Non-Convex Economic Dispatch of a Virtual Power Plant via a Distributed Randomized Gradient-Free Algorithm

Abstract: The economic dispatch problem of a virtual power plant (VPP) is becoming non-convex for distributed generators’ characteristics of valve-point loading effects, prohibited operating zones, and multiple fuel options. In this paper, the economic dispatch model of VPP is established and then solved by a distributed randomized gradient-free algorithm. To deal with the non-smooth objective function, its Gauss approximation is used to construct distributed randomized gradient-free oracles in optimization iterations. A projection operator is also introduced to solve the discontinuous variable space problem. An example simulation is implemented on a modified IEEE-34 bus test system, and the results demonstrate the effectiveness and applicability of the proposed algorithm.


Introduction
Distributed energy resources (DERs) include distributed generators (DGs), renewable energies (REs), and energy storage systems (ESs) [1].Optimizing the output of DERs can greatly improve the efficiency of their energy utilization.The centralized dispatch methods adopt lambda iterative algorithms [2] and interior point lambda iterative algorithms [3], both of which need the objective functions to be smooth and derivable.Actually, the cost functions of DGs are non-convex because of their features of valve-point loading effects, prohibited operating zones, and multiple fuel options [4].The charging and discharging of ESs also aggravates the complexity of economic dispatch problems [5].Based on advanced management concepts and software technologies, the virtual power plant (VPP) [6,7] has been developed to be a new DERs management tool.Additionally, it is necessary to solve the non-convex economic dispatch of VPPs in order to realize the optimal scheduling of DERs.Intelligent optimization algorithms can solve non-convex optimization problems effectively, including the genetic algorithm (GA) [8], particle swarm optimization (PSO) [9], and differential evolution [10,11].However, these algorithms rely on the dispatch center or controller to collect and process DERs' information, which may lead to a higher communication cost; they also need a centralized communication structure with high bandwidth, which has poor system reliability and is more vulnerable to a single point of failure [12].
At present, the optimization decision process of VPPs is changing from centralized ways to distributed ones [13].According to the local communication mechanism, the operation information of the DERs is collected through communication lines built among DER units and their adjacent units, and then the real-time scheduling process can be carried out.The distributed consensus algorithm employed in the distributed scheduling of [14] can greatly reduce communication costs and communication delays.Compared with centralized solutions, the distributed gradient algorithm in [15] can not only obtain comparable optimums but also respond in a timely manner when the operation conditions of the system change.To solve the problem of the objective function of ESs being not smooth, the sub-gradient is calculated as an alternative of the gradient [16,17].The economic dispatch of a VPP is a non-convex optimization problem for the consideration of valve-point loading effects, prohibited operating zones, and multiple fuel options.The algorithms mentioned in [14][15][16][17] cannot be applied to the non-convex optimization problem because it is difficult to estimate the gradient or sub-gradient.The distributed auction-based algorithm designed in [12] can realize the optimal power output by sharing units' bidding and then determining the auction results in the process of consensus.However, it introduces too many intermediate variables, which will make the iteration format more complex.
This paper adopts a distributed randomized gradient free algorithm (DRGF) [18] to solve a VPP's non-convex economic dispatch problem considering DGs' valve-point loading effects, prohibited operating zones, and multiple fuel options.The algorithm is established on a distributed communication structure that has a higher operation reliability and lower communication cost [12].In addition, the DRGF approach calculates randomized gradient-free oracles, instead of gradients or sub-gradients, to implement the distributed optimization, which makes the iteration formula simple and easily solved.The modified IEEE-34 bus system is employed to verify the effectiveness of the proposed method.The simulation results show that the DRGF algorithm can formulate an economical scheme for a VPP's non-convex economic dispatch.
This paper is organized as follows.In Section 2, a VPP's non-convex economic dispatch with constraints of valve-point loading effects, prohibited operating zones, and multiple fuel options is discussed.Section 3 introduces the DRGF algorithm, and the simulation results presented in Section 4 show its effectiveness.Finally, the paper is summarized in Section 5.

