
205.211
177.000
250.000
250.000
457.000
163.000
282.078
254.494
345.626
264.559
15
205.220
177.000
310.646
265.000
457.000
224.225
277.774
254.494
406.509
273.310
16
194.144
177.000
274.350
265.000
457.000
199.127
262.903
250.463
304.055
246.630
17
178.354
177.000
261.510
261.510
457.000
162.985
250.000
247.775
266.450
246.924
18
167.277
177.000
250.000
250.000
233.617
163.000
250.000
179.579
253.000
246.382
19
163.876
177.000
250.000
250.000
451.287
163.000
250.000
177.557
248.712
250.000
20
171.953
177.000
250.000
250.000
235.047
160.543
250.000
184.512
253.000
250.000
21
168.889
177.000
250.000
250.000
234.691
163.000
250.000
181.563
253.000
250.000
22
178.361
177.000
252.512
252.512
240.000
171.466
250.000
192.440
263.232
246.630
23
165.864
177.000
250.000
250.000
240.000
156.074
250.000
183.479
253.000
246.925
24
176.904
177.000
250.000
250.000
457.000
158.553
250.000
181.986
251.996
250.000Table 2
Fuel switching of generators for case study 1.
Hour
Unit
1
2
3
4
5
6
7
8
9
10
1
2,1
,
1
1,1
,
1
1,1
,
1
3,3
,
2
3,3
,
1
3,1
,
1
2,1
,
1
3,2
,
2
3,1
,
1
1,1
,
1
2
2,1
,
1
1,1
,
1
1,1
,
1
3,3
,
2
3,3
,
1
3,1
,
1
1,1
,
1
3,3
,
2
3,1
,
1
1,1
,
1
3
1,1
,
1
1,1
,
1
1,1
,
1
2,3
,
2
1,3
,
1
1,1
,
1
1,1
,
1
2,2
,
2
1,1
,
1
1,1
,
1
4
1,1
,
1
1,1
,
1
1,1
,
1
2,2
,
2
1,3
,
1
1,1
,
1
1,1
,
1
2,2
,
2
1,1
,
1
1,1
,
1
5
1,1
,
1
1,1
,
1
1,1
,
1
3,2
,
2
1,3
,
3
1,1
,
1
1,1
,
1
3,3
,
2
1,1
,
1
1,1
,
1
6
1,1
,
2
1,1
,
1
1,1
,
1
3,3
,
3
1,3
,
3
1,3
,
1
1,1
,
1
2,3
,
3
1,3
,
1
1,1
,
1
7
1,1
,
2
1,1
,
1
1,3
,
1
3,3
,
3
3,3
,
3
1,3
,
1
1,3
,
1
2,3
,
3
1,3
,
1
1,1
,
1
8
1,1
,
2
1,1
,
1
1,2
,
1
2,3
,
2
3,3
,
3
1,3
,
1
1,3
,
1
2,3
,
2
1,3
,
1
1,2
,
1
9
2,1
,
2
1,1
,
1
1,2
,
1
3,3
,
2
1,3
,
1
3,3
,
1
1,3
,
1
3,3
,
2
3,3
,
1
1,2
,
1
10
2,1
,
2
1,1
,
1
1,2
,
1
3,3
,
2
3,3
,
1
3,3
,
1
1,3
,
1
3,3
,
2
1,3
,
1
1,3
,
1
11
2,1
,
2
1,1
,
1
1,1
,
1
3,3
,
1
3,3
,
3
3,3
,
1
1,3
,
1
3,3
,
2
1,3
,
1
1,1
,
1
12
2,1
,
2
1,1
,
1
1,1
,
1
3,3
,
2
3,3
,
1
3,3
,
1
1,1
,
1
3,3
,
2
1,3
,
1
1,1
,
1
13
2,1
,
2
1,1
