1. Introduction
Efforts have been made to handle the energy crisis and environmental issues by exploiting distributed energy resources (DERs) [
1,
2]. It is acknowledged that DERs possess the characteristics of cleanliness, renewability, and diversification. DERs mainly contain micro-gas generators (MGGs), wind generators (WGs), photovoltaic systems (PVs), and batteries (BEs) [
2]. Natural conditions (e.g., wind speeds, light intensity, etc.) will inevitably give rise to the intermittent and randomness of the DERs’ power outputs. In addition, some non-ideal communication network factors may also interfere with DERs’ scheduling [
3]. Communication time delays slow down the speeds of scheduling information and channel noises fluctuate DERs’ power outputs, which disobey the power system’s requirements for rapidity and stability. DERs’ over-limit, plug-and-play, and channel faults are the common time varying topology events that disrupt the normal operation of economic dispatch, and even damage the system. If these large-scale and small-capacity DERs have access to the power system, they will pose challenges to the economic dispatch, power quality (e.g., frequency harmonics, voltage flicker, etc.), and the electricity market. Therefore, to realize the DERs’ organized regulation is an urgent research task.
As is known, the micro-grid can coordinate DERs within a self-control and management system [
4]. Unlike the micro grid, depending on intelligent software systems and advanced management techniques, the virtual power plant (VPP) can aggregate DERs and controllable loads into a virtual whole to participate in the power grid’s operation and electricity market’s transactions [
2]. VPP can also stabilize the fluctuation of DERs’ generation and even provide the power system with auxiliary services with high reliability, quality, and safety [
5,
6]. The VPP has fewer region limits and better market interactivity than the microgrid, which gives it broad application prospects.
VPP economic dispatch (VPED) strategies can adopt centralized dispatch, using centralized scheduling algorithms (e.g., genetic algorithm [
7] and particle swarm optimization [
8]), as well as the distributed dispatch algorithms in [
9,
10,
11]. The centralized scheduling approaches have encountered a great number of problems in practical application [
9]. The centralized dispatch must obtain all DERs’ information, including the power outputs, profits and costs, and other parameters [
12]. Accordingly, it is essential to establish a dispatch center and broadband communication channels between the dispatch center and DERs. Then, it may result in higher communication costs and more sophisticated communication networks, which will exert serious communication time delays, channel noises, and dimension disasters during the optimization progress [
13]. Additionally, to make VPP an open system in the electricity market, the communication network with high security and stability is required. However, the centralized dispatch is susceptible to a single point of failure because of its access to each DER’s information. Owing to the mechanism of limited communication, each DER only needs its own and adjacent DERs’ information to implement the optimization. Hence, the distributed dispatch can encourage DER owners to participate in the VPP’s operation actively on the premise of keeping their private data secret. In summary, the distributed dispatch has a broad application prospect in the VPED due to the advantages of economy, flexibility, agility, information security, and strong robustness.
A VPP’s distributed scheduling exerts a tremendous fascination for researchers. A distributed VPP scheduling model composed of WGs and electric vehicles is set up in [
14] and solved by linear programming. However, the advanced industrial control often uses fuzzy-model-based nonlinear networks [
15], which cannot be addressed well by the linear programming. In [
12], a distributed center-free algorithm is developed to coordinately control the power outputs of DERs in a VPP. Although the algorithm converges fast, introducing auxiliary variables will weaken the stability of the system. A distributed gradient algorithm is presented in [
9], which can be used to deal well with the equality and inequality constraints under the topology reconstruction situation, but obvious fluctuation will emerge when the number of DERs is large. In [
10], a distributed primal-dual sub-gradient method (DPDSM) is designed to solve the optimal VPED model. Based on limited information exchange among DERs, the algorithm can still achieve the global convergence within a less optimization time. The DPDSM is also used to handle the VPED in [
11] and simulation results show that the algorithm has a good convergence even in solving a more complex model. The DPDSM in [
10,
11] employs the negative sub-gradient of the power and the multiplier during the distributed optimization process. However, it has no engagement with the consensus algorithm and effects of non-ideal communication conditions on VPED are also neglected.
