Opposition-based differential evolution for hydrothermal power system

Abstract

This paper presents opposition-based differential evolution to determine the optimal hourly schedule of power generation in a hydrothermal system. Differential evolution (DE) is a population-based stochastic parallel search evolutionary algorithm. Opposition-based differential evolution has been used here to improve the effectiveness and quality of the solution. The proposed opposition-based differential evolution (ODE) employs opposition-based learning (OBL) for population initialization and also for generation jumping. The effectiveness of the proposed method has been verified on two test problems, two fixed head hydrothermal test systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed opposition-based differential evolution based approach is able to provide better solution.

Introduction

Optimal scheduling of power plant generation is of great importance to electric utility systems. Because of insignificant marginal cost of hydroelectric power, the problem of minimizing the operational cost of hydrothermal system essentially reduces to that of minimizing the fuel cost of thermal plants under the various constraints on the hydraulic, thermal and power system network.

The hydrothermal scheduling problem has been the subject of investigation for several decades. Several classical methods such as Newton’s method [1], mixed integer programming [2, 3], dynamic programming (DP) [4], etc. have been widely used to solve hydrothermal scheduling problem. Among these methods, DP appears to be the most popular. However, major disadvantages of DP method are computational and dimensional requirements which grow drastically with increasing system size and planning horizon.

Recently, stochastic search algorithms such as simulated annealing (SA) [5], evolutionary programming (EP) [6], genetic algorithm (GA) [7, 8], evolutionary programming technique [9], differential evolution (DE) [10,11,12], particle swarm optimization [13], artificial immune system [14], clonal selection algorithm [15] and teaching learning based optimization [16] have been successfully used to solve hydrothermal scheduling problem.

Since the mid 1990s, many techniques originated from Darwin’s natural evolution theory have emerged. These techniques are usually termed by “evolutionary computation methods” including evolutionary algorithms (EAs), swarm intelligence and artificial immune system. Differential evolution (DE) [17,18,19,20], a relatively new member in the family of evolutionary algorithms, first proposed over 1995–1997 by Storn and Price at Berkeley is a novel approach to numerical optimization. It is a population-based stochastic parallel search evolutionary algorithm which is very simple yet powerful. The main advantages of DE are its capability of solving optimization problems which require minimization process with nonlinear, non-differentiable and multi-modal objective functions.

The basic concept of opposition-based learning (OBL) [21,22,23] was originally introduced by Tizhoosh. The main idea behind OBL is for finding a better candidate solution and the simultaneous consideration of an estimate and its corresponding opposite estimate (i.e., guess and opposite guess) which is closer to the global optimum. OBL was first utilized to improve learning and back propagation in neural networks by Ventresca and Tizhoosh [24], and since then, it has been applied to many EAs, such as differential evolution [25], particle swarm optimization [26] and ant colony optimization [27].

Opposition-based harmony search algorithm [28] has been applied to solve combined economic and emission dispatch problems. In [29] oppositional real coded chemical reaction optimization has been used for solving economic dispatch problems. Opposition-based gravitational search algorithm [30] has been applied for solving reactive power dispatch problem.

This paper proposes opposition-based differential evolution (ODE) for optimal scheduling of generation in a hydrothermal system. This paper considers fixed head as well as variable head hydrothermal system. In case of fixed head hydro plants, water discharge rate curves are modeled as a quadratic function of the hydropower generation and thermal units with nonsmooth fuel cost function. Here, scheduling period is divided into a number of subintervals each having a constant load demand. In case of variable head hydrothermal system, multi-reservoir cascaded hydro plants having prohibited operating zones and thermal units with valve point loading are used. The proposed method is validated by applying it to two test problems, two fixed head hydrothermal test systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading. The test results are compared with those obtained by other evolutionary methods reported in the literature. From numerical results, it is found that the proposed ODE based approach provides better solution.

Problem formulation

Fixed head hydrothermal system

Fixed head hydrothermal scheduling problem with Ν _h hydro units and Ν _s thermal units over M time subintervals is described as follows:

Objective function

The fuel cost function of each thermal generator, considering valve-point effect, is expressed as a sum of quadratic and sinusoidal function. The superimposed sine components represent rippling effect produced by steam admission valve opening. The problem minimizes following total fuel cost

$$ {f}_{FH}={\displaystyle \sum_{m=1}^{\mathrm{M}}{\displaystyle \sum_{i=1}^{{\mathrm{N}}_s}{t}_m\left[{a}_{s i}+{b}_{s i}{\mathrm{P}}_{s i m}+{c}_{s i}{\mathrm{P}}_{s i m}^2\kern0.5em +\kern0.5em \right.}}\left.{d}_{s i}\times \sin \left\{{e}_{s i}\times \left({\mathrm{P}}_{s i}^{\min }-{\mathrm{P}}_{s i m}\right)\right\}\right] $$

