Solving Power Economic Dispatch Problem with a Novel Quantum-Behaved Particle Swarm Optimization Algorithm

+is paper proposes the shrink Gaussian distribution quantum-behaved optimization (SG-QPSO) algorithm to solve economic dispatch (ED) problems from the power systems area. By shrinking the Gaussian probability distribution near the learning inclination point of each particle iteratively, SG-QPSOmaintains a strong global search capability at the beginning and strengthen its local search capability gradually. In this way, SG-QPSO improves the weak local search ability of QPSO and meets the needs of solving the ED optimization problem at different stages. +e performance of the SG-QPSO algorithm was obtained by evaluating three different power systems containing many nonlinear features such as the ramp rate limits, prohibited operating zones, and nonsmooth cost functions and compared with other existing optimization algorithms in terms of solution quality, convergence, and robustness. Experimental results show that the SG-QPSO algorithm outperforms any other evaluated optimization algorithms in solving ED problems.


Introduction
Solving economic dispatch (ED) problem is to ensure that the power production is safe, high-quality and meets the customer's electricity demand by using various technical and management measures to make the power production equipment in the best working state and reach the lowest cost of the power system. Simultaneously, the nonlinear characteristics of the generator, such as ramp rate limit, prohibited operating area, and nonsmooth cost function, should be considered. erefore, the ED problem is a complex nonlinear problem with many constraints.
Besides, some improved algorithms are proposed. For example, Kaboli proposes an artificial cooperative search (ACS) [19] optimization algorithm, which is provided by balancing exploration of the problem's search space and exploitation of better results through the use of two advanced evolutionary operators and only one control parameter. Pandey proposes an improved FWA with Chaotic Sequence Operator (IFWA-CSO) [20], in which the global search ability of FWA has been strengthened. Sun uses the improved particle swarm optimization algorithm RDPSO (Random Drift Particle Swarm Optimization) to solve the power optimization problem [21]. In order to improve the local search capability of PSO, KHAMSAWANG and Grag adopt a hybrid method of differential evolution or genetic algorithm to enhance the local capability [22,23]. Besides, Khan et al. control the diversity of particle swarm to avoid the algorithm falling into the local optimal [24][25][26]. Besides, other improved algorithms such as the iterated-based optimization method [27], dynamically controlled particle swarm optimization method [28] are also performed well on ED problems.
However, for these methods, the major deficiencies still are highly sensitive to the initial value of the control parameters and a large number of control parameters or some situations such as trap into local optima and premature easily occur. erefore, it is difficult to obtain satisfactory and feasible solutions for multiconstrained, nonlinear optimization problems. e quantum-behaved particle swarm optimization (QPSO) algorithm is a variant PSO algorithm that has strong and robust global search ability but has relatively low convergence speed and local search ability. erefore, this paper proposes a shrink Gaussian distribution quantum-behaved optimization (SG-QPSO) algorithm to solve ED problems from the power systems area. By shrinking the Gaussian probability distribution near the learning inclination point of each particle iteratively, SG-QPSO not only maintains a strong global search capability at the early search stage but also strengthens the local search capability at the later stage. In this way, the proposed SG-QPSO improves the weak local search ability of QPSO and meets the needs of solving the ED optimization problem at different stages. Besides, SG-QPSO has fewer parameters than other optimization algorithms, such as genetic algorithms, differential evolution, or other one-dimensional search algorithms like the Powell algorithm, which is easier to control. e remaining chapters of this paper are arranged as follows: Section 2 describes the proposed SG-QPSO in detail. Section 3 describes the mathematical formulation of the ED problem in detail. Section 4 shows the experimental results obtained by SG-QPSO on three power systems, compares its results with previous algorithms, and analyzes its merits and disadvantages. Section 5 summarizes this article and introduces the focus of future work.

The Proposed Algorithm
In this section, we first introduce the theoretical aspect of the canonical QPSO algorithm and then analyze its advantages and disadvantages when dealing with ED problems. Based on the analysis above, an improved QPSO algorithm called SG-QPSO is proposed, and its process of solving the ED problem is given as a flowchart.

