Solving economic dispatch problem with valve-point effects using swarm-based mean–variance mapping optimization (MVMO S )

: Mean–variance mapping optimization (MVMO) is a new population-based metaheuristic technique which is successfully applied for different power system optimization problems. The special feature of MVMO is the mapping function applied for the mutation based on the mean and variance of n -best population. Recently, the modified version of MVMO has been developed to become more powerful, named as swarm-based mean–variance mapping optimization (MVMO S ). This paper proposes MVMO S as a new approach for solving the economic dispatch (ED) problem considering valve-point effects. To validate the performance of the proposed method, the MVMO S is tested on three systems including 3, 13, and 40 thermal generating units with valve-point effects and the obtained results from MVMO S are compared to those from other existing meth-ods in the literature. Test results have indicated that the proposed MVMO S is more robust and produces better solution quality than many other methods. Therefore, the MVMO S is efficient for solving the ED with valve-point effects.


PUBLIC INTEREST STATEMENT
In recent years, swarm intelligence has been widely applied to a variety of fields in engineering due to its outstanding characteristics for solving optimization problems with complex objective function and constraints. This paper presents a swarm intelligent approach for solving the nonconvex economic dispatch problems which is one of the important tasks in power generation systems. This problem is often related to fuel cost saving. Real-world economic dispatch problems have nonconvex objective functions with complex constraints. This leads to difficulty in finding the global optimal solution. Over the past decades, various optimization techniques have been applied to economic dispatch problems. In general, these techniques can be classified into classical calculus based methods, Artificial Intelligence techniques and hybrid methods. However, nonconvex optimization problems are still a challenge for engineers and decision-makers in the industry. Hence, there is always a need for developing new techniques for solving nonconvex problems.

Introduction
In power system, thermal generating units are supplied with multiple fuel sources such as coal, natural gas, and oil. The price of these fuels is highly volatile and faces depletion. Hence, the economical operations of the power systems gained importance. The economic dispatch (ED) is defined as the process of allocating the real power output of generating units to meet required load demand so as their total fuel cost is minimized while satisfying all physical and operational constraints (Dieu, Schegner, & Ongsakul, 2013).
Traditionally, the fuel cost function of each generating unit is presented as the quadratic function approximations and is solved using mathematical programming techniques such as lambda iteration method, Newton's method, gradient search, dynamic programming (Wollenberg & Wood, 1996), and quadratic programming (Fan & Zhang, 1998). However, most of these techniques are not capable of dealing with nonconvex and nonlinear ED problems. The ED problem is more practical when considering the effects of valve-point loadings. The valve-point effects (VPE) can cause the input-output curve of thermal generators to become more complicated. Therefore, the ED should be represented as nonconvex or nonsmooth optimization problem. This leads to difficulty in finding global optimum solution. More advanced optimization methods based on artificial intelligence concepts are implemented effectively to deal with ED problems such as genetic algorithm (GA) (Chiang, 2005), evolutionary programming (EP) (Sinha, Chakrabarti, & Chattopadhyay, 2003), artificial bee colony (ABC) (Hemamalini & Simon, 2010;Le Dinh, Vo Ngoc, & Vasant, 2013;Secui, 2015), ant colony optimization (ACO) (Pothiya, Ngamroo, & Kongprawechnon, 2010), evolutionary strategy optimization (ESO) (Pereira-Neto, Unsihuay, & Saavedra, 2005), and differential evolution (DE) (Noman & Iba, 2008). Recently, particle swarm optimization (PSO) is the most popular method applied for solving the ED problems, especially for nonconvex problems (Lin, Chen, Tsai, Yuan, et al., 2015;Shahinzadeh, Nasr-Azadani, & Jannesari, 2014;Vasant, Ganesan, Elamvazuthi, 2012). Several improvements of PSO method are developed for solving.
The MVMO is a novel optimization algorithm which is conceived and developed by István Erlich (Erlich, Venayagamoorthy, & Worawat, 2010). This algorithm also falls into the category of the so-called "population-based stochastic optimization techniques". Recently, the extensions of MVMO has been developed by Rueda and Erlich (2013), which is named MVMO S . The search process of MVMO S starts with a set of particles. In addition, two parameters of MVMO including the scaling factor and variable increment parameters are extended to enhance the mapping. Hence, the ability for global search of MVMO S is more powerful than the single particle version. In this paper, MVMO S is proposed for solving the ED problem with valve-point effects.
The remaining organization of this paper is as follows. Section 2 presents the formulation of the ED problem with valve-point effects. The review of MVMO, extension of MVMO-MVMO S , and implementation of the proposed MVMO S to ED problem are exhibited in Section 3. The numerical test and results discussion are shown in Sections 4 and 5, respectively. The paper is concluded in Section 6.