(1)

Constraints

(i) Power balance constraints:

$$ {\displaystyle \sum_{i=1}^{{\mathrm{N}}_s}{\mathrm{P}}_{s im}+}{\displaystyle \sum_{j=1}^{{\mathrm{N}}_h}{\mathrm{P}}_{h jm}}-{\mathrm{P}}_{Dm}-{\mathrm{P}}_{Lm}=0\kern1.0em m\in \mathrm{M} $$

(2)

and

$$ {P}_{Lm}={\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s}{\displaystyle \sum_{r=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s}{\mathrm{P}}_{l m}{\mathrm{B}}_{l r}{\mathrm{P}}_{r m}}}\kern1.0em m\in \mathrm{M} $$

(3)

(ii) Water availability constraints:

$$ {\displaystyle \sum_{m=1}^{\mathrm{M}}\left[{t}_m\left({a}_{0 hj}+{a}_{1 hj}{\mathrm{P}}_{h j m}+{a}_{2 hj}{\mathrm{P}}_{h j m}^2\right)\right]-{W}_{h j}=0}\kern1.0em j\in {\mathrm{N}}_h $$

(4)

(iii) Generation limits:

$$ {\mathrm{P}}_{h j}^{\min}\le {\mathrm{P}}_{h j m}\le {\mathrm{P}}_{h j}^{\max}\kern2.5em j\in {\mathrm{N}}_h,\kern0.5em m\in \mathrm{M} $$

(5)

and

$$ {\mathrm{P}}_{s i}^{\min}\le {\mathrm{P}}_{s i m}\le {\mathrm{P}}_{s i}^{\max}\kern2.5em i\in {\mathrm{N}}_s,\kern0.5em m\in \mathrm{M} $$

(6)

Determination of generation level of slack generator

Thermal generators and hydro generators deliver their power output subject to the power balance constraint (2), water availability constraint (4) and respective capacity constraints (5) and (6). Assuming the power loading of Ν _Ρ and first (Ν _s - 1) generators are known, the power level of the Ν _s th generator (i.e. the slack generator) is given by

$$ {\mathrm{P}}_{{\mathrm{N}}_s m}={\mathrm{P}}_{Dm}+{\mathrm{P}}_{Lm}-{\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\mathrm{P}}_{l m}}\kern2.5em m\in \mathrm{M} $$

(7)

The transmission loss Ρ _Lm is a function of all the generators including the slack generator and it is given by

$$ {\mathrm{P}}_{Lm}={\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\displaystyle \sum_{r=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\mathrm{P}}_{l m}{\mathrm{B}}_{l r}{\mathrm{P}}_{r m}}}\kern0.5em +\kern0.5em 2{\mathrm{P}}_{{\mathrm{N}}_s m}\left({\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\mathrm{B}}_{{\mathrm{N}}_s l}{\mathrm{P}}_{l m}}\right)+{\mathrm{B}}_{{\mathrm{N}}_s{\mathrm{N}}_s}{\mathrm{P}}_{{\mathrm{N}}_s m}^2\kern2.5em m\in \mathrm{M} $$

(8)

Expanding and rearranging, equation (7) becomes

$$ {\mathrm{B}}_{{\mathrm{N}}_s{\mathrm{N}}_s}{\mathrm{P}}_{{\mathrm{N}}_s m}^2+\left(2{\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\mathrm{B}}_{{\mathrm{N}}_s l}{\mathrm{P}}_{l m}-1}\right){\mathrm{P}}_{{\mathrm{N}}_s m}+{P}_{Dm}+\kern0.5em {\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\displaystyle \sum_{r=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\mathrm{P}}_{l m}{\mathrm{B}}_{l r}{\mathrm{P}}_{r m}-{\displaystyle \sum_{l=1}^{{\mathrm{N}}_h+{\mathrm{N}}_s-1}{\mathrm{P}}_{l m}}=0}}\kern2.5em m\in \mathrm{M} $$

(9)

The loading of the slack generator (i.e. Ν _s th) can then be found by solving equation (9) using standard algebraic method.

Variable head hydrothermal system

The variable head hydrothermal scheduling problem is aimed to minimize the fuel cost of thermal plants, while making use of the availability of hydro power as much as possible. The objective function and associated constraints of the hydrothermal scheduling problem are formulated as follows.