QPSO Algorithm.
e crucial issue of QPSO is how to design a reasonable potential energy field [29]. Clerc analyzed the dynamic evolution process and showed that each particle gradually converges to a point [30]. In other words, those points attract the particles swarm during the search process.
ose points are named learning inclination points (LIPs) in QPSO, and its current position is calculated as follows: where φ j i,t ∈ (0, 1) is a random variable generated by uniform distribution, P j i,t is the value that denotes the j-th dimension of the current personal best position of particle i, and G j t represents the value of the jth dimension of the current global best position. e updated formulation of each particle in QPSO is as follows: where u j i,t ∈ (0, 1) is a random variable generated by uniform distribution, L j i,t is the length between the current position of each particle and the mean personal best position, and its definition is as follows: e mean personal best position is calculated by the following: By simulating the strong uncertainty of the superposition of states in the quantum system, QPSO makes it possible to cover the whole probability search space during the search process. Simultaneously, the algorithm uses the mean personal best position to guide the particles to gradually aggregate to LIPs.
is delay strategy makes the algorithm convergence slowly and helps the algorithms enhance their global search ability. e details of the QPSO algorithm can be found in [29].

e SG-QPSO Algorithm.
When the area near the global or local optima is tiny, particles in QPSO are easier to skip this area for the range of update area of each particle is large. At the same time, considering that Gaussian distribution is introduced to generate random variables sequence may weaken the global search ability, a shrink Gaussian distribution quantum-behaved particle swarm optimization (SG-QPSO). In SG-QPSO, the variance of Gaussian distribution declines linearly to shrink the area of each particle near its LIP, which enhances the local search ability gradually and maintains the global search ability of QPSO during the search process. e update formula of the SG-QPSO algorithm is as follows: where k is random values generated by using the uniform probability distribution functions in the range [0, 1]. e learning inclination point p and mean personal best position C is calculated by equations (1) and (4). Note that the number of particles in a particle swarm is M: In equation (6), σ denotes the variance of Gaussian distribution, declining linearly from the initial value σ 1 and the end value σ 0 in the search process. T is the maximum number of fitness evaluations and t represents the current iteration step. e Pseudocode of the SG-QPSO algorithm is shown in Algorithm 1.

Mathematical Model of Power System Economic Dispatch.
e ED problem can be reduced to an optimization problem. Its goal is to determine the power output level of the online generator and further minimize the total fuel cost of all generators within a period while satisfying various nonlinear constraints.

Objective Function.
e objective function of ED problem can be defined as follows: where F j (P j ) is the cost function of jth generator set, P j Is the actual output of the jth generator set, and N g is the total number of generators in the power system. e cost function of each generator set is related to the actual power put into the system and is usually modeled with a smooth quadratic function: a j , b j , and c j is the cost correlation coefficient of the jth generator set.

Constrains of ED Problem.
In this work, we consider the following constraints of the ED problem: (a) Power balance constraints e power balance constraints are expressed as follows: e total power generation of the system is equal to the load demand of the system plus transmission loss. In other words, the total power generation N g j�1 P j should be equal to the total power demand P D plus transmission network loss P L while minimizing total power generation costs. P L is usually approximated by the Krone loss formula, which represents the relationship between the transmission loss and the output level of the system generator set: According to the design requirements of the generator, the amount of power generated by each unit must vary between its minimum P min j and maximum P max j production limits. (c) Ramp rate limitation During the actual operation of the generator set, the operating range of all online units is limited by their ramp rate limitation. According to [5], the inequality constraint due to the slope limitation is as follows: (i) If the amount of power generation increases (ii) If the amount of power generation decreases (d) Prohibited operating areas Since the steam valve operates in the bearing (i.e., vibration), the system contains some prohibited operating zones. In the actual power system, the load demand of the power system must avoid prohibited zones. erefore, if the constraints in (11) are considered, the feasible operating area of the jth generator set can be described in the following way: where P l j,k and P u j,k is e upper and lower boundaries of the kth prohibited zone of the jth generator set, n j is the number of prohibited zones of the jth generator set.

3.1.3.
e Mathematical Formulation of ED Problem. Combining the equations (11)-(13), we obtained the following: Mathematical Problems in Engineering 3 erefore, considering the feasible operation zones, we can express the ED problem as the following constrained optimization problem:

Solving ED Problem Using SG-QPSO.
Before applying the SG-QPSO algorithm to the ED problem, make the following provisions: Each component in a single particle represents a generation unit, so each particle represents a candidate solution for a given ED problem.
e current position of the tth particle P g,i with N g generation units can be given by the following: (17) where M is the population size, which is the index to generate jth unit, and P is the output power of the ith generating unit in the jth particle.

