2. School of Science, University of Science and Technology Liaoning, Anshan 114051, China;
3. Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA;
4. College of Computer and Communication Engineering, Liaoning Shihua University, Fushun 113001, China;
5. Department of Computer Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China;
6. Department of Computer Science and Technology, Tongji University, Shanghai 201804, China
Economic dispatch (ED) is one of the most fundamental optimization problems in electric power systems with the objective to minimize the total cost for power generation. It aims at economically allocating the load demand among the generators while satisfying several equality and inequality constraints in the systems.
As a classical optimization problem, ED with smooth cost functions has been solved by numerous traditional programming methods such as gradient methods [1], lambda iteration method [2], quadratic programming [3], linear programming [4], dynamic programming [5] and Lagrangian method [6]. In recent years, several ED problems with some complex and nonsmooth functions are proposed by considering transmission network losses and plant constraints such as valvepoint effects, multiple fuel options, generation ramp rates, and prohibited operation zones. Most of the traditional techniques are not capable of efficiently solving such problems that have nonconvex, noncontinuous or nonlinear solution spaces. Over the past two decades, evolutionary computation developed rapidly [7][14]. Many modern metaheuristic algorithms and their variants were successfully used to solve such problems. According to their characteristics, they can be divided into three types: evolutionary algorithms [15][19], simulated ecosystem algorithms [20][24], and swarm intelligence algorithms [25][31].
Cuckoo search (CS) algorithm is a natureinspired swarm intelligence technique based on the brood parasitism of some cuckoo species, as introduced by Yang and Deb in 2009 [32]. Due to its simple concept and easy implementation, CS has been successfully applied to tackle unimodal and multimodal numerical optimization and engineering problems [33][41]. Many researchers have applied CS to solve ED problems in power systems [9], [42][48]. Several studies show that CS can always find the optimal results, but it may not guarantee a fast convergence because its searching process relies entirely on a random walk [9], [36], [47]. Meanwhile, a small or regular step may cause it to be trapped in a local optimal solution [48]. To overcome this deficiency, a modified cuckoo search algorithm (MCSA) is proposed in this paper, where a neighbor study strategy is designed and a selfadaptive parameter selection strategy is formulated. Compared with the existing studies, we have made three contributions:
1) A new selfadaptive step size strategy is proposed such that the step decreases in different speeds as iterations proceed. In the beginning, it maintains a high value in order to enhance the exploration ability. Then, it declines rapidly to its minimum value in order to make MCSA converge steadily to a refined solution.
2) A neighbor study strategy is adopted. When the best solution is no longer updated after a number of iterations, each solution can exchange the information with others randomly.
3) A new lambda iteration method is designed to generate feasible solutions at the initial stage. In MCSA, all solutions must be feasible. A relaxation method is thus designed to handle the equality constraint that may lead to infeasible solutions.
The rest of paper is organized as follows: Section Ⅱ describes the ED problem. The standard CS and MCSA are introduced in Section Ⅲ. Section Ⅳ implements MCSA to solve the ED problem. Section Ⅴ is dedicated to numerical simulations and results. Conclusions and future work are given in Section Ⅵ.
Ⅱ. PROBLEM FORMULATIONThe problem discussed in this paper is the same as those in literature [15][20], [26], [43]. The objective of ED problem is to minimize the fuel cost of generators in electric power systems for a given load demand subject to various constraints.