The Operation Models of the DER's Units
In traditional economic dispatch, the cost function of a DG is a standard quadratic function [19].If valve-point loading effects, simulated by sinusoidal terms, and multiple fuel options are taken into account, the cost function can be described as [4,12]: where n g is the total number of DGs and the power output of DGi is P i , yielding to the upper limit P max i and lower limit P min i .When P i exceeds the value of P (q-1),i , unit i chooses the fuel q and its cost function Because the power output is usually concentrated in some areas, the operational efficiency can be greatly promoted by prohibiting units from running in low productivity areas.The power output constraints considering DGs' prohibited operating zones can be expressed as [4,12]: where unit i has Mi prohibited operating zones, the mi th of which subjects to the upper boundary U mi and lower boundary L mi .
ESs can work in charging or discharging modes, and their cost functions and operation constraints are as follows [20,21]: where n e is the total number of ESs, and the power output of ESi is P ei .C (P ei ) represents the charging (ch)/discharging (dch) cost, and E i is the charging/discharging efficiency.The maximum charging and discharging power of ESi is P max ch , P max dch .Additionally, the minimum of both is 0. For the value of the state-of-charge (S OC ), S max , S min , S up , S down are the maximum, minimum, upper, and lower values, respectively [20].
Compared with operating at maximum power, REs will be more flexible in a schedulable mode.However, this will also cause some profit losses, that is, the schedulable cost C (P ri ) [21]: where n r is the total number of REs, and the power output of REi is P ri .The ρ V (photovoltaic systems) and ρ W (wind turbine) are the grid-connected prices, including electricity prices and generation compensations.The maximum available power of REi is P max ri , and it can be either the maximum photovoltaic system tracking power P V [22] or the maximum wind power P W [23]: where P Vmax represents the maximum output under standard test conditions.G C means the actual light intensity; G Cmax is the reference intensity under standard test conditions.The conversion coefficient of temperature to power is depicted by K. T c and T r are the environment temperature and the reference temperature under standard test conditions, respectively.P r is the rated power output of the wind generators (WGs).v, v ci , v co , and v r are the wind speeds, cut-in wind speeds, cut-out wind speeds, and the rated wind speeds, respectively.

Dispatch Objectives and Constraints
According to the operation mode of a VPP, the economic dispatch objective function and the system constraints can be defined as [16,21]: ng where F VPP denotes the total income of the VPP, and C VPP represents its total generation cost.The maximum generation capacity of the VPP is P max VPP , that is, the sum of the maximum power output of each DER unit.The total load demand is P D and P S is the interface power at the point of common coupling (PCC).If P S is negative, the power will flow from the VPP into the main grid.ρ d is the bidding of the VPP and ρ S is the electricity price of the main grid.γ 1 , γ 2 , and γ 3 are the reserved capacity coefficients of the load demand, photovoltaic systems, and wind turbines, respectively.

Distributed Randomized Gradient-Free (DRGF) Algorithm
The DERs' power output vector P = P 1 , P 2 , • • • P ng , P e1 , P e2 , • • • P ne , P r1 , P r2 , • • • P nr ] T can be denoted as a variable vector x = x 1 , x 2 , • • • x n ] T , and there exists n = n g + n e + n r .Accordingly, the upper and lower bounds of the variables are represented by x max i , x min i .Thus, the active power balance constraint of the original VPP economic dispatch problem can be denoted as: Substituting formula ( 14) into formula (10), one can obtain: It can be seen that the F VPP is only dependent on variables x i when the values of ρ d , ρ S , and where f (x) denotes the objective function, and X is the feasible space of x, that is, x ∈ X may represent the power output constraints of DERs.
Based on a distributed optimization framework, the DERs can collect neighboring units' information and the information at the PCC, and then this information will be calculated by weighted mean values [14,18] x where ε i is the power regulation factor, and x i is the weighted mean value of x i .Unit j is connected to unit i with the communication weight W ij [k], and its calculation format is shown in [12].This paper employs a DRGF algorithm [18] to solve the problems of a discontinuous solution space and the non-convex objective function.Although the objective function of a VPP is not smooth, it is Lipschitz continuous in the variable space [24], and its smooth form can be written as: where µ i is the smoothing coefficient of the objective function, and f µi i (x i ) is its smooth form, calculated by: where the conversion coefficient denotes as γ = ( √ 2π) n , and the random sequence τ i meets the Gaussian distribution.The theory of Gauss approximation is shown in the Appendix A. Now, distributed randomized gradient-free oracles can be constructed to implement the optimal iteration: where g µi (x i [k]) represents the distributed randomized gradient-free oracle of x i at the kth iteration.By the above steps, the iterative form of the optimization variables can be derived as: where the projection operator P X is defined as in [25].The iteration step-size satisfies the following conditions: The implementation process of a VPP's non-convex economic dispatch via the DRGF algorithm is shown in Figure 1, and the specific procedures are as follows: 1.
The input data includes the coefficients of cost functions, various limits of the DERs' power output, the total load demand, etc.The maximum available power output of the REs is calculated by formula ( 8) and ( 9). 2.
The optimization variable x i [0] is initialized according to references [8,10,15].Then, set up the smoothing coefficient of the objective function and generate the random sequence.

