A Distributed Randomized Gradient-Free Algorithm for the Non-Convex Economic Dispatch Problem

: In this paper, a distributed randomized gradient-free algorithm (DRGF) is employed to solve the complex non-convex economic dispatch problem whose non-convex constraints include valve-point loading effects, prohibited operating zones, and multiple fuel options. The DRGF uses the Gauss approximation, smoothing parameters, and a random sequence to construct distributed randomized gradient-free oracles. By employing a consensus procedure, generation units can gather local information through local communication links and then process the economic dispatch data in a distributed iteration format. Based on the principle of projection optimization, a projection operator is adopted in the DRGF to deal with the discontinuous solution space. The effectiveness of the proposed approach in addressing the non-convex economic dispatch problem is demonstrated by simulations implemented on three standard test systems.


Introduction
The general optimal economic dispatch is realized by designing unit commitment [1] for obtaining the minimum overall generation cost [2]. The centralized optimization methods are generally used in the economic dispatch problems, such as lambda iterative algorithms [3] and interior point algorithms [4], which require the objective function to be derivative. In practice, there exist conditions of valve-point loading effects in units' startup stages [5], prohibited operating zones during the operation periods [6], and choices of the several economical fuels according to the actual power outputs [5]. These conditions make the economic dispatch problem non-convex, and in this paper, we investigate a non-convex optimization algorithm to address its non-convex characteristic.
Modern metaheuristic algorithms can solve non-convex optimization problems efficiently [7], mainly including genetic algorithm (GA [5,6,8]), particle swarm optimization (PSO [9][10][11][12]), differential evolution algorithms [13,14], and ant colony optimization (ACO, [15][16][17]). They all depend on dispatch centers to manage the information gathering from the whole system and will also use a centralized controller that will bring high communication costs. In addition, these controllers are prone to modeling errors and need a high-bandwidth communication structure with miscellaneous links [18]. Moreover, the reliability of the system is easily vulnerable to the single-point of failure [19]. Recently, the topology of power grids and communication networks are getting time-varying and extensible, but heuristic algorithms suffer a great challenge in meeting the changes.
At present, the distributed framework for economic dispatch has become a hot research topic [20][21][22]. Through building communication lines among the generation units and their neighbors,

Non-Convex Economic Dispatch
This paper studies an optimal loads allocation problem within a single time period. Under the premise of satisfying the system's active power balance constraint and units' power outputs limits, the total loads can be distributed among all of the units by a most economical scheme.
In conventional economic dispatch, the cost of a thermal generator unit is a quadratic function, and the dispatch center is aimed to minimize the total costs of all generation units.
where C(P) represents the total costs of all units whose total number is n; P i is the power outputs of unit i and C i (P i ) is the production cost. The production characteristic coefficients are a i , b i , and c i . If the ramp rate limits are considered, the power outputs limits can be expressed as [10] Max(P min where P o i is the previous power output, the upper and lower power output limits are P max i , P min i respectively, and DR i and UP i are the down and up ramp limits, respectively. The system must meet the active power balance constraints.
where P d is the total load demand. The network losses of the system are P loss calculated by the Kron formula [18].
where B ij , B 0,i , and B 00 are the losses coefficients.
(a) Valve-point loading effects. To formulate the valve-point loading effects, a sinusoidal term is added to the original cost function [5] where d i , e i are the valve-point loading coefficients of unit i.
(b) Prohibited operating zones. Considering this constraint can ensure the stable operation of generation units. Therefore, the units' power outputs constraints will be modified as follows [9]: where L m,i and U m,i are the lower and upper boundaries of prohibited operating zone m of unit i, respectively. The number of prohibited operating zones of unit i is M i .
(c) Multiple fuel options. Since units are practically provided with multiple fuel sources, each unit selects the most economical fuel in different operation range. When considering multiple fuel options, the cost function of unit i with valve-point loading effects is formulated as [10] i are the cost coefficients of unit i under fuel q. P (q−1),i is the lower power outputs limit when the fuel q is in use.