A. Objective FunctionThe fuel function without valvepoint loading of generators is given below.
$ \begin{align} \min F_{t} =\sum _{i=1}^{D}F_{i} (P_{i} )= \sum _{i=1}^{D}a_{i} +b_{i} P_{i} +c_{i} P_{i}^{2} \end{align} $  (1) 
where
Practical largesize generators usually have multivalve steam turbines. When each steam valve is on or off, it may produce a ripple. Usually, a sinusoidal term is added in (1) with consideration of valvepoint effects (VPE) [2], thus leading to
$ \begin{align} \min F_{t} =\sum _{i=1}^{D}a_{i} +b_{i} P_{i} +c_{i} P_{i}^{2} +\lefte_{i} \sin \left(f_{i} \left(P_{i}^{\min } P_{i} \right)\right)\right \end{align} $  (2) 
where
In order to balance the power, the total generated power should meet the power demand and transmission loss (TL)
$ \begin{align} \label{GrindEQ__3_} \sum _{i=1}^{D}P_{i} =P_{T} +P_{L} \end{align} $  (3) 
where
$ \begin{align} \label{GrindEQ__4_} P_{L} =\sum _{i=1}^{D}\sum _{j=1}^{D}P_{i} B_{ij} P_{j} +\sum _{i=1}^{D}B_{0i} P_{i} +B_{00} \end{align} $  (4) 
where
The output power of each generator has a lower limit and an upper one
$ \begin{align} \label{GrindEQ__5_} P^{\min}_{i} \le P_{i} \le P^{\max}_{i} , ~~~ i=1, 2, \ldots , D \end{align} $  (5) 
where
Each generator may have certain prohibited operating zones (POZ) caused by opening or closing its steam valve. The feasible operating zones of generator
$ \begin{align} \label{GrindEQ__6_} P_{i} \in \begin{cases} {P_{i}^{\min } \le P_{i} \le P_{i, 1}^{l} } \\[1mm] P_{i, j1}^{u} \le P_{i} \le P_{i, j}^{l} \\[1mm] P_{i, n_{j} }^{u} \le P_{i} \le P_{i}^{\max } \end{cases}, \ \ {j=2, 3, \ldots , n_{j} } \end{align} $  (6) 
where
Practically, all generators should satisfy the physical limitation of starting up and shutting down by using ramp rate limits (RRL). The increase and reduction of power generation in each generator are limited by
$ \begin{align} \label{GrindEQ__7_} &P_{i} P_{i}^{0} \le U_{i} \\[1mm] \end{align} $  (7) 
$ P_i^0  {P_i} \le {L_i} $  (8) 
where
Combining (7) and (8) with (5) results in the change of the generation limits to
$ \begin{align} \label{GrindEQ__9_} \underline{P}_{i} \le P_{i} \le \overline{P}_{i}, ~~~ i=1, 2, \ldots , D \end{align} $  (9) 
where
$ \begin{align*} &\underline{P}_{i} =\max (P_{i}^{\min }, P_{i}^{0} L_{i} ), ~~~\overline{P}_{i} =\min (P_{i}^{\max }, P_{i}^{0}+ U_{i} ). \end{align*} $ 
CS is a populationbased swarm intelligence algorithm inspired by the interesting breeding behavior of cuckoo [32]. It is enhanced by the socalled Lévy flight, rather than by simple isotropic random walks. There are mainly three rules during its searching process [49]. Its main steps are given below:
1) Initialization
Suppose that there are
$ \begin{align} \label{GrindEQ__10_} X_{k}^{i} =P_{i}^{\min } +r_{1} \times \left(P_{i}^{\max } P_{i}^{\min } \right) \end{align} $  (10) 
where
2) New solution generation via Lévy flight
After initialization, CS uses a Lévy flight random walk to search a new solution, denoted by
$ \begin{align} X_{k\left({\rm new}\right)}^{i} =X_{k}^{i} +\alpha \times r_{2} \times \Delta X_{k\left({\rm new}\right)}^{i} \end{align} $  (11) 
where
$ \Delta X_{k\left({\text{new}}\right)}^{i} =\frac{u}{\leftv\right^{{\frac{1}{\beta} } } } \times \frac{\sigma _{u} \left(\beta \right)}{\sigma _{v} \left(\beta \right)} \times \left(X_{k}^{i} X_{\rm gbest}^{i} \right) $  (12) 
$ {\sigma _{u} \left(\beta \right)= \left\{\frac{\Gamma (1+\beta )\times \sin \left(\pi \times \frac{\beta}{2}\right)}{\Gamma \left[\frac{1+\beta}{2} \right]\times \beta \times 2^{\frac{\beta 1}{2}}} \right\}^{\frac{1}{\beta}}, } ~~~ {\sigma _{v} \left(\beta \right)=1} $  (13) 
where
3) A new solution generated via a crossover operation
The discovery of an alien egg in a nest of a host bird has a probability denoted by
$ \begin{align} &X_{k'}^{i} =\begin{cases} X_{k}^{i}+r_{3} \times \left(X_{c1}^{i}X_{c2}^{i} \right), &{\rm if}~ {r_{4} >P_{a} } \\[1mm] X_{k}^{i} , &{\rm otherwise} \end{cases} \notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ {i=1, 2, \ldots, D} \end{align} $  (14) 
where
The pseudo code of CS is given as Algorithm 1.
