Interactive DE for solving combined security environmental economic dispatch considering FACTS technology

Abstract

This paper presents an efficient interactive differential evolution (IDE) to solve the multi-objective security environmental/economic dispatch (SEED) problem considering multi shunt flexible AC transmission system (FACTS) devices. Two sub problems are proposed.The first one is related to the active power planning to minimize the combined total fuel cost and emissions, while the second is a reactive power planning (RPP) using multi shunt FACTS device based static VAR compensator (SVC) installed at specified buses to make fine corrections to the voltage deviation, voltage phase profiles and reactive power violation. The migration operation inspired from biogeography-based optimization (BBO) algorithm is newly introduced in the proposed approach, thereby effectively exploring and exploiting promising regions in a space search by creating dynamically new efficient partitions. This new mechanism based migration between individuals from different subsystems makes the initial partitions to react more by changing experiences. To validate the robustness of the proposed approach, the proposed algorithm is tested on the Algerian 59-bus electrical network and on a large system, 40 generating units considering valve-point loading effect. Comparison of the results with recent global optimization methods show the superiority of the proposed IDE approach and confirm its potential for solving practical optimal power flow in terms of solution quality and convergence characteristics.

Keywords

optimal power flow (OPF) / economic dispatch / environment / differential evolution / migration operator / multi-objective / flexible AC transmission systems (FACTS) / static VAR compensator (SVC)

Introduction

Combined security environmental economic dispatch, a complex problem with nonlinear constraints, is a vital research area for power system operation and control. In recent years and with the growth in electricity demand, and due to the pressing public demand for clean air, environmental considerations have become one of the major management concerns for organizations and country governments, and forced the utilities to modify their operational strategies to reduce the emissions of toxic gases related to thermal power plants [1-3].

The main objective of an optimal power flow (OPF) strategy is to determine the optimal operating state of a power system by optimizing a particular objective function while satisfying certain specified physical and security constraints. In its most general formulation, a security constrained OPF is a nonlinear, non-convex, large-scale power flow with both continuous and discrete control variables. It becomes even more complex when many conflicting objectives are considered such as fuel cost, gas emission, real power loss, voltage deviation, and voltage stability [4].

Many conventional optimization techniques [5-8] have been applied to solve the OPF problem. All these techniques rely on initial conditions and on the form of the objective function. In general, the methods based on these assumptions do not guarantee to find the global optimum when taking into consideration the practical generators constraints (prohibited zones, valve point effect). Zehar et al. [9,10] have presented a recent review of the major contributions in this area.

The difficulties associated with using mathematical optimization on complex engineering problems have contributed to the development of alternative solutions. During the last two decades, the interest in applying new metaheuristic optimization methods in power system field has grown rapidly. In the literature many standard optimization methods and hybrid variants based metaheuristic algorithms have been proposed and applied with success for solving many complex problems related to power system operation and control such as quantum genetic algorithm (QGA) [11], artificial immune system (AIS) [12], adaptive particle swarm optimization (APSO) [13], improved PSO (IPSO) [14], improved chaotic particle swarm optimization (ICPSO) [15], hybrid multi agent based particle swarm optimization (HMPSO) [16], enhanced cross-entropy method [17], and gravitational search algorithm [18]. These methods and many other techniques have a better searching ability in finding near global optimal solution compared to mathematical methods and to standard evolutionary algorithms (EAs). Actually and with the wide integration of many types of new technology known as flexible ac transmission systems (FACTS) first introduced by Hingorani [19] in modern deregulated power systems become flexible strategy to control practical power systems at critical situations. Many papers have been proposed for solving the power system planning problem considering FACTS technology [20]. Very recently Frank et al. [21,22] have proposed a significant review of recent non-deterministic and hybrid methods applied for solving the multi objective OPF problem.