Objective Function and Constraint
Handling. e equality constraints in the formula can be handled by adding penalty terms. e objective function becomes as follows: (18) where K n is called the penalty coefficient and is a positive real number, which increases with the number of iterations. e penalty term in equation (18) is the equality constraint in equation (9). When the ED problem is restricted, it is solved by a population-based search method (such as SG-QPSO). If the equality constraint is violated, the value of the penalty term is nonzero.
On the one hand, when the candidate solution violates the equation-constrained candidate solution, equation (18) gives a larger objective function value so that the candidate solution has a greater probability of being discarded. On the other hand, when the equality constraint is not violated, the penalty term is zero. No matter how large the penalty coefficient is, the final penalty term value is zero. erefore, the final objective function value is obtained by adding the value of the penalty term to the given objective function value so as to control each candidate solution in the population to approach the feasible solution area.

e Summary of ree Power Systems.
ree real power systems are used to verify the effectiveness of SG-QPSO, with considering the ramp rate limit and the prohibited zones. Other optimization methods are also tested on these three systems for comprehensive performance comparison, including binary-coded GA [5], PSO with inertial weights [6], DE [7], Ant Colony Search Algorithm (ACSA) [13], artificial immune system (AIS) [14], bee colony optimization (BCO) [16], firefly algorithm (FA) [17], standard PSO (SPSO) with shrinkage and inertial weights [31], chaotic PSO (CPSO) [32], antipredatory PSO (APSO) [33], mixed gradient descent PSO (HGPSO) [31], mixed PSO with mutation (HPSOM) [31], QPSO [29], and GQPSO [34]; the Hopfield neural network (NN) was also tested. Note that, for each system, all test methods use the same objective function.  Tables 1 and 2. In the normal operation of the system, the loss factor with a basic capacity of 100 MVA is shown in Table 3. It is a small system and is the easiest problem among three test systems, and the dimension of the ED problem is 6. As shown in Table 2, there are 12 prohibited zones in this system, and 13 inequality constraints are generated according to these prohibited zones. System 2: this system has 15 thermal units, the characteristics of which are given in Tables 4 and 5. e load demand of the system is 2630 MW. Due to space constraints, the loss coefficient matrix is not listed. is system is a medium-scale system, and its ED problem has 15 dimensions. As shown in Table 5, the power generating units 2, 5, 6, and 12 have 11 prohibited zones. erefore, according to the inequality constraints described above regarding its ED problem, the ED problem of this system is relatively difficult to optimize compared with System 1. System 3: the system contains 40 units in a large-scale hybrid power generation system named Tai power system. e load demand of the system is 8550 MW. Due to space constraints, unit parameters and loss factors are not listed. e dimension of the ED problem of this system is 40. In the ED problem of this system, each power generation unit has no prohibited zone, so there are fewer unequal constraints, but this does not significantly reduce the difficulty of the problem. e large size and multiple fuel options attribute of this system make the ED problem to be one of the most difficult to solve among the three systems.