Algorithm 1 The Standard CS Algorithm 
1. Initialize a population of host nests 
2. Get the global best solution 
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4. Update the solution via Lévy flight, compare with the last generation, and keep the current individual as the best nest 
5. Replace a fraction of eggs to generate a new solution via a crossover operation 
6. Keep the current individual solution as the best one by comparing the solutions before and after a crossover operation 
7. Get the global solution 
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B. SelfAdaptive Step Size
$ \begin{align} \label{GrindEQ__15_} \alpha \left(t\right)=ba\times \frac{\exp \left(\frac{10\times \left(t1\right)}{t^{M} 1} \right)1}{\exp \left(10\right)1}, ~~~ t=1, 2, \ldots , t^{M} \end{align} $  (15) 
where
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Fig. 1 Value of 
In a standard CS, new solutions are generated via (14). It is obvious that solutions may easily be trapped in a local optimum if the search environment is complex with local optima. A solution may have the same values as the global optimum in some dimensions, but it has a high fitness value for a minimization problem because of the poor solutions in the other dimensions [30]. In order to make a better use of the beneficial information, a neighbor study strategy is proposed to improve CS as inspired by the work [30], [31], [51].
$ \begin{align} \label{GrindEQ__16_} X_{k'}^{i} =\left(1K\right)\times X_{k}^{i} +K\times\gamma _{k}^{i}, ~~~ i=1, 2, \ldots , D \end{align} $  (16) 
where
$ \begin{align} K=\begin{cases} 1, &{\rm if}~r_{5} >P_{\alpha } \\ 0, &\mbox{otherwise}\end{cases} \end{align} $  (17) 
where
Every solution in the nest is able to learn from other solutions' best experiences in different dimensions. Thus, the ability of exploration is enhanced by such information sharing mechanism. Algorithm 2 describes the method of generating an exemplar
Algorithm 2 Generate Exemplar Dimensions for 
1. Input 
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Ⅳ. MCSA TO SOLVE ED PROBLEMS
In this section, MCSA is applied to solve the ED problem. First, a new solutiongenerated method is introduced. Then, inequality and equality constraints' handling techniques are proposed. Finally, the main steps of MCSA are described.
A. Modified Lambda Iteration MethodIn the standard CS, a solution generated via (10) is usually infeasible and difficult to repair especially in largescale systems. Thus it is necessary to propose a fast and reasonable method to solve this infeasibility problem. The classical lambda iteration was introduced in literature and applied in many software packages [2]. Although it is widely used by power utilities for ED, improper selection of the initial value may cause slow convergence and sometimes leads to divergence [52][54]. In this section we propose a method to generate a new solution effectively. Although the new solution is sometimes infeasible, it is easier to repair compared to the traditional methods especially in a system with many generators.