Differential evolution (DE) is one of the most EAs proposed by Storn and Price [23], which is a simple evolutionary algorithm showing attractive performance in global optimization. The main advantages of the DE include simple program, few control parameters, and high convergence characteristics. Many variants based DEs have been proposed to improve the optimal solution. Coelho et al. [24] have presented a hybrid method which combines DE and EAs with the cultural algorithm (CA) to solve the economic dispatch problems associated with the valve-point effect. Mahdad and Srairi [25] have proposed a DE based dynamic decomposed strategy for solution of large practical economic dispatch with consideration of valve point loading effect. An adaptive hybrid differential evolution (AHDE) has been proposed [26] for solving the practical dynamic economic dispatch problem considering valve-point effects.

Very recently, a new optimization concept, called biogeography, has been proposed by Simon [27]. The historical background of biogeography is very interesting. Biogeography describes the migration of species from one island to another, the arise of new species, and the extinction of species. Recently many researchers have become interested in the application of this technique in solving a variety of problems related to power system operation and control. Bhattacharya and Chattopadhyay [28] have proposed a hybrid differential evolution with biogeography-based optimization for solving the economic load dispatch. Roy et al. have suggested a biogeography-based optimization for multi-constraint optimal power flow with emission and non-smooth cost function [29].

This paper proposes an approach based IDE implemented with Matlab program to minimize the total fuel cost considering environmental constraints and to maintain an acceptable system security in terms of limits on generator reactive power outputs, bus voltages, voltage angle profiles and overload in transmission lines considering multi SVC compensators. The advantages of the proposed approach over other traditional optimization techniques and global optimization methods have been demonstrated using the Algerian 59-bus test system with smooth cost function considering environment effect and with 40 generating units considering valve point effect. Figure 1 shows the basic strategy of security combined environment economic dispatch considering FACTS technology.

Mathematical formulation of optimal power flow problem

The optimal power flow problem is considered as a general minimization problem with constraints, and can be written as

(1)\min f(x,u),

(2)\mathrm{s}\mathrm{t}g(x,u)=\mathrm{0},

(3)h(x,u)\le \mathrm{0},

where f(x,u) is the objective function,g(x,u)and h(x,u)are respectively the set of equality and inequality constraints, x is the state variables, and u is the vector of control variables. The control variables are generator active and reactive power outputs, bus voltages, shunt capacitors/reactors and transformers tap-setting. The state variables are voltage and angle of load buses.

Active power planning with smooth cost function

For optimal active power dispatch, the simple objective functionfis the total generation cost expressed in a quadratic form as

(4)\min f={\displaystyle \sum _{i=\mathrm{1}}^{N_g}(a_i+b_i{P}_{gi}+c_i{P}_{gi}^{\mathrm{2}})},

where N_g is the number of thermal units, P_{gi} is the active power generation at unit i and a_i, b_i and c_i are the cost coefficients of the ith generator. Figure 2 illustrates the vector control structure for active power planning.

Emission objective function

An alternative dispatch strategy to satisfy the environmental requirement is to minimize operation cost under environmental requirement. Emission control can be included in conventional economic dispatch by adding the environmental cost to the normal dispatch. The objective function that minimizes the total emissions can be expressed as the sum of all the three pollutants (NO_x, CO₂, SO₂) resulting from generator real power [4].

In this study, NO_x emission is taken as the index from the viewpoint of environment conservation. The amount of NO_x emission is given as a function of generator output (in t/h), which is the sum of quadratic and exponential functions [9].

(5)f_e={\displaystyle \sum _{i=\mathrm{1}}^{N_g}{\mathrm{10}}^{-\mathrm{2}}\times ({\alpha }_i+{\beta }_i{P}_{gi}+{\gamma }_i{P}_{gi}^{\mathrm{2}}+{\omega }_i\exp ({\mu }_i{P}_{gi}))},

where {\alpha }_i, {\beta }_i, {\gamma }_i, {\omega }_i and {\mu }_i are the parameters estimated based on the results of unit emissions.