Problem formulation
In this study, the VPE is considered as practical operation of generators. The VPE is a natural characteristic of a thermal turbine. The turbine of generating unit has many admission steam valves. The opening of these steam valves increase the throttling losses rapidly, leading to rise the incremental heat rate suddenly. The VPE is the direct result of the practical operation of thermal generating unit which produces ripples effects on the input-output curve as seen in Figure 1.
The VPE makes the fuel cost function highly nonlinear and nonsmooth containing multiple minima. The fuel cost function is described as the superposition of sinusoidal functions and quadratic functions.
The model of ED problem with VPE is formulated as follows (Dieu et al., 2013): subject to (a) Real power balance constraint: The total active power output of generating units must be equal to total active power load demand plus power loss: where the power loss P L can be approximately calculated by Kron's formula (Wollenberg & Wood, 1996): (b) Generator capacity limits: The active power output of generating units must be within the allowed limits:

MVMO and MVMO S
Mean-variance mapping optimization (MVMO) is a new optimization algorithm which falls into the category of the so-called "population-based stochastic optimization technique". The similarities between MVMO and the other known stochastic algorithms are in three evolutionary operators including selection, mutation, and crossover. The major differences between MVMO and other existing techniques are summarized in Erlich et al. (2010) as follows: (1) (4) P i, min ≤ P i ≤ P i, max .
• The key feature of MVMO is a special mapping function which applied for mutating the offspring. The mapping function is described by the mean and variance of n best solutions stored in the archive.
• The total space for searching of all variables is limited within the range from 0 to 1. The output of mapping function is always inside [0, 1]. However, the function evaluation is carried out always in the original scales.
• MVMO is a single-agent search algorithm because only a single offspring is generated in each iteration. Therefore, the number of fitness evaluations is identical to the number of iterations.
• A compact and dynamically updated solution archive serves as the knowledge base for guiding the search direction (i.e. adaptive memory). The normalized n-best are filled up in the archive progressively over iterations and sorted in a descending order of fitness.
Swarm-based mean-variance mapping optimization (MVMO S ) is an extension of the original version MVMO. The difference between MVMO and MVMO S is the initial search process with particles. MVMO starts the search with single particle while MVMO S starts the search with a set of particles. At the beginning of the optimization process of MVMO S , each particle performs m steps independently to collect a set of reliable individual solutions. Then, the particles start to communicate and to exchange information. MVMO is extended two parameters including the scaling factor f s and variable increment Δd parameter to enhance the mapping. Therefore, the search global ability of MVMO S is strengthened.

Handing of constraints
Neglecting the transmission power loss, the equality constraint (2) is rewritten by: By using the slack variable method (Kuo, 2008) to guarantee that the equality constraint (5) is always satisfied. The power output of the slack unit is calculated as follows: The fitness function for the proposed MVMO S will include the objective function (1) and penalty terms for the slack unit if inequality constraint (4) is violated. The fitness function is as follows: The penalty factor K for the slack unit is large enough and set to 1,000 for all systems.