At the
$ F_{i} \left(P_{i} \right) = a_{i} + b_{i} P_{i} + c_{i} P_{i}^{2} \notag\\ \ \ \ \ \ \qquad\qquad P_{i, \min } \le P_{i} \le P_{i, \max } , ~\, i = 1, 2, \ldots, D $  (18) 
$ \frac{dF_{i} }{dP_{i} } =\lambda _{i} =b_{i} +2c_{i} P_{i}. $  (19) 
Then, we calculate
$ \lambda _{i, \min } =b_{i} +2c_{i} P_{i}, ~~~ {\text{at}} ~{P_{i} =P_{i, \min } } $  (20) 
$ \lambda _{i, \max } =b_{i} +2c_{i} P_{i}, ~~~ {\text{at}} ~ {P_{i} =P_{i, \max } }. $  (21) 
At last, we calculate
$ \lambda _{\rm mean} =\frac{1}{2D} \sum\limits _{i=1}^{D}\left(\lambda _{i, \min } +\lambda _{i, \max } \right) $  (22) 
$ \lambda _{\rm var} =\frac{1}{2D} \sum\limits _{i=1}^{D}\left(\left(\lambda _{i, \min } \lambda _{\rm mean} \right)^{2} +\left(\lambda _{i, \max } \lambda _{\rm mean} \right)^{2} \right). $  (23) 
The method to generate a new solution is given in Algorithm 3.
Algorithm 3 Generate a New Solution 
1. Calculate 
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In Algorithm 3, sqrt represents the square root, randn represents a normally distributed pseudorandom number, and rand represents a uniformly distributed pseudorandom number.
B. Constraint Handling Mechanism1) Inequality Constraints
The global solutions should satisfy inequality constraint (9). During the searching process, if there are some solutions that are not in the scope of the feasible solution region, MCSA may stop at the region boundary.
For constraint (6), when a unit operation is in one of its POZ, a repairing strategy is activated [43].
$ \begin{align} \label{GrindEQ__24_} &P_{i}^{\rm new} = \begin{cases} P_{i, j}^{l} , &{\rm if}~P_{i} \le P_{i, j}^{m} \\[1mm] P_{i, j}^{u}, &{\rm if}~P_{i}>P_{i, j}^{m} \end{cases} \notag\\ & \qquad\qquad\qquad\qquad i = 1, 2, \ldots , D, \ \ j = 2, 3, \ldots , n_{j} \end{align} $  (24) 
where
2) Equality Constraints
Although a solution satisfies all inequality constraints, it may be infeasible if it does not satisfy the power balance constraint (3). When it happens, the simplest approach for handling such infeasible solutions is to use the penalty function. However, it is well known that defining a proper penalty coefficient is difficult. Hence, a slack approach [42], [43] is used in this paper.
Assume that a slack generator
$ \begin{align} \label{GrindEQ__25_} P_{s} =P_{T} +P_{L} \sum _{i=1, i\ne s}^{D}P_{i}. \end{align} $  (25) 
The transmission loss
$ \begin{align} \label{GrindEQ__26_} P_{L} =&\ B_{ss} P_{s}^{2} +\left(2 \sum _{i=1, i\ne s}^{D}B_{si} P_{i} +B_{0s} \right)\times P_{s} \notag\\[1mm] &\, + \sum _{i=1, i\ne s}^{D}\sum _{j=1, j\ne s}^{D}P_{i} B_{ij} P_{j} + \sum _{i=1, i\ne s}^{D}B_{0i} P_{i} + B_{00} . \end{align} $  (26) 
By substituting (26) into (25), we have
$ \begin{align} \label{GrindEQ__27_} A\times P_{s}^{2} +B\times P_{s} +C=0 \end{align} $  (27) 
where coefficients
$ A=\ B_{ss} $  (28) 
$ B= \ 2\sum\limits _{i=1, i\ne s}^{D}B_{si} P_{i} +B_{0s} 1 $  (29) 
$ C = \ \sum\limits _{i=1, i\ne s}^{D}\sum\limits _{j=1, j\ne s}^{D}P_{i} B_{ij} P_{j} + \sum\limits _{i=1, i\ne s}^{D}B_{0i} P_{i}\notag\\ \ \ \ \ \ \ \ + B_{00} +P_{T} \sum\limits _{i=1, i\ne s}^{D}P_{i}. $  (30) 
The power output of can be calculated by solving (27). The pseudo code is shown in Algorithm 4.