Combined economic and load dispatch

Fuel cost and emission are conflicting objective functions. The solutions may be obtained in which fuel cost and emission are combined in a single function with different weighting factors. This objective function is described by

(6)Minimize{F}_T=\alpha f+(\mathrm{1}-\alpha )f_{ce},

where \alpha is a weighting factor that satisfies \mathrm{0}\le \alpha \le \mathrm{1}.

In this model, when the weighting factor \alpha = 1, the objective function becomes a classical economic dispatch; when the weighting factor \alpha=0, the problem becomes a pure minimization of the pollution control level.

(7)f_{ce}=\omega f_e,

where \omega is the emission control cost factor in $/t.

Active power planning with valve-point loading effect

The valve-point loading is taken into consideration by adding a sine component to the cost of the generating units [9]. Typically, the fuel cost function of the generating units with valve-point loadings is represented as

(8)f_T={\displaystyle \sum _{i=\mathrm{1}}^{NG}(a_i+b_i{P}_{gi}+c_i{P}_{gi}^{\mathrm{2}})}+\left|d_i\sin (e_i(P_{gi}^{\min }-P_{gi}))\right|,

where d_i and e_i are the cost coefficients of the unit with valve-point effects.

Reactive power planning (RPP)

In general the reactive power planning strategy is associated to the improvement of the indices of power quality. The main role of RPP is to adjust dynamically the control variables individually or simultaneously to reduce the total power loss, power flow in lines, voltages deviation, and to improve voltage stability, while still satisfying specified constraints (generators constraints and security constraints). Figure 3 demonstrates the structure of the control variables to be optimized.

Power loss objective function

The objective function in this work is to minimize the total active power loss (P_{loss}) in the transmission system. It is given as

(9)P_{loss}={\displaystyle \sum _{k=\mathrm{1}}^{N_l}{g}_k}[{(t_k{V}_i)}^{\mathrm{2}}+V_j^{\mathrm{2}}-\mathrm{2}{t}_k{V}_i{V}_j\cos {\delta }_{ij}],

where N_l is the number of transmission lines; g_k, the conductance of branch k between buses i and j; t_k, the tap ratio of transformer k; V_i, the voltage magnitude at bus i; and {\delta }_{ij}, the voltage angle difference between buses i and j.

Voltage deviation objective function

One of the important indices of power system security is the bus voltage magnitude. The voltage magnitude deviation from the desired value at each load bus must be as small as possible. The deviation of voltage is given as

(10)\mathrm{\Delta }V={\displaystyle \sum _{k=\mathrm{1}}^{N_{PQ}}\left|V_k-V_k^{des}\right|},

where N_{PQ} is the number of load buses and V_k^{des}, the desired or target value of the voltage magnitude at load bus k.

Constraints

Equality constraints

The equality constraints g(x) are the real and reactive power balance equations, expressed as

(11)P_{gi}-P_{di}-V_i{\displaystyle \sum _{j=\mathrm{1}}^N{V}_j(g_{ij}\cos {\delta }_{ij}+b_{ij}\sin {\delta }_{ij})}=\mathrm{0},

(12)Q_{gi}-Q_{di}-V_i{\displaystyle \sum _{j=\mathrm{1}}^N{V}_j(g_{ij}\sin {\delta }_{ij}-b_{ij}\cos {\delta }_{ij})=\mathrm{0}},

where N is the number of buses, P_{gi} and Q_{gi} are the active and the reactive power generation at bus i, P_{di} and Q_{di} are the real and the reactive power demand at bus i; V_i and V_j are the voltage magnitude at bus i and j respectively, {\delta }_{ij} is the phase angle difference between buses i and j respectively, and g_{ij} and b_{ij} are the real and imaginary part of the admittance (Y_{ij}).

Inequality constraints

The inequality constraints h reflect the constraints of the generators and the security limits of the power system.

Generator constraints

1) Upper and lower limits of the active power generations:

(13)P_{gi}^{\min }\le P_{gi}\le P_{gi}^{\max },i=\mathrm{1},\mathrm{2},\cdots ,NPV.