Parameters Setting.
For each of these three systems, the maximum number of iterations to execute the objective function of each optimization algorithm is set to 20,000. Simultaneously, two sets of experiments are performed on each system for each algorithm. One has a population size of M � 100 and a maximum number of generation of G max � 200, another population size of M � 20 and G max � 1000. On each system, each algorithm performed 100 independent experiments with a given maximum generation G max and population size M. e penalty coefficient in the objective function is set to where t is the current number of generations. e other experimental configuration settings are as follows: e size of the crossover probability p c � 0.8, and the size of the mutation probability p m � 0.1 in GA; the constant mutation factor used by the DE algorithm is f m � 0.4, and the size of the crossover rate is CR � 0.8; for PSO with inertia weight, the inertia weight decreases linearly from 0.9 to 0.4 during the search process, the acceleration coefficients c 1 � c 2 � 2.0 and V max � ((P max j − P min j )/2); for SG-QPSO, during the search process, the variance of the Gaussian distribution σ decreases linearly from 5 to 0.001. Input: Input parameters: σ 1 and σ 0 Output: Fitness value of EDPs Step 1: Initialize the current position of each particle, and set its personal best position to be its current position; Step 2: Set t � 0; Step 3: While the termination condition is not met, perform the following steps; Step 4: Set t � t+1 and compute the mean best position C i t ; Step 5: From i � 1 executes the following steps; Step 6: Evaluate the value of X, and update G t and P i t ; Step 7: Update position of each particle according to (2); Step 8: set i � i+1, and return to Setp 5 until i � M; Step 9: Return to Step 3; Step 10: Return result ALGORITHM 1: e pseudocode of SG-QPSO algorithm.    Table 6 lists the total cost of each method for the ED problem of System 1. From Table 6, the average cost and standard deviation of 100 runs of SG-QPSO are better than other methods, which shows that the performance and robustness of SG-QPSO on System 1 is better than other algorithms. Under the two experimental configurations, the CPSO algorithm is the second best method of the system in terms of the mean cost. When M � 100 and G max � 200, the worst performance optimization algorithm is APSO, and the mean cost obtained in 100 runs is 15473.3164$/h. For this system, Hopfield NN produces the worst results. When M � 20 and G max � 1000, the mean cost of SPSO performs the worst, although the lowest cost found in 100 runs is 15442.9130 $/h, which is better than other comparison algorithms other than SG-QPSO. When M � 100 and G max � 200, the SG-QPSO algorithm obtained the best solution, the lowest standard deviation, and the best mean cost. Table 7 lists the solution vector P j (j � 1, 2, . . . , 6) relative to the best solution. e minimum cost of SG-QPSO running 100 times is 15442.7831 $/h when M � 100 and G max � 200. To prove that the equality constraints in (16) are satisfied, we add the power loss (12.4173 MW) to the load demand (1263 MW) for a total of 1275.4173 MW. By comparing the sum to the total power output (1276.4183 MW), we can find that the equality constraints (i.e., power balance constraints) are well satisfied. Figure 1 visualizes the convergence of all test methods on the ED problem of System 1 for an average of 100 experiments, indicating that SG-QPSO has better convergence than other algorithms. Table 8 lists the mean cost, minimum and maximum cost, and the standard deviation values obtained by performing 100 experiments with each algorithm of the ED  problem on System 2. Obviously, in any experiment configurations, SG-QPSO obtained the lowest cost. It can be seen that when M � 100 and G max � 200, QPSO ranks second with a mean cost of 15455.6220$/h. When M � 20 and G max � 1000, the second-best method is HPSOM, and its average cost is 32811.3701$/h. In the two experimental settings, the worst-performing optimization algorithms are BCO (mean cost 33113. 0149$/h) and GA (mean cost 33188.5443$/h). Table 9 lists the results of the ED problem of each algorithm on System 3. From this table, in all the algorithms of the two experiment configurations, SG-QPSO achieved the best results. When M � 100, G max � 200, the second-best performing algorithm is HPSOM, and the average cost obtained in 100 experiments is 131614.7211 $/h. In this set of experiments, GA performed the worst among all the algorithms participating in the test.

Experiment Results and Analysis.
In two different experiments configurations, the best solution to the system is obtained through the SG-QPSO algorithm. e minimum cost of the algorithm in 100 runs was 32663.2635 $/h. Table 10 shows the corresponding total power generation cost for the best solution is 32663.2635 $/h when M � 20 and G max � 1000. From Table 9, the difference between the available load (2659.5748 MW-29.5683 MW) and the load demand (2630 MW) is 0.0065, which proves that the power balance equation constraint in (9) and (18) is satisfied. Figure 2 shows that for this ED problem, the SG-QPSO method has better convergence than other    algorithms, providing faster convergence speed and the best final mean fitness value.
From Table 9, when M � 20, G max � 1000, the SG-QPSO algorithm can obtain the best solution to the ED problem of the system. Due to the space limitations of this article, we only list the best results of total power output and system transmission loss obtained when the minimum total cost is 129078.4705 $/h in Table 11. In order to prove that the equality constraints in (9) and (18) are satisfied, we will combine the left side of (9) (that is, total output power (8631.9425 MW)) and the right side of (9) (that is, power loss (81.9390 MW)) and the sum of the load demand (8550 MW) (8631.9425 MW) is compared, and both satisfy the equality constraints. In addition, as shown in Figure 3, SG-QPSO also has the best convergence for the fitness value of the ED problem of the system.    Mathematical Problems in Engineering 9

Conclusion
A shrink Gaussian distribution Quantum-behaved particle swarm optimization (SG-QPSO) algorithm is proposed to effectively solve the power economic dispatch problem by considering the nonlinear characteristics of the generator. SG-QPSO yields better solutions of different systems compared to any other tested algorithms, and highly similar optimization results among 100 independent trails of each system confirmed its robustness. In addition, the performance of SG-QPSO shows a stronger global search performance, which can be seen from the relatively low system cost obtained in 100 runs. erefore, the SG-QPSO method is a promising tool for solving ED problems and other optimization problems in the industrial field. Our future work will focus on the application of the SG-QPSO method on other industrial problems, as well as the theoretical analysis of the algorithm search mechanism.
Data Availability e experimental data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.