Algorithm 4 Calculate 
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If the return value is 0, Algorithm 4 is reexecuted until solution
$ \begin{align} \label{GrindEQ__31_} P_{s} =P_{T} \sum _{i=1, i\ne s}^{D}P_{i}. \end{align} $  (31) 
3)Application of MCSA for ED
In MCSA, each nest represents a solution and a population of nests are used for finding the best solution of the problem. The main steps of MCSA are illustrated in Fig. 2.
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Fig. 2 Flow chart of MCSA. 
In order to demonstrate the efficiency and robustness of MCSA for solving the ED problem, some cases are conducted on 6, 13, 20, and 40unit systems, and the results are compared with several stateoftheart ED algorithms in the literature. All case studies are implemented in MATLAB R2016a, on a personal computer with Intel i5 2.3 GHz processor, 4 GB of RAM and Windows 10 Professional. Due to the stochastic nature of an evolutionary algorithm in each case, 50 independent trials are conducted to calculate the best, mean, and worst fuel costs, and its standard deviation for each test system.
A. Parameter SelectionThe performance of MCSA is sensitive to parameter settings. In this work, the parameters of MCSA described in Table Ⅰ are selected based on a rigorous empirical study for each case. In this paper, the value of the selfadaptive step size
The first tested system is a 6unit system which has a demand of 1263 MW with POZ, TL and RRL. Its input data are taken from [55], [56]. The objective function for this system is smooth and no convexity is given by the prohibited operating zones and ramp rate limits. The best generation values, transmission losses and optimal cost obtained by MCSA are presented in Table Ⅱ. Note that all system constraints, such as POZ and RRL are satisfied. The total generation cost and the corresponding transmission loss are 15 449.8995 ＄/h and 12.9582 MW, respectively.
Table Ⅲ shows the obtained best cost, mean cost, worst cost, standard deviation and time for this test system after 50 trial runs. To show the differences among the results, we use four digits after the decimal point. These results are compared with other algorithms that have been reported recently such as modified artificial bee colony (MABC) [57], backtracking search algorithm (BSA) [58], differential harmony search (DHS) [59], new particle swarm optimization with local random search (NPSOLRS) [60], particle swarm optimization (PSO) [55], multiple tabu search (MTS) [61], chaotic bat algorithm (CBA) [62], and CS [63].
$ \begin{align} \label{GrindEQ__32_} TOL=\sum _{i=1}^{D}P_{i} P_{T} P_{L}. \end{align} $  (32) 
Through comparison, we can find that the best, mean, worst costs and standard deviation obtained by MCSA are the least. Although the best, mean and worst costs are same for MABC and MCSA, the standard deviation, time and obtained by MCSA are much better than those by MABC. The standard CS and MCSA have almost the same effect on Case 1. Moreover, it shows that the MCSA is more consistent and stable than the other algorithms.
Fig. 3 shows the generation cost convergence of the best solution with iterations for CS ans MCSA for a typical run. It can be seen that both CS and MCSA enjoy smooth convergence, but MCSA is faster than CS.
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Fig. 3 Convergence characteristic of MCSA for Case 1. 
System 2 includes 13 generators and supplies a total load demand of 2520 MW with valvepoint effects but without ramp rate limits and prohibited operating zones. Moreover, this system is analyzed for both "without transmission losses (Case 2)" and "with transmission losses (Case 3)". The generator data of this system are taken from [64], and the loss coefficient is from [65] with correction
It can be seen from Table Ⅴ that the statistical results obtained by MCSA are highly competitive compared to these by MABC [57], shuffled differential evolution (SDE) [65], one rank cuckoo search algorithm (ORCSA) [67], distributed sobol particle swarm optimization and tabu search algorithm (DSPSOTSA) [68], cross entropy method and sequential quadratic programming (CESQP) [66], tabu search algorithm (TSA) [68], ant colony optimization (ACO) [69] and CS. MABC, SED, ORDSA and MCSA can get the best cost, but only MABC and MCSA get the optimal mean and the least cost. Although MABC has a good performance on standard deviation value, it usually takes much more time than MCSA.