2) Upper and lower limits of the reactive power generations:

(14)Q_{gi}^{\min }\le Q_{gi}\le Q_{gi}^{\max },i=\mathrm{1},\mathrm{2},\cdots ,NPV.

3) Upper and lower limits of the generator bus voltage magnitude:

(15)V_{gi}^{\min }\le V_{gi}^{}\le V_{gi}^{\max },i=\mathrm{1},\mathrm{2},\cdots ,NPV.

Security constraints

1) Constraint on transmission line loadings:

(16)S_{li}\le S_{li}^{\max },i=\mathrm{1},\mathrm{2},\cdots ,NBR.

2) Constraints on voltage at loading buses (PQ buses):

(17)V_{li}^{\min }\le V_{li}^{}\le V_{li}^{\max },i=\mathrm{1},\mathrm{2},\cdots ,NPQ.

3) Constraints on voltage phase angle.

(18){\theta }_i^{\min }\le {\theta }_i\le {\theta }_i^{\max },i=\mathrm{1},\mathrm{2},\cdots ,NB.

where NB is the number of total buses, NPV is the number of generator buses, NPQ is the number of load buses, and NBR is the number of transmission lines.

Transformers and FACTS controller constraints

1) Upper and lower limits on the tap ratio (t) of transformer

(19)t_i^{\min }\le t_i^{}\le t_i^{\max },i=\mathrm{1},\mathrm{2},\mathrm{...},NT.

where NT is the number of regulating transformers.

2) Parameters of shunt FACTS controllers must be restricted within their upper and lower limits.

(20)X^{min}\le X_{FACTS}\le X^{max}.

Overview of DE technique

DE, a branch of EAs proposed by Storn and Price in 1995 [23], has proven to be a promising candidate in solving real valued optimization problems. Like many metaheuristic optimization methods, DE is based on stochastic searches; it utilizes the differential information to get the new candidate solution. In each step, DE mutates vectors by adding weighted, random vector differentials to them. If the fitness function of the trial vector is better than that of the target, the target vector is replaced by the trial vector in the next generation.

DE works through a simple mechanism search as presented in Fig. 4. The DE mechanism search was conducted by applying the three basic operations of mutation, crossover and selection.

Step 1 Initialization: This is the first step in DE, where the initial population of individuals is initialized with random values generated according to a uniform probability distribution in the n-dimensional space.

(21)X_i^{(G)}=x_{ij}^{(L)}+rand[\mathrm{0},\mathrm{1}](x_{ij}^{(U)}-x_{ij}^{(L)}),

where rand[\mathrm{0},\mathrm{1}] denotes a uniformly distributed random value within [0,1] and x_{ij}^{(L)} and x_{ij}^{(U)} are lower and upper boundaries of the parameters x_{ij} respectively for

j=\mathrm{1},\mathrm{2},\mathrm{...},n.

Step 2 The role of mutation operation (or differential operation) is to avoid search stagnation by introducing new parameters into the population according to

(22){\mathit{v}}_i^{(G+\mathrm{1})}={\mathit{x}}_{r\mathrm{3}}^{(G)}+f_m({\mathit{x}}_{r\mathrm{2}}^{(G)}-{\mathit{x}}_{r\mathrm{1}}^{(G)}).

Two vectors {\mathit{x}}_{r\mathrm{2}}^{(G)} and {\mathit{x}}_{r\mathrm{1}}^{(G)} are randomly selected from the population and the vector difference between them is established. f_m>\mathrm{0}, usually taken from the range of [0,1], is a real parameter, called mutation factor.

Step 3 Evaluation of the fitness function value: For each individual, the fitness (objective function) value is evaluated.

Step 4 Following the mutation operation, the crossover operator creates the trial vectors, which are used in the selection process. A trial vector is a combination of a mutant vector and a parent vector which is formed based on probability distributions.