Objective function

$$ \mathrm{Minimize}\kern0.5em {f}_{VH}={\displaystyle \sum_{t=1}^{\mathrm{T}}{\displaystyle \sum_{i=1}^{{\mathrm{N}}_s}\left[{a}_{s i}\right.}}+{b}_{s i}{\mathrm{P}}_{s i t}+{c}_{s i}{\mathrm{P}}_{s i t}^2+\kern0.5em \left.\left|{d}_{s i}\times \sin \left\{{e}_{s i}\times \left({\mathrm{P}}_{s i}^{\min }-{\mathrm{P}}_{s i t}\right)\right\}\right|\right] $$

(10)

Constraints

(i) Power balance constraints:

The total active power generation must balance the predicted power demand and transmission loss, at each time interval over the scheduling horizon

$$ {\displaystyle \sum_{i=1}^{{\mathrm{N}}_s}{P}_{s it}+}{\displaystyle \sum_{j=1}^{{\mathrm{N}}_h}{\mathrm{P}}_{h jt}}-{\mathrm{P}}_{Dt}-{\mathrm{P}}_{Lt}=0\kern2.5em t\in \mathrm{T} $$

(11)

The hydroelectric generation is a function of water discharge rate and reservoir water head, which in turn, is a function of storage.

$$ {\mathrm{P}}_{h jt}={C}_{1 j}{V}_{h jt}^2+{C}_{2 j}{Q}_{h jt}^2+{C}_{3 j}{V}_{h jt}{Q}_{h jt}+{C}_4{V}_{h jt}\kern0.5em +\kern0.5em {C}_{5 j}{Q}_{h jt}+{C}_{6 j}\kern0.5em j\in {\mathrm{N}}_h\kern2.5em t\in \mathrm{T} $$

(12)

The transmission loss Ρ _Lt is given by

$$ {P}_{Lt}={\displaystyle \sum_{i=1}^{{\mathrm{N}}_s+{\mathrm{N}}_h}{\displaystyle \sum_{j=1}^{{\mathrm{N}}_s+{\mathrm{N}}_h}{\mathrm{P}}_{i t}{\mathrm{B}}_{i j}{\mathrm{P}}_{j t}+{\displaystyle \sum_{i=1}^{{\mathrm{N}}_s+{\mathrm{N}}_h}{\mathrm{B}}_{0 i}{\mathrm{P}}_{i t}}}}+{\mathrm{B}}_{00} $$

(13)

(ii) Generation limits:

$$ {\mathrm{P}}_{h j}^{\min}\le {\mathrm{P}}_{h j t}\le {\mathrm{P}}_{h j}^{\max },\kern2.5em j\in {\mathrm{N}}_h,\kern0.5em \mathrm{t}\in T $$

(14)

and

$$ {\mathrm{P}}_{s i}^{\min}\le {\mathrm{P}}_{s i t}\le {\mathrm{P}}_{s i}^{\max },\kern2.5em i\in {\mathrm{N}}_s,\kern0.5em t\in \mathrm{T} $$

(15)

(iii) Hydraulic network constraints

The hydraulic operational constraints comprise the water balance equations for each hydro unit as well as the bounds on reservoir storage and release targets. These bounds are determined by the physical reservoir and plant limitations as well as the multipurpose requirements of the hydro system. These constraints include:

(a) Physical limitations on reservoir storage volumes and discharge rates,

$$ {V}_{h j}^{\min}\le {V}_{h j t}\le {V}_{h j}^{\max },\kern2.5em j\in {\mathrm{N}}_h,\kern2.5em t\in \mathrm{T} $$

(16)

$$ {Q}_{h j}^{\min}\le {Q}_{h j t}\le {Q}_{h j}^{\max },\kern2.5em j\in {\mathrm{N}}_h,\kern2.5em t\in \mathrm{T} $$

(17)

b) The continuity equation for the hydro reservoir network

$$ {V}_{h j\left( t+1\right)}={V}_{h j t}+{\mathrm{I}}_{h j t}-{Q}_{h j t}-{S}_{h j t}+\kern0.5em {\displaystyle \sum_{l=1}^{R_{uj}}\left({Q}_{h l\left( t-{\tau}_{l j}\right)}+{S}_{h l\left( t-{\tau}_{l j}\right)}\right)},\kern1.0em j\in {\mathrm{N}}_h,\kern1.0em t\in \mathrm{T} $$

(18)

(iv) Prohibited operating regions of water discharge-rates

$$ {Q}_{hj}\in \left\{\begin{array}{l}{Q}_{hj}^{\min}\le {Q}_{hj}\le {Q}_{hj,1}^L\\ {}{Q}_{hj, k-1}^U\le {Q}_{hj}\le {Q}_{hj, k}^L\\ {}{Q}_{hj,{n}_j}^U\le {Q}_{hj}\le {Q}_{hj}^{\max}\end{array}\right., k=2,\dots {n}_j $$

(19)