,
1
1,1
,
1
3,3
,
3
3,3
,
3
3,3
,
1
1,1
,
1
3,3
,
2
3,3
,
1
1,1
,
1
14
2,2
,
1
1,1
,
1
1,1
,
1
3,2
,
3
3,3
,
3
3,1
,
1
1,1
,
1
3,2
,
3
3,1
,
1
1,1
,
1
15
2,2
,
1
1,1
,
1
1,1
,
1
3,3
,
3
3,1
,
3
3,3
,
3
1,1
,
1
3,2
,
3
3,1
,
1
1,1
,
1
16
2,1
,
1
1,1
,
1
2,1
,
1
3,2
,
3
3,3
,
3
3,1
,
1
2,1
,
1
3,3
,
3
3,1
,
1
1,1
,
1
17
2,1
,
1
1,1
,
1
2,1
,
1
3,3
,
2
3,3
,
3
3,3
,
1
3,1
,
1
3,3
,
3
3,1
,
1
1,1
,
1
18
2,1
,
2
1,1
,
1
2,1
,
1
3,3
,
2
3,3
,
1
3,3
,
1
3,1
,
1
3,3
,
2
3,1
,
1
2,1
,
1
19
2,1
,
2
11,1
1,3
,
1
2,3
,
2
3,3
,
3
3,3
,
1
2,3
,
1
3,3
,
2
3,3
,
1
1,1
,
1
20
2,1
,
2
1,1
,
1
1,1
,
1
2,3
,
2
3,3
,
1
3,3
,
1
1,1
,
1
3,3
,
2
3,3
,
1
1,1
,
1
21
2,1
,
2
1,1
,
1
1,1
,
1
2,3
,
2
3,3
,
1
3,3
,
1
1,1
,
1
3,3
,
2
3,3
,
1
1,1
,
1
22
1,1
,
2
1,1
,
1
1,1
,
1
2,3
,
3
3,3
,
1
1,3
,
1
1,1
,
1
3,3
,
2
1,3
,
1
1,1
,
1
23
1,1
,
1
1,1
,
1
1,1
,
1
2,3
,
2
1,3
,
1
1,3
,
1
1,1
,
1
3,3
,
2
1,1
,
1
1,1
,
1
24
1,1
,
1
1,1
,
1
1,1
,
1
2,3
,
2
1,3
,
3
1,1
,
1
1,1
,
1
2,3
,
2
1,1
,
1
1,1
,
1Figure 2
Load demand patterns for case study 1.The best cost obtained by the DE-NM method considering spinning reserve for pattern 1 is $10649.68, for pattern 2 is $8103.326, and for pattern 3 is $11054.83. The worst cost obtained by the DE-NM method for pattern 1 is $11952.07, pattern 2 is $9163.973, and pattern 3 is $12036.89. The average cost of pattern 1, pattern 2, and pattern 3 for all the 30 trials is $11451.82, $9711.448, and $11497.85.  Figures3(a), 3(b), and 3(c) shows the distribution of the cost obtained for load patterns 1, 2, and 3, respectively. A straight line passing through each distribution area denotes the average cost. The percentage of producing quality solutions by DE-NM is above 60% as seen from Figure 4.(a) Cost distribution obtained by DE-NM for case study 1 with pattern 1. (b) Cost distribution obtained by DE-NM for case study 1 with pattern 2. (c) Cost distribution obtained by DE-NM for case study 1 with pattern 3.
(a)
(b)
(c)Figure 4
Frequency of convergence for case study 1.