For each mutate vector,{\mathit{v}}_i^{(G+\mathrm{1})}, a trail vector, {\mathit{u}}_i^{(G+\mathrm{1})}={[u_{i\mathrm{1}}^{(G+\mathrm{1})},u_{i\mathrm{2}}^{(G+\mathrm{1})},\mathrm{...},u_{i\mathrm{2}}^{(G+\mathrm{1})}]}^T is generated according to

(23)$$\begin{gathered} u_{ij}^{(G+\mathrm{1})}=\{v_{ij}^{(G+\mathrm{1})}if(rand[\mathrm{0},\mathrm{1}]\le CR)or(j=rnbr(i)), \\ {x}_{ij}^{(G)}otherwise. \end{gathered}$$

where the index rnbr(i)\in \{\mathrm{1},\mathrm{2},\mathrm{...},n\} is randomly chosen using a uniform distribution.

Step 5 The selection operator chooses the vectors that are going to compose the population in the next generation. These vectors are selected from the current population and the trial population. Each individual of the trial population is compared with its counterpart in the current population.

Step 6 Verification of the stopping criterion (convergence): Loop to Step 3 until a stopping criterion is satisfied, usually a maximum number of iterationsG_{\max }.

Shunt facts modelling

The static VAR compensator (SVC) [17-19] is a shunt connected var generator or absorber whose output is adjusted dynamically to exchange capacitive or inductive current so as to maintain or control bus voltages. It includes separate equipment for leading and lagging VARs. The steady state model proposed in Ref. [30] was used in this work to incorporate the SVC on power flow problems. The basic principle operation of the SVC controller is shown in Fig. 5. The model used was based on representing the controller as variable impedance, as shown in Fig. 5(a). Figure 5(b) shows the voltage–current (V–I) characteristic of the SVC controller

(24)I_{SVC}=j{B}_{SVC}V,

(25)\begin{array}{l}{B}_{SVC}=B_C-B_{TCR}={\displaystyle \frac{\mathrm{1}}{X_C{X}_L}}\left\{X_L-{\displaystyle \frac{X_C}{\pi }}[\mathrm{2}(\pi -\alpha )+\sin (\mathrm{2}\alpha )]\right\},\hfill \\ X_L=\omega L,X_C={\displaystyle \frac{\mathrm{1}}{\omega C}},\hfill \end{array}\}

where B_{SVC},\alpha ,X_L,X{}_C,V are the shunt susceptance, firing angle, inductive reactance, capacitive reactance of the SVC controller, and the bus voltage magnitude to which the SVC is connected, respectively.

The exchange reactive power Q_i^{SVC} with the bus i can be expressed as

(26)Q_i^{SVC}=B_i^{SVC}{V}_i^{\mathrm{2}}.

Strategy of interactive DE

Principle of proposed approach

The proposed algorithm decomposes the solution of such a modified OPF problem into two linked sub problems. The first sub problem is an active power generation planning solved by the proposed IDE, and the second one is a RPP [25-31] to make fine adjustments on the optimum values obtained from the first stage. This will provide updated voltages, angles, and point out that generators have exceeded reactive limits. Figure 6 exhibits the flowchart of the proposed strategy based OPF considering FACTS controllers.

Decomposition mechanism

Problem decomposition is an important task for large-scale OPF problem. This section gives answers to the following two technical questions:

1) How many efficient partitions needed?

2) Where to practice and generate the efficient sub-systems?

The decomposition procedure decomposes a problem into several interacting sub-problems that can be solved with reduced sub-populations, and coordinate the solution processes of these sub-problems to achieve the solution of the whole problem.

Migration procedure

New migration operation inspired from the BBO algorithm was introduced into the IDE approach, thereby effectively exploring and exploiting promising regions in a search space. This new mechanism based migration between individuals associated to different subsystems makes the initial partitions to react more by changing experiences. The mechanism search of migration operation is presented in Fig. 7.