3.
According to formula (14), calculate the initial P S at PCC.

4.
Correct the iteration step by k = k + 1, where the initial number of iteration steps is k = 1.

5.
According to formula (17), calculate the weighted mean values; and according to formula (19), calculate the Gauss approximation.When DGs have multiple fuel options, as shown in equation ( 2), select the cost function curves on the basis of the DG's actual power output.6.
According to formula (20), calculate distributed randomized gradient-free oracles; according to formula (22), calculate the current iteration step by α According to formula (21), implement the optimal iteration of the power output variables.8.
Determine whether the current variables are within the available space.If they satisfy, proceed to the next step; otherwise, the variables take the upper ) limits of the constraints.When variable x i falls into prohibited zone mi during the decreasing process, such as , its value will be set at the upper boundary U mi .Additionally, when x i falls into prohibited zone mi during the increasing process, such as , the value will be set at the lower boundary L m .9.
According to formula (14), update the initial P S at PCC. 10.Determine whether the current power imbalance satisfies the allowable value.If it satisfies, proceed to the next step; otherwise, return to step (5) to recalculate the weighted mean values.11.Calculate the iteration error.12. Determine whether the iteration error satisfies the allowable value.If it satisfies, proceed to the next step; otherwise, return to step (4) for the next iteration.13.Output the optimal solution vector.

Start Input parameters and constraints
Calculate the initial PS at PCC k=k+1 Update the PS |P S | < 5 kW ?

Numerical Examples
Based on a modified IEEE 34 bus system, a VPP system is built to verify the effectiveness of the proposed algorithm.It mainly investigates DGs' valve-point loading effects, prohibited operating zones, and multiple fuel options.The reference [18] shows that the convergence coefficient has little effect on the convergence of the algorithm, so the smoothing coefficient of the cost function is set to 0.0005 in this example.The communication topology is shown in Figure 2. The operation parameters are listed in Figure 1.The flowchart for a virtual power plant's (VPP's) non-convex economic dispatch via the distributed randomized gradient free algorithm (DRGF) algorithm.

Numerical Examples
Based on a modified IEEE 34 bus system, a VPP system is built to verify the effectiveness of the proposed algorithm.It mainly investigates DGs' valve-point loading effects, prohibited operating zones, and multiple fuel options.The reference [18] shows that the convergence coefficient has little effect on the convergence of the algorithm, so the smoothing coefficient of the cost function is set to 0.0005 in this example.The communication topology is shown in Figure 2. The operation parameters are listed in Tables 1 and 2. The total load demand is 650 kW, and the initial power outputs of the DGs, REs, and ESs are 75, 75, and 0 kW, respectively.For solving non-convex economic dispatch problems, a PSO solution used in [9] has achieved the lowest cost among numerous centralized algorithms.
As contrast, we will also adopt the PSO [9] (one of the centralized dispatch method) to deal with the VPP's economic dispatch model.Table 3 provides the optimization results when one of the centralized dispatch method (PSO) is adopted, and Table 4 shows the VPP's average profits made by PSO and DRGF.This section sets up three simulation scenarios as follows: (A) the VPP's distributed economic dispatch with valve-point loading effects; (B) the VPP's distributed economic dispatch with prohibited operating zones; (C) the VPP's distributed economic dispatch with multiple fuel options.