Description of opposition-based differential evolution

A brief description of differential evolution

Differential Evolution (DE) is a type of evolutionary algorithm originally proposed by Price and Storn [19] for optimization problems over a continuous domain. DE is exceptionally simple, significantly faster and robust. The basic idea of DE is to adapt the search during the evolutionary process. At the start of the evolution, the perturbations are large since parent populations are far away from each other. As the evolutionary process matures, the population converges to a small region and the perturbations adaptively become small. As a result, the evolutionary algorithm performs a global exploratory search during the early stages of the evolutionary process and local exploitation during the mature stage of the search. In DE the fittest of an offspring competes one-to-one with that of corresponding parent which is different from other evolutionary algorithms. This one-to-one competition gives rise to faster convergence rate. Price and Storn gave the working principle of DE with simple strategy in [19]. Later on, they suggested ten different strategies of DE [18]. Strategy-7 (DE/rad/1/bin) is the most successful and widely used strategy. The key parameters of control in DE are population size (Ν _Ρ ), scaling factor (F) and crossover rate (C _R ). The optimization process in DE is carried out with three basic operations: mutation, crossover and selection. The DE algorithm is described as follows:

Initialization

The initial population of Ν _Ρ vectors is randomly selected based on uniform probability distribution for all variables to cover the entire search uniformly. Each individual Χ _i is a vector that contains as many parameters as the problem decision variables D. Random values are assigned to each decision parameter in every vector according to:

$$ {\mathrm{X}}_{ij}^0\sim U\left({\mathrm{X}}_j^{\min },{\mathrm{X}}_j^{\max}\right) $$

(20)

where i =1,…., Ν _Ρ and j =1,…., D; ${X}_j^{\min }$ and ${X}_j^{\max }$ are the lower and upper bounds of the j th decision variable; $U\left({\mathrm{X}}_j^{\min },{\mathrm{X}}_j^{\max}\right)$ denotes a uniform random variable ranging over $\left[{\mathrm{X}}_j^{\min },{\mathrm{X}}_j^{\max}\right]$. ${X}_{ij}^0$ is the initial j th variable of i th population. All the vectors should satisfy the constraints. Evaluate the value of the cost function $f\left({\mathrm{X}}_i^0\right)$ of each vector.

Mutation

DE generates new parameter vectors by adding the weighted difference vector between two population members to a third member. For each target vector ${X}_i^k$ at k th iteration the noisy vector ${X}_i^{/ k}$ is obtained by

$$ {\mathrm{X}}_i^{/ k}={\mathrm{X}}_a^k+ F\left({\mathrm{X}}_b^k-{\mathrm{X}}_c^k\right),\kern2.5em i\in {\mathrm{N}}_P $$

(21)

where $X{}_a{}^k$, ${X}_b^k$ and ${X}_c^k$ are selected randomly from Ν _Ρ vectors at k th iteration and a ≠ b ≠ c ≠ i. The scaling factor (F), in the range 0 < F ≤ 1.2, controls the amount of perturbation added to the parent vector. The noisy vectors should satisfy the constraint.

Crossover

Perform crossover for each target vector ${X}_i^k$ with its noisy vector ${X}_i^{/ k}$ and create a trial vector ${X}_i^{// k}$ such that

$$ {X}_i^{// k}=\left\{\begin{array}{llll}\hfill & \hfill & \hfill & \hfill \\ {}{X}_i^{/ k},\hfill & \mathrm{if}\hfill & \rho \le {C}_R\hfill & \hfill \\ {}\hfill & \hfill & \hfill &, i\in {N}_P\hfill \\ {}{X}_i^k,\hfill & \mathrm{otherwise}\hfill & \hfill & \hfill \end{array}\right. $$

(22)

where ρ is an uniformly distributed random number within [0, 1]. The crossover constant (C _R ), in the range 0 ≤ C _R  ≤ 1, controls the diversity of the population and aids the algorithm to escape from local optima.

Selection

Perform selection for each target vector, ${X}_i^k$ by comparing its cost with that of the trial vector, ${X}_i^{// k}$. The vector that has lesser cost of the two would survive for the next iteration.

$$ {X}_i^{k+1}=\left\{\begin{array}{ccc}\hfill {X}_i^{// k},\hfill & \hfill \mathrm{if}\hfill & \hfill f\left({X}_i^{// k}\right)\le f\left({X}_i^k\right)\hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill i\in {N}_P\hfill \\ {}\hfill {X}_i^k,\hfill & \hfill \mathrm{otherwise}\hfill & \hfill \hfill \end{array}\right. $$

(23)

The process is repeated until the maximum number of iterations or no improvement is seen in the best individual after many iterations.

Figure 1 shows the flowchart of differential evolution.