Treatment of Active Power Balance Constraints
The power imbalance among economic dispatch can be described as where P ∆ is the power imbalance of the system, and it is the difference between the total power and the total loads plus the network active power losses. Then, each unit should adjust its power outputs to meet the supply-demand balance by keeping P ∆ within an acceptable level [20].
where ε i represents the power outputs adjustment factor of unit i [20]. It can be calculated by 1/n in this paper. Assuming that each unit evaluates the network losses associated with its own power outputs [18] as where P loss (i) denotes the evaluated transmission losses by unit i. Based on the actual power outputs and the losses coefficients, the proposed approximation defines the power outputs including the losses, denoted by − P i , as [18] − P i = P i + P loss (i) (11) where − P i represents the proximate power generation that contains the information of network losses. The rationale for this choice is that the losses for the currently known solution P i are very close to the losses for the unknown solution − P i [18], that is P loss (i) ≈ − P loss (i).

The Mathematical Reformulation of the Objective Function
The non-convex economic dispatch objective Function (5) can be rewritten as: where the vector x = [x 1 , x 2 , · · · x n ] T represents the power output vector P = [P 1 , P 2 , · · · P n ] T . f (x) is the total operation cost, and f i (x i ) is the production cost of unit i. X i is the set of x i , which represents the feasible power output space. When considering network transmission losses, Hence, the original non-convex economic dispatch problem can be rewritten as a non-convex optimization problem. Because the objective function is non-convex and the constraint conditions are discontinuous, this paper employs a distributed randomized gradient-free algorithm to address the problem. Although the objective function in the distributed scheduling model is not smooth, it satisfies the Lipschitz continuity in the variable space [23], and the smoothed version of (12) is given by where µ i is the smoothing parameter of function f µi i (x i ), which represents the Gaussian approximation of f i (x i ), explicitly given by [23] where the constant γ = ( √ 2π) n is a conversion factor. The random sequence τ i satisfies independent distribution, according to the Gaussian distribution for each i. The theory of using Gauss approximation to replace the objective function is briefly shown in the Appendix A.

The Optimization Process of the Distributed Randomized Gradient-free Algorithm
The DRGF belongs to the distributed scheduling based on a distributed communication network structure, assuming that n units communicate over a network with a time-varying topology that can be denoted by a directed graph via a nodes set V = 1, 2, · · · , n. We define the matrix W[k] as the communication network structure at the time k (not negative), and the communication weight Assumption 2. For all k, there exists a positive integer T such that the graph is strongly connected.
Since the objective function is non-convex, it can use µ i , f µi i (x i ), τ i to produce the random gradient-free oracles instead of gradient or sub-gradient in the iteration optimization where ) is the random gradient-free oracle of the iteration variable x i at iteration k. Then, the units collect neighbors' information by calculating the average weight.
where x i is the average weight of x i . The unit j communicates with unit i in a communication weight W ij [k] in the kth interation. By integrating the above steps, the iteration form of optimization variables can be derived as where P X is a projection operator defined in [24]. Iteration step-sizes satisfy the following conditions: According to Formulas (13)- (19), the flowchart of the DRGF can be given by Figure 1. Further, the implementation process of the non-convex economic dispatch via the DRGF can be designed, as shown in Figure 2.
The specific procedures of Figure 2 are as follows: (1) The input data of operation parameters include the production characteristic coefficients (a i , b i , c i , d i , and e i ), total number of generation units (n), the total load demand (P d ), and the losses coefficients (B ij , B 0,i , and B 00 ). The operation constraints contain the ramp rate limits, the lower and upper boundaries of prohibited operating zones, and the operation ranges of different fuel options. (2) The initial power outputs may meet the initial supply-demand balance in Formula (3). So, all of them can be set to the same value P d n and P d is given in references [5,9,25]. (3) After setting up the initialization power output, the P loss (i) will be calculated by Formula (10)  (7) When all the equality and inequality constraints are satisfied, it will form the objective functions and feasible solution space, such as shown in Formula (12). (8) Next, implement the DRGF optimization (shown in Figure 1) and perform the kth optimization. (9) Re-examine the power outputs constraints, and if the constraints meet the Formula (6), it will get into the next step; otherwise, it will go back to step (3). (10) Calculate the power imbalance P ∆ , and if it exceeds the allowable value, a step of adjustment will be implemented, which is P i ← P i + ε i · P ∆ , then the optimization process will restart from step (3) until the power imbalance P ∆ less than 0.01 MW, and finally, the optimum solution can be obtained.  The pseudo code of the distributed randomized gradient-free algorithm for non-convex economic dispatch problem (Algorithm 1) is shown as follows: Algorithm 1: the distributed randomized gradient-free algorithm for non-convex economic dispatch problem