Basic definitions

1) Parallel DE: is a procedure (algorithm) used to optimize sub systems based on the initial decomposed mechanism without exchanging information between sub systems.

2) Candidate emission partition (CEP): a source containing good (strong) individuals (in this study, individuals represent the active power generation).

3) Candidate reception partition (CRP): a source containing bad (weak) individuals.

4) Migration mechanism: migration operation between CEP and CRP consists of making exchange of experience between these two categories to realize the following objectives:

(1) Improve the proprieties of the CRP by integrating new strong individuals;

(2) Make a flexible diversity to the CEP by integrating new weak individuals.

The following steps summarize the basic mechanism search of migration operator within the standard PDE:

1) Select one CEP, Par{t}_k;

2) Select randomlyN_kcandidate emission individuals (CEI) from partition Par{t}_k;

3) Select randomly CRP, Par{t}_j(j=\mathrm{1},\mathrm{...},N_{part,}j\ne k);

4) For each CEP choose randomly candidates reception individuals (CRI);

5) Make migration: replace the CEI with CRI using the mechanism search as illustrated in Fig. 7;

6) Evaluate the fitness function for new partitions generated: If new total cost ‘’: save new partitions generated;

7) Repeat until stopping criterion is reached. In this study, maximum number of iterations is taken as the stopping criterion.

Stages of proposed algorithm

The basic interactive DE algorithm is presented based on the following stages:

Stage 1 Generate initial feasible partitions: The main idea of this first stage is to optimize the active power demand for each partitioned network to minimize the total fuel cost. An initial candidate solution was generated for the global N population size. Figure 8 shows the mechanism search partitioning for active power generation, whereas Fig. 9 shows the corresponding equivalent mechanism partitioning search related to active power demand.

1) For each decomposition level, the initial active power demand was estimated using

(27)P{d}_{\mathrm{1}}={\displaystyle \sum _{i=\mathrm{1}}^{M\mathrm{1}}{P}_{Gi}},

(28)P{d}_{\mathrm{2}}={\displaystyle \sum _{i=\mathrm{1}}^{M\mathrm{2}}{P}_{Gi}=PD-P{d}_{\mathrm{1}}},

where P{d}_{\mathrm{1}} is the active power demand for the first initial partition; P{d}_{\mathrm{2}}, the active power demand for the second initial partition; and PD, the total active power demand for the original network.

The following equilibrium equations were verified for each decomposed level:

For level 1:

(29)P{d}_{\mathrm{1}}+P{d}_{\mathrm{2}}=PD+P_{loss}.

(1) Each sub-population contained Mi control variables (active power generation) to be optimized.

(2) Each sub-population was updated based on the DE operators.

2) Fitness evaluation based load flow. For all sub-systems, generated fitness function based economic dispatch was executed to evaluate their performance. A candidate solution formed by all sub-systems was better if its fitness was higher.

3) Consequently, under this concept, the final value of active power demand should satisfy

(30){\displaystyle \sum _{i=\mathrm{1}}^{N_g}(P_{gi})}={\displaystyle \sum _{i=\mathrm{1}}^{Par{t}_i}(P{d}_i)+P_{loss}},

(31)P_{\mathrm{g}i}^{\min }\le P_{gi}\le P_{\mathrm{g}i}^{\max }.

Stage 2 Interactive DE: generate dynamically new partitions based migration operation: The migration operation was newly introduced into the parallel DE mechanism, thereby effectively exploring and exploiting the promising regions in a search space. This new mechanism based migration between individuals from different subsystems (partitions) made the initial partitions to react more by changing experiences. Figure 10 shows the mechanism of migration operation between individuals from different subsystems.

Stage 3 Final search mechanism based RPP:

1) All the sub-systems were collected to form the original network. Global data base generated based on the best results U_best were found from all sub-populations.

2) The final solution U_best (global) is found out after RPP procedure to adjust the reactive power generation limits, and voltage deviation, the final optimal cost is modified to compensate the reactive constraint violations.