Fig. 1

Flowchart of differential evolution

Opposition-based learning

Opposition-based learning (OBL) was developed by Tizhoosh to improve candidate solution by considering current population as well as its opposite population at the same time.

Evolutionary optimization methods start with some initial population and try to improve them toward some optimal solution. The process of searching terminates when some predefined criteria are satisfied. The process is started with random guesses in the absence of a priori information about the solution. The process can be improved by starting with a closer i.e. fitter solution by simultaneously checking the opposite solution. By doing this, the fitter one (guess or opposite guess) may be chosen as an initial solution. According to the theory of probability, 50% of the time, a guess is further from the solution than its opposite guess. Therefore, process starts with the closer of the two guesses. The same approach can be applied not only to the initial solution but also continuously to each solution in the current population.

Definition of opposite number

If x be a real number between [lb, ub], its opposite number is defined as

$$ \overline{x}= lb+ l u- x $$

(24)

Similarly, this definition can be extended to higher dimensions [21] as stated in the next sub-section.

Definition of opposite point

Let X = (x ₁, x ₂,...., x _n ) be a point in n - dimensional space where x _i  ∈ [lb _i , ub _i ] and i ∈ 1, 2, …, n. The opposite point $\overline{X}=\left(\overline{x_1},\overline{x_2},.....,\overline{x_n}\right)$ is completely defined by its components as in (25).

$$ \overline{x_i}= l{b}_i+ u{b}_i-{x}_i $$

(25)

By employing the definition of opposite point, the opposition-based optimization is defined in the following sub-section.

Opposition-based optimization

Let X = (x ₁, x ₂,...., x _n ) be a point in n - dimensional space i.e. a candidate solution. Assume f = (•) is a fitness function which is used to measure the candidate’s fitness. According to the definition of the opposite point, $\overline{X}=\left(\overline{x_1},\overline{x_2},.....,\overline{x_n}\right)$ is the opposite of X = (x ₁, x ₂,...., x _n ). Now, if $f\left(\overline{X}\right)< f(X)$ (for a minimization problem), then point X can be replaced with $\overline{X}$; otherwise, the process is continued with X. Hence, the point and its opposite point are evaluated simultaneously in order to continue with the fitter one.

Opposition-based differential evolution

In the present work, the concept of the opposition-based learning [21] is incorporated in differential evolution. The original DE is chosen as a parent algorithm and the opposition-based ideas are embedded in DE.

Figure 2 shows the flowchart of ODE algorithm.

Fig. 2

Flowchart of ODE

Simulation results

Two test problems, two fixed head hydrothermal systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading are investigated. The computational results have been used to compare the performance of the proposed ODE method with that of other evolutionary methods. The proposed ODE algorithm and DE algorithm used in this paper are implemented by using MATLAB 7.0 on a PC (Pentium-IV, 80 GB, 3.0 GHz).

Simple examples

Example 1: Consider the maximization problem [31]

$$ \underset{x_1,{x}_2}{ \max } f\left({x}_1,{x}_2\right)=21.5+{x}_1 \sin \left(4\pi {x}_1\right)+{x}_2 \sin \left(20\pi {x}_2\right) $$

(26)

where − 3.0 ≤ x ₁ ≤ 12.1 and 4.1 ≤ x ₂ ≤ 5.8

This function is multimodal. The problem is solved by using ODE.

Here, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected 10, 0.3, 1.0 and 50 respectively. The best optimum value, the variables corresponding to the best optimum value, average and worst value and average CPU time among 100 runs of solutions obtained from proposed ODE and DE for example 1 have been shown in Table 1. Figure 3 shows the nature of convergence obtained from ODE and DE for example 1.

Table 1 Best optimum value, the variables corresponding to the best optimum value, average value, worst value and average CPU time for example 1

Fig. 3

Convergence characteristic of example 1

Example 2: Consider the minimization problem [31]

$$ \underset{x_1,{x}_2}{ \min } f\left({x}_1,{x}_2\right)={\displaystyle \sum_{i=1}^5 i \cos \left[\left( i+1\right){x}_1+ i\right]}{\displaystyle \sum_{i=1}^5 i \cos \left[\left( i+1\right){x}_2+ i\right]} $$

(27)

where − 10 ≤ x ₁ ≤ 10 and − 10 ≤ x ₂ ≤ 10

This function has 760 local minima, 18 of which are global minima with −186.73. The problem is solved by using ODE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected 10, 0.3, 1 and 100 respectively for the example under consideration.

To validate the proposed ODE based approach, the same example is solved by using DE.