1.
INPUT: The operation parameters and constraints and the initial power outputs P i [0].
for i = 1 to n do 5.
Calculate the network losses by Formula (10) P loss (i) = P i · n ∑ j=1 (B ij P j ) + P i · B 0,i + B00 n

7.
Deal with the prohibited operating zones according to Formula (6) (5): 13. Select the fuels according to power outputs according to Formula (7):

25.
if P ∆ ≤ 0.01 then output the optimum solution 26. else do P i ← P i + ε i · P ∆ and then return to step 4. 27. end if 28. end for

Numerical Examples
Three test systems are employed for simulation to verify the effectiveness of the proposed DRGF in solving non-convex economic scheduling problems. The solution space in this paper is compared with the ones get by existing methods.

Description of the Test Systems
The first test system consists of six units whose cost functions are all quadratic, with prohibited operating zones for four units and network losses. The total load demand is 1263 MW, and the details of the six units are reported in [9].
The second system consists of 10 units with considering both valve-point loading effects and multiple fuel options. The unit-1 has two fuel options, while others have three. The total load demand is 2700 MW, and the details of the ten units are shown in [5]. The third test system contains 40 units with cost functions modeling valve-point loading effects. The load demand is 10,500 MW, and the details of the 40 units are given in [25].

Setting of Simulation
The DRGF iterates over the power outputs based on the randomized derivative-free oracles. According to the flow chart, the parameters of µ i and α[k] may affect the convergence of the algorithm. In [26], the influence of the number of optimization variables and µ i on the convergence of the algorithm is studied by several examples. It is a fact that, with the number of optimization variables increasing, the DRGF algorithm requires more iteration to achieve optimal result. µ i is set at 0.0005 in the test examples. A dynamic iteration step-size is put to use in this paper, such as α[k] = 1/ √ k + 1. The coding is written with MATLAB 2012 programming language and executed on the Intel (R) Core (TM) i5, 2.20-GHz, 8.0-GB RAM processor.

Solution Quality
Tables 1-3 provides the optimization results of six units, ten units and forty units test systems by using the DRGF algorithm, including the optimal power outputs, the total power outputs (TP), the total network losses (TL), and the total costs (TC). Figures 3-5 show the optimization curves of TC by various algorithms.    Table 1. Optimization results of six units test system. PSO-LRS: particle swarm optimization-simple local random search; NPSO: new particle swarm optimization; NPSO-LRS: new particle swarm optimization-simple local random search. DRGF: distributed randomized gradient-free algorithm.