## 6.4. A Short Overview of the Indian Utility System

Indian power system is a geographically dispersed network of generators. For ease of operation, the entire Indian power system has been divided into five regions, namely, Northern Region (NR), Western Region (WR), Southern Region (SR), Eastern Region (ER), and North Eastern Region (NER). While NR, WR, ER, and NER are interconnected as a single grid, the SR had not been interconnected with the national grid. This prevents the SR from receiving benefits from national power market. The SR covering 6, 51,000 Sq. Km of area comprises the states of Andhra Pradesh, Karnataka, Kerala, Pondicherry and Tamil Nadu. The Indian utility system considered in this paper belongs to the state of Tamil Nadu. The Electricity Board of Tamil Nadu state was recently restructured and the board is in the move of handing over transmission and distribution to the private operators.The South Indian network comprises of 19 thermal units, located at various parts of the state of Tamil Nadu. Since the system under consideration belongs to the state department, the technical data of the system cannot be published is available with the corresponding author, and will be provided on request. Also the data of system network is approximated neglecting several local feeder data this work neglects estimation of transmission losses for simplicity. Even then, as per the board’s advice, this work assumes a uniform 20% loss in the transmission and hence it is added to the 24-hour load demand data for scheduling.


## 6.5. Case Study 2

This system involves a standard Indian utility system with 19 units. All the 19 units include nonlinear characteristics such as valve point effect, ramp rate limit, multiple fuels, prohibited operating zones and spinning reserve constraint. This system is applied for three different load patterns as shown in Figure5. The demand for each hour includes approximately 20% of transmission losses. A generation schedule is obtained using the DE-NM method where each unit contributes to the demand and the transmission loss.  The maximum demand for pattern 1 is 4400 MW, for pattern 2 is 4186 MW, and for pattern 3 is 4173 MW. Table 3 gives the best generation schedule obtained by the DE-NM method for pattern 2. Few units in the system are provided with multiple fuel option. The fuel types include 1, 2, 3, 4, 5 and 6. Table 4 shows the fuel switching for units 3, 5, 7, 13, and 19 and all the remaining units utilize fuel 1 for every hour of time interval.Table 3
Best power generation schedule obtained by DE-NM for case study 2.
Hour
Power output (MW)
1
2
3
4
5
6
7
8
9
10
1
126.623
176.332
126.942
25.000
113.750
340.000
166.554
150.000
242.127
15.000
2
124.189
165.000
144.530
08.000
146.176
340.000
177.498
138.874
254.528
15.000
3
127.640
161.742
140.000
25.000
177.500
334.685
177.500
152.687
283.000
15.000
4
132.015
287.062
368.136
25.000
175.995
340.000
177.499
165.069
260.000
40.000
5
148.315
421.843
400.000
25.000
177.500
340.000
174.898
193.407
280.290
40.000
6
227.936
438.000
440.000
25.000
177.500
340.000
176.290
321.681
302.381
40.000
7
292.692
438.000
400.000
25.000
177.499
363.000
177.499
452.292
516.598
40.000
8
300.000
438.000
400.000
25.000
177.500
371.145
177.499
500.000
532.465
40.000
9
120.045
204.725
374.718
25.000
177.500
338.122
177.499
144.607
283.000
40.000
10
129.676
196.189
202.088
25.000
177.500
340.000
177.500
153.581
257.593
40.000
11
125.834
137.702
136.903
25.000
177.499
340.000
177.499
152.633
254.420
15.000
12
126.297
149.825
140.000
25.000
177.497
335.679
177.484
294.088
238.824
15.000
13
131.871
176.309
129.436
25.000
177.499
340.000
177.499
137.506
234.751
15.000
14
119.616
177.057
103.519
25.000
177.499
340.000
177.499
142.830
260.000
15.000
15
124.985
165.000
290.000
25.000
177.499
340.000
177.499
152.687
249.272
40.000
16
114.362
309.274
337.442
25.000
177.499
340.000
177.499
152.334
239.253
40.000
17
126.154
324.1193
377.603
25.000
177.499
340.000
177.499
158.274
254.610
40.000
18
126.332
308.015
275.423
25.000
166.567
340.000
177.499
152.908
258.483
40.000
19
125.370
187.353
288.980
25.000
177.499
340.000
177.499
123.743
260.000
15.000
20
129.077
185.934
140.000
22.658
177.499
338.663
177.499
129.939
260.000
15.000
21
103.540
154.576
301.215
25.000
167.760
332.659
177.500
149.915
267.546
15.000
22
128.134
151.202
140.000
08.000
177.499
340.000
157.499
151.103
250.829
15.000
23
118.221
165.000
140.000
08.000
177.499
331.385
168.524
147.996
291.043
15.000
24
131.233
165.000
140.000
25.000
176.952
340.000
177.493
142.536
250.797
15.000
Hour
Unit (MW)
11
12
13
14
15
16
17
18
19
1
050.036
75.000
177.499
56.908
032.620
80.000
80.000
079.895
460.000
2
050.000
40.000
130.805
95.000
043.997
15.000
80.000
073.000
420.219
3
150.000
75.000
177.500
95.000
049.824
80.000
80.000
230.000
443.210
4
150.000
75.000
177.499
95.000
141.701
80.000
80.000
230.000
473.308
5
150.000
75.000
177.500
95.000
220.000
80.000
80.000
230.000
466.529
6
150.000
75.000
177.500
95.000
220.000
80.000
80.000
230.000
483.000
7
150.000
75.000
177.499
95.000
220.000
80.000
80.000
230.000
491.740
8
150.000
75.000
177.499
95.000
220.000
80.000
80.000
230.000
487.647
9
050.000
75.000
177.499
95.000