In case of DE, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected as 10, 0.3, 1.0 and 100 respectively. Table 2 summarizes the best optimum value, the variables corresponding to the best optimum value, average and worst value and average CPU time among 100 runs of solutions obtained from proposed ODE and DE for example 2. Figure 4 depicts the nature of convergence obtained from ODE and DE for example 2.

Table 2 Best optimum value, the variables corresponding to the best optimum value, average value, worst value and average CPU time for example 2

Fig. 4

Convergence characteristic of example 2

Figure 4 depicts the nature of convergence obtained from ODE and DE for example 2.

Case study of fixed head hydrothermal system

Test system 1

This system consists of two hydro plants and two thermal plants whose characteristics and load demands are given in Tables 13, 14 and 15 respectively in Appendix 1. Transmission loss formula coefficients are also given in the Appendix 1. Hydro plant data is taken from [32].

The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover rate (C _R ) and the maximum iteration number (Ν _max) have been selected as 100, 1.0, 1.0 and 100 respectively for the test system under consideration.

The optimal hydrothermal generation obtained by the proposed ODE and DE are provided in Tables 3 and 4 respectively. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed ODE and DE method are summarized in Table 5. The cost obtained from artificial immune system (AIS) [14], particle swarm optimization (PSO) [14] and evolutionary programming (EP) [14] are also shown in Table 5. The cost convergence characteristic obtained from proposed ODE and DE is shown in Fig. 5. It is seen from Table 5 that the cost found by using ODE is the lowest among all other methods.

Table 3 Results obtained from ODE of test system 1 of fixed head hydrothermal system

Table 4 Results obtained from DE for test system 1 of fixed head hydrothermal system

Table 5 Comparison of performance for Test System 1 of fixed head hydrothermal system

Fig. 5

Cost convergence of test system 1 of fixed head hydrothermal system

Test system 2

This system comprises of two hydro plants and four thermal plants whose characteristics and load demands are given in Tables 16, 17 and 18 respectively in Appendix 2. Transmission loss formula coefficients are also given in the Appendix 2.

The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover rate (C _R ) and the maximum iteration number (Ν _max) have been selected as 100, 1.0, 1.0 and 200 respectively for the test system under consideration. The optimal hydrothermal generation obtained by the proposed ODE and DE are provided in Tables 6 and 7 respectively. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed ODE and DE are summarized in Table 8. The cost obtained from artificial immune system (AIS) [14], particle swarm optimization (PSO) [14] and evolutionary programming (EP) [14] are also shown in Table 8. The cost convergence characteristic obtained from proposed ODE and DE is depicted in Fig. 6. It is seen from Table 8 that the cost found by using ODE is the lowest among all other methods.

Table 6 Results obtained from ODE of test system 2 of fixed head hydrothermal system

Table 7 Results obtained from DE of test system 2 of fixed head hydrothermal system

Table 8 Comparison of performance for Test System 2 of fixed head hydrothermal system

Fig. 6

Cost convergence of test system 2 of fixed head hydrothermal system

Case study of variable head hydrothermal system

Three variable head hydrothermal test systems are considered to inspect and verify the proposed ODE method.

Test system 1

This test system considers a multi-chain cascade of four reservoir hydro plants and an equivalent thermal plant. The entire scheduling period is 1 day and divided into 24 intervals. Here, two cases are considered.

Case 1: Here fuel cost is considered as a quadratic function of the power from the composite thermal plant. The detailed parameters for this case come from [7].

The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected 100, 1, 1 and 300 respectively for this case.

The optimal hourly discharges and hydrothermal generation obtained by the proposed ODE method are provided in Tables 19 and 20 in Appendix 3 respectively. Figure 7 depicts the reservoir storage volumes of four hydro plants obtained from ODE. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed ODE and DE are summarized in Table 9. The cost obtained from modified differential evolution (MDE) [10], improved particle swarm optimization (IPSO) [13], teaching learning based optimization (TLBO) [16], improved fast evolutionary programming (IFEP) [9] and genetic algorithm (GA) [7] methods are also shown in Table 9. The cost convergence characteristic obtained from proposed ODE and DE is shown in Fig. 8. It is seen from Table 9 that the cost found by using ODE is the lowest among all other methods.

Fig. 7

Hydro reservoir storage volumes for case of test system 1 of variable head hydrothermal system

Table 9 Comparison of performance for case 1 of Test System 1 of variable head hydrothermal system

Fig. 8

Cost convergence characteristics for case 1 of the system 1 of variable head hydrothermal system

Case 2: Here prohibited operating zones of hydro plants and valve point loading of thermal generator are considered. The detailed parameters for this case come from [9].

The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected 100, 1, 1 and 400 respectively for this case.