The Units
Power  A varieties of centralized algorithms have been successful presented to solve the six units test system and three improved PSO algorithms have achieved the best result with a minimum cost of 15,450 $/h. The new particle swarm optimization (NPSO) [10] and new particle swarm optimizationsimple local random search (NPSO-LRS) [10] are almost the same in convergence rate and show their superiority over the particle swarm optimization-simple local random search (PSO-LRS) algorithm. The NPSO algorithm performs well due to the extra diversification provided by the worst experience component. As shown in Figure 3, the DRGF algorithm can also realize the same minimum generation cost. And it is slightly better than NPSO-LRS due to the distributed searching ability.
The 10 units test system mainly considers valve-point loading effects and multiple fuel options. As shown in Figure 4, a varieties of centralized algorithms have been successful presented to solve the 10 units test system, and three improved PSO algorithms have achieved the best result with a minimum cost of 624.13 $/h. Here, DRGF is faster in convergence rate than that of PSO algorithm. The behavior of AA falls between those of DRGF and NPSO-LRS. For all the three test cases, DRGF runs first in the performance ladder. The proposed worst experience component in the velocity update equation, and the random gradient-free search procedure strengthens the DRGF. From Figure 4, it can be found that the DRGF algorithm can achieve the optimal solution (about 623.95 $/h) as the AA algorithm does, which indicates that the distributed algorithm DRGF is superior to the NPSO-LRS.
When only considering valve-point loading effects, the 40 units test system can be optimized by the DRGF algorithm whose optimization results are provided in Table 3. As a comparison, Table 4 presents the optimal cost under different scheduling methods, and Figure 5 shows the cost curves of the NPSO-LRS, DRGF, and AA. It can be seen that the two distributed economic dispatch schemes (DRGF and AA) obtain a cost of 121,788.70 $/h within 100 iterations, and NPSO-LRS gets a cost of 121,664.43 $/h, the minimum in 100 trials, at about the 570th iteration. In terms of costs, the NPSO-LRS is slightly better than the distributed methods DRGF and AA. But it is found that the convergence speed of the distributed algorithm is obviously better than the centralized algorithm. The DRGF approach is simple and adaptive to large-scale generation systems.

Robustness Test of the Algorithm
Due to the randomness of the calculation process, results acquired from intelligent optimization algorithms are not unique. A great deal of optimization trials, e.g., 100 trials, should be conducted to examine their performance and the optimal operation cost is listed in Table 4 including the minimum, the maximum, and the average data. To compare the results in a statistical manner, the frequencies of attainment of a cost within the specific ranges are presented in Table 5. As shown in Table 4, the maximum and average cost of centralized algorithms have a large difference with the minimum costs, such as PSO for 7360 $/h and 4590 $/h; NPSO-LRS has achieved better optimization results in many algorithms, with the difference of 1317.16 $/h and 544.89 $/h, respectively. In contrast, the optimal solutions of the distributed scheduling are fixed (seen in Table 5), which demonstrate that the robustness of distributed algorithm is far better than the centralized algorithms. As a result, the DRGF algorithm is more suitable for solving complex non-convex economic dispatch problems.
In this paper, the average CPU (Central Processing Unit) running time is used to analyze the computation efficiency of the DRGF algorithm. We examine the average optimization time T o and the operation time T f with the fixed iteration number (200 steps) to achieve the optimal results under each test system, respectively. The algorithms run 100 times on the MATLAB 2012a, and the average results are shown in Table 6. As is known from the above figure, the algorithm NPSO-LRS will take more time to get the optimal results when the number of units is lager. By analysis and comparison with NPSO-LRS, it can be seen that the distributed scheduling schemes (AA and DRGF) for optimizing non-convex scheduling need less time. Due to different algorithms frameworks, centralized dispatches need to collect all the information of the optimization variables and then process and transmit them in a centralized way. The computational efficiency of the distributed DRGF algorithm is better than NPSO-LRS and AA algorithm.

Conclusions
This paper establishes a non-convex economic dispatch model after considering the constraints of valve-point loading effects, prohibited operating zones, and multiple fuel options. According to these complex non-convex characteristics, this paper introduces a DRGF with randomized derivative-free oracles instead of sub-gradients to solve the economic dispatch model. Simulation scenes are implemented on three test systems, and optimization results obtained by different algorithms have been made further comparisons. The analysis validates that the DRGF algorithm can effectively address the complex economic dispatch problem with valve-point loading effects, prohibited operating zones and multiple fuel options. In the next study, effort should be spent on improving the iteration efficiency of the proposed algorithm.