Implemention of MVMO S to ED
The flowchart of MVMO S is depicted in Figure 2:

Initialization of algorithm
The parameters for MVMO S have to be initialized including iter max , n_var, n_par, mode, Since different parameters of the proposed method have effect on the performance of MVMO S , it is important to determine an optimal set of parameters of the proposed methods for ED problem. For each selection, one parameter is varied from the low value to higher value while the other parameters are fixed. The obtained result after one run is compared with the previous one. Multiple runs are carried out to choose the suitable set of parameters.

Normalization and de-normalization of variables
The search process of the MVMO S starts with a set of particles. Initial variables is normalized to the range [0, 1] as follows: Check for the global best Identification of redundant particle Deletion of particle?
Offspring generation (Parent assignment and mutation via mapping function) However, the function evaluation is carried out always in the original scales of the problem space.
The de-normalization of optimization variables is carried using (9): where After that, the power output for the slack generator is calculated using (6) to evaluate fitness function in (7), store f best and x best in archive.
MVMO S utilizes swarm implementation to enhance the power of global searching of the classical MVMO by starting the search with a set of n p particles, each having its own memory and represented by the corresponding archive and mapping function. At the beginning of the optimization process, each particle performs m steps independently to collect a set of reliable individual solutions. Then, the particles start to communicate and to exchange information.
It is worthless when particles are very close to each other since this would entail redundancy. To avoid closeness between particles, the normalized distance of each particle's local best solution x lbest, i to the global best x gbest is calculated by Rueda and Erlich (2013). The i-th particle is discarded from the optimization process if the distance D i is less than a certain user defined threshold D min .
where N denotes the number of optimization variables.

Solution archive
The best n individuals are stored in the archive table which is described as Figure 3. The archive size (n) is taken to be a minimum of two. If archive size is greater than two, the table of best individuals is filled up progressively over iterations in a descending order of the fitness. When the table is filled with n members, an update is performed only if the fitness of the new population is better than those in the table.
Mean x i and variance v i are computed from the archive where the n best populations are stored as follows (Erlich et al., 2010): (11) where j goes from 1 to n (archive size). At the beginning x i corresponds with the initialized value of x i and the variance is set to v i = 1.0.

Parent assignment
The individual with the best fitness f best and its corresponding optimization values, x best , are stored in memory as the parent of the population for that iteration. This parent is used for creation of offspring .

Offspring creation
Creation of an offspring, of N dimensions involves three common evolutionary computation algorithms' operations including selection, mutation and crossover.  (13) and depicted as Figure 4. The transformation mapping function, h, is calculated by the mean x and shape variables s i1 and s i2 as in (15) (Erlich et al., 2010): where h x , h 1 , and h 0 are the outputs of transformation mapping function based on different inputs given by: where The scaling factor f s in (16) is a MVMO parameter which can be used to change the shape of the function during iteration. In MVMO S , this factor is extended for the need of exploring the search space at the beginning more globally, whereas at the end of the iterations, the focus should be on the exploitation. It is determined by (Rueda & Erlich, 2013): where (13) for controlling the f s factor is not used (Rueda & Erlich, 2013).
The shape variables s i1 and s i2 in (15) are determined using the following algorithm (Rueda & Erlich, 2013): At the start of the algorithm, the initial values of d i (typically between 1 and 5) are set for all variables. Sometimes, the variance can oscillate over a wide range. Using the factor d i instead of s i which is a function of variance a smoothing effect is achieved. At every iteration, if s i > d i , d i will be multiplied by Δd leads to increased d i . In case s i < d i , the current d i is divided byΔd which is always greater than 1.0 resulting in reduced value of d i . Therefore, d i will always oscillate around the current shape factor s i . Furthermore, Δd is randomly varied around the value (1 + Δd 0 ) with the amplitude of Δd 0 adjusted in accordance to (19), where Δd 0 can be allowed to decrease from 0.4 to 0.01 (Rueda & Erlich, 2013).

Crossover:
For the remaining un-mutated dimensions, the genes of the parent, x best , are inherited. In other words, the values of these un-mutated dimensions are clones of the parent. Here, crossover is by direct cloning of certain genes. In this way, the offspring is created by combining the vector x best , and vector of m mutated dimensions.