Scenario A: The VPP's Distributed Economic Dispatch with Valve-Point Loading Effects
The characteristic of valve-point loading effects makes the DGs' cost function have many nondifferentiable points.The operation coefficients of DGs are shown in Tables 1 and 2. Figures 3 and 4 provide the optimal scheduling process of this scenario.It can be seen that the DGs' power output will have a great fluctuation during the initial stage of optimization, and then the ESs will change their power output to reduce the impact of valve-point loading effects on the system's active power balance.In addition, the VPP also exchanges power with the main grid through the PCC to stabilize the supply-demand balance.After a certain number of iterations, the DERs' optimization curves tend to be stable, and the DRGF algorithm finally achieves the same results as centralized algorithms (shown in Tables 3 and 4).However, if considering the communication cost and operability, the distributed dispatch will be more economical and practical.

Scenario A: The VPP's Distributed Economic Dispatch with Valve-Point Loading Effects
The characteristic of valve-point loading effects makes the DGs' cost function have many non-differentiable points.The operation coefficients of DGs are shown in Tables 1 and 2. Figures 3  and 4 provide the optimal scheduling process of this scenario.It can be seen that the DGs' power output will have a great fluctuation during the initial stage of optimization, and then the ESs will change their power output to reduce the impact of valve-point loading effects on the system's active power balance.In addition, the VPP also exchanges power with the main grid through the PCC to stabilize the supply-demand balance.After a certain number of iterations, the DERs' optimization curves tend to be stable, and the DRGF algorithm finally achieves the same results as centralized algorithms (shown in Tables 3 and 4).However, if considering the communication cost and operability, the distributed dispatch will be more economical and practical.

Scenario A: The VPP's Distributed Economic Dispatch with Valve-Point Loading Effects
The characteristic of valve-point loading effects makes the DGs' cost function have many nondifferentiable points.The operation coefficients of DGs are shown in Tables 1 and 2. Figures 3 and 4 provide the optimal scheduling process of this scenario.It can be seen that the DGs' power output will have a great fluctuation during the initial stage of optimization, and then the ESs will change their power output to reduce the impact of valve-point loading effects on the system's active power balance.In addition, the VPP also exchanges power with the main grid through the PCC to stabilize the supply-demand balance.After a certain number of iterations, the DERs' optimization curves tend to be stable, and the DRGF algorithm finally achieves the same results as centralized algorithms (shown in Tables 3 and 4).However, if considering the communication cost and operability, the distributed dispatch will be more economical and practical.

Scenario B: The VPP's Distributed Economic Dispatch with Prohibited Operating Zones
The DGs' power output is discontinuous for some prohibited operating zones.Figures 5 and 6 show the simulation results under this scenario.In Figure 5a, the platforms in the curve indicate the situation where DGs fall into prohibited operating zones.When DGs jump out of these areas, they will resume normal operation.ESs can flexibly charge and discharge, greatly reducing the impact of prohibited operating zones on the active power balance.As can be seen from Table 3, the DRGF algorithm can get the same results as the centralized algorithms, which demonstrates that the DRGF algorithm can effectively deal with DGs' prohibited operating zones.

Scenario C: the VPP's Distributed Economic Dispatch with Multiple Fuel Options
According to actual power output, DGs will choose the most economical fuel selection with different cost coefficients, leading to some non-differentiable points.Figures 7 and 8 provide this

Scenario B: The VPP's Distributed Economic Dispatch with Prohibited Operating Zones
The DGs' power output is discontinuous for some prohibited operating zones.Figures 5 and 6 show the simulation results under this scenario.In Figure 5a, the platforms in the curve indicate the situation where DGs fall into prohibited operating zones.When DGs jump out of these areas, they will resume normal operation.ESs can flexibly charge and discharge, greatly reducing the impact of prohibited operating zones on the active power balance.As can be seen from Table 3, the DRGF algorithm can get the same results as the centralized algorithms, which demonstrates that the DRGF algorithm can effectively deal with DGs' prohibited operating zones.