Table Ⅵ shows a comparison among the results obtained by MCSA and other recently published stochastic methods such as oppositional real coded chemical reaction optimization (ORCCRO) [70], MABC [57], SDE [65], biogeography based optimization (BBO) [70], differential evolution with biogeography based optimization (DEBBO) [70], improved coordinated aggregation with particle swarm optimization (ICAPSO) [71], selftuning hybrid differential evolution (STHDE)
[16], hybrid differential evolution (HDE) [16], differential evolution (DE) [16] and CS. The best cost obtained from MCSA is the least in comparison with other methods except ORCCRO. It must be mentioned that the best cost value of ORCCRO with 24 513.91 (＄/h) is obtained by adopting a higher tolerance (
Figs. 4 and 5 show the generation cost convergence of the best solution with iterations for CS and MCSA. It can be seen that both CS and MCSA enjoy smooth convergence, but MCSA converges to the optimal solution faster.
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Fig. 4 Convergence characteristic of the MCSA for Case 2. 
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Fig. 5 Convergence characteristic of the MCSA for Case 3. 
In this subsection, the ED problem is solved for a system with 20 generators considering the transmission losses and a demand of 2500 MW. VPE, RRL and POZ are neglected. The data of this test system can be found in [72]. Table Ⅶ shows the optimal generations and cost obtained by MCSA for Case 4. The optimal cost and corresponding transmission loss are 62 456.6331 ＄/h and 91.9667 MW, respectively.
Table Ⅷ shows the comparison among MCSA and other methods such as CBA [62], continuous quick group search optimizer (CQGSO) [73], ORCSA [67], group search optimizer (GSO) [73], backtracking search algorithm (BSA) [74], biogeography based optimization (BBO') [75], Hopfield modeling (HM) [72],
Fig. 6 shows the generation cost convergence of the best solution among the 50 trial runs of Case 4.
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Fig. 6 Convergence characteristic of the MCSA for Case 4. 
In order to further demonstrate the efficiency and scalability of MCSA, a larger system with 40 units and VPE is considered. The demand of the system is 10 500 MW neglecting TL, RRL and POZ. The fuel costs and power generation limits are taken from [64]. Table Ⅸ presents the optimal generation values and cost obtained by MCSA. The optimal cost is 121 412.5355 ＄/h. It can be seen that the generations satisfy the generation limit constraints.
Table Ⅹ shows the comparison of the results obtained by MCSA and other recently reported algorithms in the literature such as exchange market algorithm (EMA) [76], ORCSA [67], CSA [43], SDE [65], MABC [57], CBA [62], new adaptive particle swarm optimization (NAPSO) [77], speciesbased quantum particle swarm optimization (SQPSO) [78], CESQP [66], modified differential evolution (MDE) [79], firefly algorithm (FA) [80], BSA [74], QPSO [78], BBO' [75] and CS. It shows that the best cost result of MCSA is the same or better than those obtained with other methods. Also, the mean and worst costs obtained with MCSA are better than those from its peers. Moreover, its obtained standard deviation and time are better than those from most of other methods. In summary, MCSA is more consistent and stable than the other optimization techniques.
Convergence characteristic of MCSA for Case 5 is presented in Fig. 7. It shows that MCSA has better convergence characteristic in comparison with the standard CS.
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Fig. 7 Convergence characteristic of the MCSA for Case 5. 
In this paper, a modified CS algorithm is proposed. It is implemented to solve both convex and nonconvex economic dispatch problems by considering ramp rate limits, valvepoint effects, transmission losses and prohibited operating zones. A slack method is used to handle equality constraints. A modified lambda iteration method is used to generate new solutions. Statistical results are compared with the reported results in literature. It is found that MCSA is capable of yielding a suitable balance between exploitation and exploration and has a better performance in terms of efficiency and robustness. All the experimental results confirm its high capability in solving ED problems. In the future, we need to develop more advanced efficient optimization methods [81][83] to solve power system problems involving renewable energy sources and parallel dispatch.
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