Numerical results and analysis

Test system 1: Algerian electrical network planning

Active power dispatch without SVC compensators

The proposed algorithm was developed in the Matlab programming language using the 6.5 version. The proposed approach was tested on the Algerian electrical network (Sonelgaz) which consisted of 59 buses, 83 branches (lines and transformers) and 10 generators. Table 1 lists the technical and economic parameters of the ten generators. It should be mentioned that the generator of bus No. 13 was not in service. Table 2 presents the emission coefficients of the generators taken from Ref. [31]. Figure 11 shows the topology of the Algerian electrical network with 59 buses. To verify the efficiency of the proposed approach, a comparison was made of the proposed algorithm with other competing OPF algorithms. Zehar and Sayah [8] proposed a fast successive linear programming algorithm applied to the Algerian electrical network. Mahdad et al. [31,32] proposed an efficient decomposed GA for the solution of large-scale OPF with consideration of shunt FACTS devices under severe loading conditions and a fuzzy controlled genetic algorithm.

To demonstrate the effectiveness and the robustness of the proposed approach, two cases were considered with and without consideration of SVC controller installation.

Case 1 Minimum total operating cost (α = 1).

Case 2 Minimum total emission (α = 0).

Initially, the general database was generated. Several runs were done with different values of DE parameters such as mutation constant f_m, crossover constant (CR), size population NP (the number of partition), and maximum number of generations G_max which was used in this study as convergence criteria. The following values are selected based on the size of the new decomposed network (subsystems): f_m=\mathrm{0.4}-\mathrm{0.95}; CR=\mathrm{0.5}-\mathrm{0.8}; NP=\mathrm{10}-\mathrm{30}; G_{\max }=\mathrm{250}.

Table 3 shows the simulation results obtained by the proposed approach for the two cases (minimum cost, and minimum emission), not taking SVC controllers into consideration. The comparison of the results obtained by the application of the proposed IDE with those found by global optimization (GA, FGA, and PGA) and conventional method (FSLP) are reported in Tables 4 and 5. The proposed approach gives more important results compared to all cases. For example at the case corresponding to the minimum total operating cost, the fuel cost is 1750.20 $/h, and power loss 16.402 MW which are better compared with the results found by the global and conventional methods, while the emission gas reduced to 0.4125 t/h, compared to the best result found by PGA (0.4213 t/h). It is important to note that all results obtained by the proposed approach do not violate the generation capacity constraints. The security constraints are satisfied for voltage magnitudes (0.9<V<1.1 pu), voltage phase profiles () and line flows. Figure 12 shows the convergence characteristic for the first case ‘Case 1’. It can be see that the minimum cost is 1672.7 $/h which is better than the result depicted in Table 3. However, this value was obtained without taking in consideration the voltage phase constraint fixed at the value 14°, which proved the impact of this important constraint in the value of the final cost. Figures 13 and 14 show the voltage phase profiles. It is clearly identified that all voltage phases are within the constraint limit. Figure 15 shows the convergence characteristic for Case 2 with consideration of all constraints. Figure 16 shows the voltage profiles related to the two cases.

RPP considering SVC controllers

For the secure operation of a power system, it is important to maintain the required security margin. System loadability, voltage magnitude and power loss are three important indices of power quality. In the second stage, dynamic shunt compensation based SVC controllers were taken into consideration. The control variables selected for reactive power dispatch (RPD) were the generator voltages and reactive power of the SVC compensators installed at critical buses.

Before the insertion of SVC devices, the system was pushed to its collapsing point by increasing both active and reactive load discretely using continuation load flow [31]. In this test system, according to the results obtained from the continuation load flow, buses 7, 14, 17, 35, 36, 39, 44, 47, 56 were the best location points for installation and coordination between SVC compensators and the network. Table 6 gives details of the SVC data.