Termination criteria
The algorithm of the proposed MVMO S is terminated when the maximum number of iterations iter max is reached.

Overall procedure
The steps of procedure of MVMO S for the ED problem are described as follows: Step 1: Setting the parameters for MVMO S including iter max , n_var, n_par, mode, d i , Δd ini 0 , Δd final 0 archive zize, f * s_ini , f * s_final , n_randomly, n_randomly_min, indep.runs(m), D min . Set i = 1, i donotes the function evaluation.
Step 4: De-normalized variables using (9), calculate power output for the slack generator using (6) to evaluate fitness function in (7), store f best and x best in archive.
Step 5: Increase i = i + 1. If i < m (independent steps), go to step 6. Otherwise, go to step 7.
Step 6: Check the particles for the global best, collect a set of reliable individual solutions. The i-th particle is discarded from the optimization process if the distance D i is less than a certain user defined threshold D min . If the particle is delected, increase k = k + 1, n p = n p − 1 and go to step 4. Otherwise, go to step 7.
Step 7: Create offspring generation through three evolutionary operators: selection, mutation, and crossover.
Step 8: if k < n p , increase k = k + 1 and go to step 4. Otherwise, go to step 9.
Step 9: Check termination criteria. If stopping criteria is satisfied, stop. Otherwise, go to step 3.

Numerical results
This part presents results of the implementation of proposed MVMO S and origional MVMO in solving the ED problem with valve-point effects. The obtained results by the proposed MVMO S are compared to those from the other optimization methods for three test cases including 3-unit system, 13-unit system, and 40-unit system. For each case, the algorithm of MVMO S is run 50 independent trials on a core i5 3.4 GHz PC with 4 GB RAM. The implementation of the proposed MVMO S is coded in the Matlab R2013a platform.

Case 1: 3-unit system
The data of 3-unit test system with valve-point effects is taken from Sinha et al. (2003). In this case, the power load demand is 850 MW, the transmission power loss is neglected. The obtained results by MVMO and MVMO S for this case are presented in Table 1. Figure 5 depicts the convergence characteristic of the MVMO and MVMO S for case 1.
The parameters for MVMO S for this system are as follows: iter max = 10,000, n_var (generators) = 3, The min, average, and max fuel cost and CPU time obtained by the proposed MVMO S are compared to the results of the other methods in Table 2. The best optimal solution for this case is 8,234.0717 ($/h) (Park et al., 2005). All methods obtain the minimum cost with 8,234.0717 ($/h) in Table 2. However, the proposed MVMO S achieves the best optimal solution with a high probability (the standard deviation is 0%). For computational time, it may not be directly comparable among the methods because these methods were run and coded on different computers and programming languages. However, a CPU time comparison is used to show the efficiency of the compared methods. The computational time of MVMO S is faster than EP, EP-PSO, PSO, CEP, FEP, MFEP, and IEEP, and close to PSO-SQP. The PSO, EP-SQP, and PSO-SQP were run on a Pentium II 500 MHz PC. There is no computer processor reported for CEP, FEP, MFEP, and IEEP.

Case 2: 13-unit system
The data of 13-unit test system with valve-point effects are referred to Sinha et al. (2003). The power load demand is 1,800 and 2,500 MW, respectively. The transmission power loss is also neglected in this case. The obtained results by the MVMO and MVMO S corresponding to the two load demand are shown in Table 3. Figures 6 and 7 show the convergence characteristic of the MVMO and MVMO S for the case of load demands 1,800 MW and the case of load demands 2,520 MW, respectively.
The parameters for MVMO S are as follows for all the cases of load demands 1,800 and 2,520 MW: iter-