187.521
80.000
80.000
230.000
430.000
10
150.000
75.000
171.776
95.000
121.250
80.000
80.000
230.000
460.000
11
150.000
75.000
177.499
81.780
040.653
80.000
80.000
230.000
469.862
12
079.616
75.000
177.488
87.838
036.233
80.000
80.000
111.209
460.000
13
050.000
25.000
177.499
65.000
041.758
80.000
80.000
092.411
460.000
14
108.595
75.000
177.499
65.000
044.047
80.000
80.000
087.993
460.000
15
150.000
75.000
177.499
95.000
028.027
80.000
80.000
184.753
433.063
16
150.000
75.000
177.499
95.000
182.323
80.000
80.000
230.000
480.285
17
150.000
75.000
177.499
95.000
198.027
80.000
80.000
230.000
483.000
18
150.000
75.000
177.499
95.000
094.367
80.000
80.000
230.000
466.193
19
150.000
75.000
177.499
95.000
091.068
80.000
80.000
230.000
464.949
20
150.000
75.000
177.499
31.432
035.402
80.000
80.000
230.000
438.682
21
060.000
75.000
113.750
65.000
044.040
80.000
80.000
073.000
460.000
22
150.000
75.000
112.426
35.000
044.592
80.000
80.000
073.000
380.409
23
50.000
75.000
113.750
94.352
033.301
15.000
44.215
073.000
460.000
24
50.000
75.000
177.499
95.000
044.034
45.000
16.283
050.000
454.875Table 4
Fuel switching of generators for case study 2.
Hour
Unit (MW)
3
5
7
13
19
1
1,1
,
2
2,4
,
2
2,6
,
2
2,2
,
2
1,1
,
2
2
2,1
,
2
2,4
,
2
2,2
,
2
2,1
,
2
1,1
,
1
3
1,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
4
1,2
,
2
2,2
,
2
2,2
,
2
6,2
,
2
1,1
,
1
5
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
6
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
7
2,2
,
2
2,2
,
2
6,2
,
2
2,2
,
6
1,1
,
1
8
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
9
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
10
2,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
11
2,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
12
2,1
,
2
2,2
,
2
2,2
,
6
2,2
,
2
2,1
,
1
13
2,1
,
1
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
14
2,1
,
1
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
15
2,2
,
1
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
16
2,2
,
1
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
1
17
2,2
,
1
2,2
,
2
2,2
,
2
2,2
,
1
2,1
,
1
18
2,2
,
2
2,6
,
2
2,2
,
6
2,2
,
2
1,1
,
1
19
2,2
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
20
2,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1
21
2,2
,
2
2,2
,
2
2,2
,
2
2,1
,
2
1,1
,
1
22
2,1
,
2
2,2
,
2
2,6
,
6
6,1
,
2
1,1
,
1
23
1,1
,
2
2,2
,
2
2,6
,
2
2,1
,
2
1,1
,
1
24
1,1
,
2
2,2
,
2
2,2
,
2
2,2
,
2
1,1
,
1Figure 5
Load demand patterns for case study 2.The best cost obtained by the DE-NM method for pattern 1, 2, and 3 is $404122.623, $324962.343, and $372140.528, respectively. The worst cost for pattern 1 is $40565.659, pattern 2 is $341440.818, and pattern 3 is $374405.694. Figures6(a), 6(b), and 6(c) shows the cost distribution for the system with pattern 1, pattern 2, and pattern 3, respectively. The straight line running through the cost distribution area represents the average cost. The average cost for pattern1 is 405515.7524, pattern 2 is $334321.1071, and pattern 3 is $33109.9977. The percentage of producing quality solutions by DE-NM for the system is above 70% as seen from Figure 7. Table 5 clearly shows the computational time of the three patterns for both the test systems. Table 6 presents the relationship between k, computation time, and result quality for a 10-unit system.Table 5
Computation time for both systems for 1000 iterations with a population size of 100.
10-unit system
19-unit system
Load pattern
Sec
Load pattern
Sec
1
13.581
1
32.939
2
12.121
2
33.782
3
11.465
3
34.566Table 6
Relationship betweenk, computation time, and result quality for a 10-unit system.
k
Time in Sec
Cost range
Pattern 1
10%
13.581
11640–11500
20%
96.32
11583–11472
30%
1134.15
11953–11678
Pattern 2
10%
12.121
8600–8000
20%
89.56
8578–7926
30%
985.23
8927–8712
Pattern 3
10%
11.465
11600–11000
20%
73.42
11578–11215
30%
921.73
11827–11749(a) Cost distribution obtained by DE-NM for case study 2 with pattern 1. (b) Cost distribution obtained by DE-NM for case study 2 with pattern 2. (c) Cost distribution obtained by DE-NM for case study 2 with pattern 3.
(a)
(b)
(c)Figure 7
Frequency of convergence for case study 2.


## 7. Conclusion

A maiden attempt has been taken to apply Differential Evolution with Neighborhood based Mutation (DE-NM) method to solve Dynamic Economic Dispatch (DED) problem including ramp rate effects, prohibited operating zones, spinning reserve, and multiple fuel options under a single frame. To show the effectiveness of the DE-NM method, an Indian utility system and a 10-unit system are presented to solve the DED problem for a given load profile. The results show that DE-NM is efficient in handling the constraints and it is applicable to larger systems. Due to enormous transmission lines in the Indian utility system, a proper model for including the losses is still under the search. The search ability of the DE-NM is improved by striking proper balance between exploration and exploitation process. The simulation results show that the DE-NM method is capable of producing solutions which are near optimal and has stable converging characteristics.



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*Source: 423426-2014-01-12.xml*