Scenario B: The VPP's Distributed Economic Dispatch with Prohibited Operating Zones
The DGs' power output is discontinuous for some prohibited operating zones.Figures 5 and 6 show the simulation results under this scenario.In Figure 5a, the platforms in the curve indicate the situation where DGs fall into prohibited operating zones.When DGs jump out of these areas, they will resume normal operation.ESs can flexibly charge and discharge, greatly reducing the impact of prohibited operating zones on the active power balance.As can be seen from Table 3, the DRGF algorithm can get the same results as the centralized algorithms, which demonstrates that the DRGF algorithm can effectively deal with DGs' prohibited operating zones.

Scenario C: the VPP's Distributed Economic Dispatch with Multiple Fuel Options
According to actual power output, DGs will choose the most economical fuel selection with different cost coefficients, leading to some non-differentiable points.Figures 7 and 8 provide this

Scenario B: The VPP's Distributed Economic Dispatch with Prohibited Operating Zones
The DGs' power output is discontinuous for some prohibited operating zones.Figures 5 and 6 show the simulation results under this scenario.In Figure 5a, the platforms in the curve indicate the situation where DGs fall into prohibited operating zones.When DGs jump out of these areas, they will resume normal operation.ESs can flexibly charge and discharge, greatly reducing the impact of prohibited operating zones on the active power balance.As can be seen from Table 3, the DRGF algorithm can get the same results as the centralized algorithms, which demonstrates that the DRGF algorithm can effectively deal with DGs' prohibited operating zones.

Scenario C: the VPP's Distributed Economic Dispatch with Multiple Fuel Options
According to actual power output, DGs will choose the most economical fuel selection with different cost coefficients, leading to some non-differentiable points.Figures 7 and 8 provide this

Scenario C: the VPP's Distributed Economic Dispatch with Multiple Fuel Options
According to actual power output, DGs will choose the most economical fuel selection with different cost coefficients, leading to some non-differentiable points.Figures 7 and 8 provide this scenario's simulation results.In Figure 7a and Table 1, when the power output is within 40-55 kW, DGs select the No. 1 fuel; if the power output is within 55-80 kW, DGs will select the No. 2 fuel.Then, the allocation of DERs should be re-optimized.The VPP's total power output may have a large fluctuation when the fuel changes, and the system will recover the active power balance quickly by absorbing some power from the main grid.The distributed dispatch can get the same result as the centralized one, which shows the DRGF algorithm's effectiveness in solving DGs' multiple fuel options.
Energies 2017, 10, 1051 10 of 12 scenario's simulation results.In Figure 7a and Table 1, when the power output is within 40-55 kW, DGs select the No. 1 fuel; if the power output is within 55-80 kW, DGs will select the No. 2 fuel.Then, the allocation of DERs should be re-optimized.The VPP's total power output may have a large fluctuation when the fuel changes, and the system will recover the active power balance quickly by absorbing some power from the main grid.The distributed dispatch can get the same result as the centralized one, which shows the DRGF algorithm's effectiveness in solving DGs' multiple fuel options.

Summary
A technology of a VPP is adopted to manage DERs by modeling its non-convex economic dispatch considering DGs' characteristics of valve-point loading effects, prohibited operating zones, and multiple fuel options.A DRGF algorithm is introduced to solve this nonlinear and nondifferentiable optimization problem.The objective function is converted to its Gauss approximation, and then used to construct distributed randomized gradient-free oracles instead of gradients or subgradients.A projection operator is also employed to deal with the discontinuous variable space.Three typical simulation scenarios are implemented on a modified IEEE 34 bus system.The results indicate that the proposed DRGF algorithm can effectively cope with a VPP's non-convex economic dispatch, and shows a good applicability.scenario's simulation results.In Figure 7a and Table 1, when the power output is within 40-55 kW, DGs select the No. 1 fuel; if the power output is within 55-80 kW, DGs will select the No. 2 fuel.Then, the allocation of DERs should be re-optimized.The VPP's total power output may have a large fluctuation when the fuel changes, and the system will recover the active power balance quickly by absorbing some power from the main grid.The distributed dispatch can get the same result as the centralized one, which shows the DRGF algorithm's effectiveness in solving DGs' multiple fuel options.