The optimal hourly discharges and hydrothermal generation obtained by the proposed ODE method are provided in Tables 21 and 22 respectively in Appendix 3. Figure 9 shows the reservoir storage volumes of four hydro plants obtained from ODE. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed ODE and DE are summarized in Table 10. The cost obtained from improved fast evolutionary programming (IFEP) [9], improved particle swarm optimization (IPSO) [13] and teaching learning based optimization (TLBO) [16] method is also shown in Table 10. The cost convergence characteristic obtained from proposed ODE and DE is shown in Fig. 10. It is seen from Table 10 that the cost found by using ODE is the lowest among all other methods.

Fig. 9

Hydro reservoir storage volumes for case 2 of test system 1 of variable head hydrothermal system

Table 10 Comparison of performance for case 2 of test system 1 of variable head hydrothermal system

Fig. 10

Cost convergence characteristics for case 2 of the system 1 of variable head hydrothermal system

Test system 2

This system considers a multi-chain cascade of four reservoir hydro plants and three thermal plants. The entire scheduling period is 1 day and divided into 24 intervals. The effect of valve point loading is considered. Transmission loss is also considered. The detailed parameters for this case are taken from [10].

The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected 100, 1, 1 and 300 respectively for this case.

The optimal hourly discharges and hydrothermal generation obtained by the proposed ODE method are provided in Tables 23 and 24 respectively in Appendix 3. Figure 11 shows the reservoir storage volumes of four hydro plants obtained from ODE.

Fig. 11

Hydro reservoir storage volumes of test system 2 of variable head hydrothermal system

The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed ODE and DE are shown in Table 11. The cost obtained from modified differential evolution (MDE) [10], clonal selection algorithm (CSA) [15] and teaching learning based optimization (TLBO) [16] is also shown in Table 11. The cost convergence characteristic obtained from proposed ODE and DE is shown in Fig. 12. It is seen from Table 11 that the cost found by using ODE is the lowest among all other methods.

Table 11 Comparison of performance of test system 2 of variable head hydrothermal system

Fig. 12

Cost convergence characteristics of test system 2 of variable head hydrothermal system

Test system 3

This system considers a multi-chain cascade of four reservoir hydro plants and ten thermal plants. The entire scheduling period is 1 day and divided into 24 intervals. The effect of valve point loading is taken into account. Here transmission loss is not considered. The detailed data for this system is taken from [12].

The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _Ρ ), scaling factor (F), crossover constant (C _R ) and maximum iteration number have been selected 100, 1, 1 and 900 respectively for this case.

The optimal hourly discharges and hydrothermal generation obtained by the proposed ODE method are provided in Tables 25 and 26 respectively in Appendix 3. Figure 13 shows the reservoir storage volumes of four hydro plants obtained from ODE. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed

Fig. 13

Hydro reservoir storage volumes of test system 3 of variable head hydrothermal system

ODE and DE are summarized in Table 12. The cost obtained from differential evolution (DE) [12] method is also shown in Table 12. The cost convergence characteristic obtained from proposed ODE and DE is shown in Fig. 14. It is seen from Table 12 that the cost found by using ODE is the lowest among all other methods.

Table 12 Comparison of performance for test system 3 of variable head hydrothermal system

Fig. 14

Cost convergence characteristics for test system 3 of variable head hydrothermal system

It is observed from in Tables 20, 22, 24 and 26 respectively in Appendix 3 that the third hydro unit has no output during some time interval. This is because of the fact that output from a particular hydro unit during a specified time interval depends on the availability of water, reservoir storage volume limit, water transport delay between cascaded reservoirs and on the system configuration as a whole. Depending on the system configuration and constraints for the present problem, this has happened in case of the third hydro unit.

Conclusion

In this paper, opposition-based differential evolution is demonstrated and presented to solve the hydrothermal scheduling problem. The proposed opposition-based differential evolution method has been successfully applied to two test problems, two fixed head hydrothermal test systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading. The results have been compared with those obtained by other evolutionary algorithms reported in the literature. It is seen from the comparisons that the proposed opposition-based differential evolution method performs better than other evolutionary algorithms in the literature.

Nomenclature

a _si , b _si , c _si , d _si , e _si : cost curve coefficients of i th thermal unit

Ρ _sim : power output of i th thermal generator during subinterval m

${P}_{si}^{\min }$, ${P}_{si}^{\max }$: lower and upper generation limits for i th thermal unit

t _m : duration of subinterval m.

Ρ _hjm : power output of j th hydro unit during subinterval m

Ρ _Dm : load demand during subinterval m

Ρ _Lm : transmission loss during subinterval m

Β _lr : loss formula coefficients.

a _0hj , a _1hj , and a _2hj : coefficients for water discharge rate function of j th hydro generator

W _hj : prespecified volume of water available for generation by j th hydro unit during the scheduling period.