Figure 5. Convergence property of MVMO and MVMO S for case 1 (3 units).
In Tables 4 and 5, the fuel cost and CPU time of proposed MVMO S are compared to those of other optimization methods for two load demands 1,800 and 2,520 MW. For the case of load demands 1,800 MW, the minimum fuel cost obtained by MVMO S is less than CEP, FEP, MFEP, IEEP, PSO, EP-SQP, PSO-SQP, HDE, and CGA-MU, and close to UHGA, Self-tuning HDE and GA-PS-SQP. It is noted that the mean cost obtained by MVMO S is less than is less than that of the others, except the UHGA. The computational time of MVMO S is faster than CEP, FEP, MFEP, IEEP, PSO, and EP-SQP, slower than UHGA, CGA-MU, HDE, self-tuning HDE, and GA-PS-SQP, and close to PSO-SQP. For the case of load demands 2,520 MW, the minimum total cost by MVMO S is less than that from the other methods in Table 5. The computational time of MVMO S is slower than ESO . The PSO, EP-SQP, and PSO-SQP were executed on a Pentium II 500 MHz PC. The HDE and self-tuning HDE were run on a Pentium 1.5 GHz with 768 MB of RAM. The computational time for ESO, UHGA, CGA-MU, GA-PS-SQP were from a Pentium IV PC, Pentium IV 2.99 GHz PC, Pentium III-700 PC, and Pentium III-1 GHz-256 MB of RAM, respectively. There is no computer processor reported for CEP, FEP, MFEP, and IEEP and no computational time for the other methods.

Case 3: 40-unit system
The data of the test system including 40 thermal generating units with VPE are from Sinha et al. (2003). The system load demand for this case is 10,500 MW neglecting transmission power loss. The obtained solutions by MVMO and MVMO S for this case are given in Table 6. The convergence characteristic of the MVMO and MVMO S are depicted in Figure 8 for case 3.
The parameters for MVMO S for this system are as follows: iter max = 150,000, n_var (generators) = 40, n p = 5, archive size = 5, mode = 4, indep.runs (m) = 2,000, n_randomly = 20, n_randomly_min = 10,  Table 7 shows the comparison of the fuel cost and CPU time of the MVMO S and the previously reported methods. As seen in Table 7, the best total cost obtained by the MVMO S is less than that from the others.

Advantages of MVMO S
The advantages of MVMO S are robustness, global solution with high probability, and easy implemention to ED problem. In this study, the MVMO and the MVMO S are run 50 independent trials. The mean cost, max cost, average cost, and standard deviation obtained by the MVMO and MVMO S to evaluate the  robustness characteristic of the proposed method for ED problems. As observed from Tables 1, 3, and 6, the power output obtained by MVMO and MVMO S are always between the minimum and maximum generator capacity limits and the total power output of generating units equals to the power load demand. It is indicated that the equality and inequality constraints always satisfy. The proposed MVMO S provides not only better solution but also more robust than the MVMO and the difference between the maximum and minimum costs from the proposed MVMO S is small. Table 8 shows the ratio between the standard deviation and the minimum cost obtained by MVMO S for all systems. The ratio between the standard deviation and the minimum cost is less than 0.149%. It clearly shows that the performance the proposed MVMO S is robust. In addition, the comparison of the total cost obtained by MVMO S and many other methods from Tables 2, 4, 5, and 7 shows that the MVMO S can obtain better total fuel costs and more robust than most of other reported methods. Consequently, the MVMO S can obtain near global solution with high probability.

Disadvantages of MVMO S
The only disadvantage of MVMO S is computational time.  time of the MVMO S is slower than the classical MVMO. This is because the MVMO S starts the search with a set of particles while the MVMO starts the search with single particle.

Conclusion
This paper has presented an application of new method for solving the ED problem. The proposed MVMO S has been successfully solved the ED problem with valve-point effects. Three test cases have been carried out to demonstrate its effectiveness and efficiency. The comparisons of numerical results have shown that the proposed MVMO S has better performance than other optimization techniques exist in the literature. It is also confirmed that the MVMO S outperformed the classical MVMO in global search for the  nonconvex problem. Therefore, the proposed MVMO S could be favorable for solving the ED problem with valve-point effects and other nonconvex ED problems as well. In the future, the MVMO S will be applied for solving dynamic ED, hydrothermal ED with cascaded hydro plants and emission constrained ED.