Table 7, shows the results of reactive power generation with SVC Compensators for the two cases (minimum cost and minimum emission). It is observed that the active power loss reduced for the two cases compared to the base case without RPP. Tables 8–9 show the details of the control variables (Q_{SVC}) reactive power of the multi SVC exchanged with the network before and after optimization of the two cases. The minimum and maximum limits of load bus voltage are 0.9 and 1.1 pu. From the base case corresponding to the first case (minimum cost) without SVC compensators, the fuel cost is reduced to 1748.3 ($/h) and power loss reduced to 15.534 MW. It is important to note that all results of power generation obtained by the proposed approach do not violate the generation capacity constraints. The security constraints are satisfied for voltage magnitudes (0.9<V<1.1 pu), voltage phase angles () and line flows (\left|P_{ij}\right|\le P_{ij}^{\max }). Figure 17 shows the voltage phase profiles distribution considering SVC compensators installed at critical buses, while Fig. 18 shows the voltage magnitude improvement using SVC compensators installed at critical buses. The system loadability improved to 2.6602 pu compared to the case without SVC compensators (1.900 pu). These results confirm clearly the ability of the proposed approach to find the accurate and efficient OPF solution with consideration of shunts FACTS compensators.

Test system 2: economic dispatch of 40 generating units considering valve point loading effect

A system with 40 generators with the effect of valve-point loading was studied in the second test. The total load demand of the system is 10500 MW. This is a larger system, the number of local optima, complexity and nonlinearity to the solution procedure is enormously increased. The system data can be retrieved from [25]. The best results of the proposed approach compared with other methods are illustrated in Tables 10 and 11 [24-34], which show clearly the superiority of the proposed approach. Figure 19 shows the convergence characteristic of the original network (40 units) using the standard DE. Figure 20 shows the sequence convergence characteristic of subsystem 1 (5 units) and the related two subsystems generated which are: subsystem 1-1 and subsystem 1-2. The number of generation is reduced greatly compared to the basic case using the standard DE.

Results and discussions

The main objective of the proposed approach named interactive differential evolution (IDE) is to find the near global solution with consideration of security constraints and shunt FACTS devices. The contributions of the proposed approach as compared with the previous methods are:

1) Decomposing the optimal constraints. The direct constraints associated to the active power planning (active power generation) and the security constraints (reactive power generation, bus voltages, voltage phase, flexible shunt compensation based multi SVC controllers, power flow of transmission lines) affect indirectly the objective function associated to the second sub problem using an efficient RPP based DE.

2) The main contribution of the proposed approach based DE compared to the decomposed parallel GA is the new migration operation inspired from the BBO algorithm introduced in the standard DE, which can effectively explore and exploit promising regions in a search space by creating dynamically new efficient partitions (subsystems).

3) The IDE algorithm proposed provides a far better solution than the conventional method FSLP, the global optimization methods such as GA based fuzzy logic and parallel GA.

Conclusion

An interactive DE approach based new migration operator was proposed to solve the multi-objective environment OPF considering multi-dynamic shunt compensators (SVC). The main objective of the proposed approach is to improve the performances of the standard DE in terms of solution quality and convergence time. In this study two coordinated stages are proposed to solve the multi-objective security optimal power flow. The first one is the active power planning based decomposed DE, the particularity of the proposed approach compared to the decomposed parallel GA is the new migration operation inspired from the BBO algorithm which can effectively explore and exploit promising regions in a search space by creating dynamically new partitions (subsystems). In the second stage, reactive power planning based multi SVC controllers was proposed to maintain an acceptable system security in terms of limits on generator reactive power outputs, bus voltages, voltage angle profiles and overload in transmission lines. The performance of the proposed approach was tested on the Algerian 59-bus electrical network (Sonelgaz) and on a large electrical system, 40 generating units considering valve point loading effects. It is found that the proposed approach can converge to the near solution and obtain a competitive solution at reduced time. Future studies can focus on the inclusion of practical generator constraints such as prohibited zones, multi fuel and the coordination control of multi types of FACTS devices to enhance the solution of multi-objective OPF under critical situations.

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