${P}_{hj}^{\min }$, ${P}_{hj}^{\max }$: lower and upper generation limits for j th hydro unit

Ρ _sit : output power of i th thermal unit at time t

Ρ _Dt : load demand at time t

Ρ _Lt : transmission loss at time t

Ρ _hjt : output power of j th hydro unit at time t

C _1j , C _2j , C _3j , C _4j , C _5j , C _6j : power generation coefficients of j th hydro unit

Q _hjt : water discharge rate of j th reservoir at time t

V _hjt : storage volume of j th reservoir at time t

${Q}_{hj}^{\min }$, ${Q}_{hj}^{\max }$: minimum and maximum water discharge rate of j th reservoir

${Q}_{hj, k}^L$, ${Q}_{hj, k}^U$: lower and upper bounds of k th prohibited zones of hydro unit j

${V}_{hj}^{\min }$, ${V}_{hj}^{\max }$: minimum and maximum storage volume of j th reservoir

Ι _hjt : inflow rate of j th reservoir at time t

R _uj : number of upstream units directly above j th hydro plant

S _hjt : spillage of j th reservoir at time t

τ _lj : water transport delay from reservoir l to j

t, Τ: time index and scheduling period

Ν _s : number of thermal generating units

Ν _h : number of hydro generating units

n _j : number of prohibited zones for hydro unit j

k: index of prohibited zones of a hydro unit

Appendices

Appendix 1

Table 13 Hydro system data of test system 1

Table 14 Thermal generator data of test system 1

Table 15 Load demands of test system 1

The transmission loss formula coefficients of test system 1 are

$$ \mathrm{B}=\left[\begin{array}{l}\begin{array}{cccc}\hfill 0.000015\hfill & \hfill 0.000140\hfill & \hfill 0.000010\hfill & \hfill 0.000015\hfill \end{array}\\ {}\begin{array}{cccc}\hfill 0.000010\hfill & \hfill 0.000060\hfill & \hfill 0.000010\hfill & \hfill 0.000013\hfill \end{array}\\ {}\begin{array}{cccc}\hfill 0.000015\hfill & \hfill 0.000010\hfill & \hfill 0.000068\hfill & \hfill 0.000065\hfill \end{array}\\ {}\begin{array}{cccc}\hfill 0.000015\hfill & \hfill 0.000013\hfill & \hfill 0.000065\hfill & \hfill 0.000070\hfill \end{array}\end{array}\right] $$

Appendix 2

Table 16 Hydro system data of test system 2

Table 17 Thermal generator data of test system 2

Table 18 Load demands of test system 2

The transmission loss formula coefficients are of test system 2

$$ \mathrm{B}=\left[\begin{array}{llllll}0.000049\hfill & 0.000014\hfill & 0.000015\hfill & 0.000015\hfill & 0.000020\hfill & 0.000017\hfill \\ {}0.000014\hfill & 0.000045\hfill & 0.000016\hfill & 0.000020\hfill & 0.000018\hfill & 0.000015\hfill \\ {}0.000015\hfill & 0.000016\hfill & 0.000039\hfill & 0.000010\hfill & 0.000012\hfill & 0.000012\hfill \\ {}0.000015\hfill & 0.000020\hfill & 0.000010\hfill & 0.000040\hfill & 0.000014\hfill & 0.000010\hfill \\ {}0.000020\hfill & 0.000018\hfill & 0.000012\hfill & 0.000014\hfill & 0.000035\hfill & 0.000011\hfill \\ {}0.000017\hfill & 0.000015\hfill & 0.000012\hfill & 0.000010\hfill & 0.000011\hfill & 0.000036\hfill \end{array}\right]\mathrm{per}\ \mathrm{M}\mathrm{W} $$

Appendix 3

Table 19 Optimal Hydro Discharge (× 10⁴ m ³) for case 1 of Test System 1 of variable head hydrothermal system

Table 20 Optimal Hydrothermal generation (MW) for case 1 of test system 1 of variable head hydrothermal system

Table 21 Optimal Hydro Discharge (× 10⁴ m ³) for case 2 of test system 1 of variable head hydrothermal system

Table 22 Optimal Hydrothermal generation (MW) for case 2 of test system 1 of variable head hydrothermal system

Table 23 Optimal Hydro Discharge (× 10⁴ m ³) of test system 2 of variable head hydrothermal system

Table 24 Optimal Hydrothermal generation (MW) of test system 2 of variable head hydrothermal system

Table 25 Optimal Hydro Discharge (× 10⁴ m ³) of test system 3 of variable head hydrothermal system

Table 26 Optimal Hydrothermal generation (MW) for test system 3 of variable head hydrothermal system