Summary
A technology of a VPP is adopted to manage DERs by modeling its non-convex economic dispatch considering DGs' characteristics of valve-point loading effects, prohibited operating zones, and multiple fuel options.A DRGF algorithm is introduced to solve this nonlinear and nondifferentiable optimization problem.The objective function is converted to its Gauss approximation, and then used to construct distributed randomized gradient-free oracles instead of gradients or subgradients.A projection operator is also employed to deal with the discontinuous variable space.Three typical simulation scenarios are implemented on a modified IEEE 34 bus system.The results indicate that the proposed DRGF algorithm can effectively cope with a VPP's non-convex economic dispatch, and shows a good applicability.

Summary
A technology of a VPP is adopted to manage DERs by modeling its non-convex economic dispatch considering DGs' characteristics of valve-point loading effects, prohibited operating zones, and multiple fuel options.A DRGF algorithm is introduced to solve this nonlinear and non-differentiable optimization problem.The objective function is converted to its Gauss approximation, and then used to construct distributed randomized gradient-free oracles instead of gradients or sub-gradients.A projection operator is also employed to deal with the discontinuous variable space.Three typical simulation scenarios are implemented on a modified IEEE 34 bus system.The results indicate that the proposed DRGF algorithm can effectively cope with a VPP's non-convex economic dispatch, and shows a good applicability.

Figure 2 .
Figure 2. The communication topology based on a modified IEEE 34 bus system.

Figure 3 .
Figure 3.The optimized scheduling scheme of scenario A: (a) Optimization results of distributed generators (DGs); (b) Optimization result of renewable energies (REs); (c) Optimization results of energy storage systems (ESs); (d) The power PS at the point of common coupling (PCC).

Figure 2 .
Figure 2. The communication topology based on a modified IEEE 34 bus system.

Figure 2 .
Figure 2. The communication topology based on a modified IEEE 34 bus system.

Figure 3 .
Figure 3.The optimized scheduling scheme of scenario A: (a) Optimization results of distributed generators (DGs); (b) Optimization result of renewable energies (REs); (c) Optimization results of energy storage systems (ESs); (d) The power PS at the point of common coupling (PCC).

Figure 3 .
Figure 3.The optimized scheduling scheme of scenario A: (a) Optimization results of distributed generators (DGs); (b) Optimization result of renewable energies (REs); (c) Optimization results of energy storage systems (ESs); (d) The power P S at the point of common coupling (PCC).

Figure 4 .
Figure 4.The active power balance of scenario A.

Figure 5 .
Figure 5.The optimized scheduling scheme of scenario B: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power PS at PCC.

Figure 6 .
Figure 6.The active power balance of scenario B.

Figure 4 .
Figure 4.The active power balance of scenario A.

Figure 4 .
Figure 4.The active power balance of scenario A.

Figure 5 .
Figure 5.The optimized scheduling scheme of scenario B: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power PS at PCC.

Figure 6 .
Figure 6.The active power balance of scenario B.

Figure 5 .
Figure 5.The optimized scheduling scheme of scenario B: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power P S at PCC.

Figure 4 .
Figure 4.The active power balance of scenario A.

Figure 5 .
Figure 5.The optimized scheduling scheme of scenario B: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power PS at PCC.

Figure 6 .
Figure 6.The active power balance of scenario B.

Figure 6 .
Figure 6.The active power balance of scenario B.

Figure 7 .
Figure 7.The optimized scheduling scheme of scenario C: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power PS at PCC.

Figure 8 .
Figure 8.The active power balance of scenario C.

Figure 7 .
Figure 7.The optimized scheduling scheme of scenario C: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power P S at PCC.

Figure 7 .
Figure 7.The optimized scheduling scheme of scenario C: (a) Optimization results of DGs; (b) Optimization result of REs; (c) Optimization results of ESs; (d) The power PS at PCC.

Figure 8 .
Figure 8.The active power balance of scenario C.

Figure 8 .
Figure 8.The active power balance of scenario C.

Table 1 .
Coefficients of the distributed generators' (DGs') production cost.DER, Distributed Energy Resources.

Table 3 .
Optimization results under one of the centralized dispatch method (PSO).

Table 4 .
The VPP's average profits obtained by two dispatch strategies (PSO and DRGF).