This paper examines the distributed VPED. The main contributions of this work include the following: The mathematical model for distributed VPED is presented and an improved DPDSM for solving the model is proposed. In the distributed optimization, the negative sub-gradient is employed in the power iteration, meanwhile the positive sub-gradient is used in the multiplier iteration [
16], and the consensus algorithm with a gain function is appropriately embedded in the sub-gradient algorithm. By introducing the Lagrangian function and projection constraint theory, the constraints are integrated into the objective function [
17,
18]. Meanwhile, the influence of non-ideal communication conditions due to time delays, channel noises, and time-varying topology are considered in the method. The modified IEEE-34 and IEEE-123 bus test systems are employed to verify the effectiveness of the distributed strategy. Simulation results from six scheduling scenarios indicate the superiority of the proposed method.
The rest of this paper is organized as follows.
Section 2 introduces the economic dispatch model of VPP. The method of DPDSM for solving the VPP economic dispatch model is presented in
Section 3.
Section 4 gives the numerical examples. Summaries are drawn in
Section 5.
2. Economic Dispatch Model
The VPED model uses the scheduling objectives including minimizing total generation cost of DERs, maximizing profits of a VPP, and maximizing energy-saving and emission-reduction of a VPP. Depending on the power outputs of various DERs, an optimal allocation model of the energy storage system whose objective function includes economy, grid supply, and voltage is constructed in [
19]. A VPP’s bidding strategy on the basis of electricity price is developed in [
20], which breaks through the routine that the day-ahead transacted electricity quantity is equal to the forecasting load demand. Then it establishes a new electricity transaction model under a unified electricity market considering both the day-ahead and real-time stochastic load demand. Different from [
20], a VPP’s three-stage stochastic bi-level bidding strategy depending on DERs’ the power outputs, loads demands and the competitor’s history price is developed in [
21].
In this paper, a VPED model with a variety of constraints is established. At the point of common coupling (PCC), the power running through PCC (recorded as Ps), which is the power exchanged between the VPP and the electricity market (or the main grid). The power collected by VPP can be sold to power users and VPP’s profit is determined by the power outputs of all DERs, the Ps, the purchase price from the main grid, and the sale price to the power users.
2.1. Objective Function
PVs and WGs cannot continuously generate power like MGGs, so it is significant to obtain its available power outputs according to the actual operation [
22]. Since this paper is aimed to study VPP’s distributed dispatch, DERs’ power outputs models will be shown in
Appendix A. To stimulate PVs’ and WGs’ scheduling potential, they may operate in the schedulable model rather than the maximum power point model [
22]. A certain number of MGGs and BEs are used to stabilize the power output fluctuations of PVs and WGs. BEs can work in the charging or discharging modes.
The operation cost function of each DER can be modeled as:
where
n is the number of DERs and the actual power output of DER
i is uniformity recorded as
PGi. The operation costs of DER
i at
PGi is denoted by
Ci(
PGi). The cost parameters are signified as
ai,
bi, and
ci.
According to VPP’s operation mode, we can get the optimization target of VPED as follows:
where
PDj is the power demand by consumer
j and the consumers’ number is
m.
θ,
β are the purchase price and the sale price, respectively. If
Ps is negative, the power will flow from VPP into the main grid.
Ps is calculated by:
2.2. Constraints
Power output constraints of DERs: Capacity constraints of all types of DERs can be formulated as inequality constraints:
where
is the minimum power output of unit
i and
is the maximum one. Here, PVs’ and WGs’ maximum power outputs are their power outputs at the maximum power point.
Transmission constraints of power lines: These constraints satisfy a set of global inequality constraints:
where
Pl means the power transmission limit of line
l,
L represents the power lines’ number and
O is the number of nodes. The power transmission coefficient of node
o and its geographically adjacent line
l is expressed as
ηol. The symbol of
i,
j →
o describes that unit
i or consumer
j may convey power and energy through the node
o.
Formula (5) is also equivalent to:
2.3. Mathematical Reformulation
The VPED is chiefly influenced by the power output of each DER and the
Ps. Some proper reformation can be done to make the optimization problem into a general economic dispatch problem. The
Ps can be eliminated by Formulas (2) and (3) and the objective function can be formulated as follows:
In this paper,
β,
θ, and the total loads are constant and are not dependent on the decision variables. Based on the principle of dual problem [
16,
18], the objective function can be reformulated as:
If power output is written as
xi, the sub-objective function will be denoted as
fi, so the VPED model is equivalent to:
where
hs represents the global inequality constraints as shown in (6) and (7). X is the set of all
x, indicates the local constraint of each DER in Equation (4) and
q is the number of constraints. The Lagrange multiplier
λ can be introduced to structure the Lagrange function:
Now, the optimization problem can be written as:
4. Numerical Examples
In this paper, to verify the validity of the proposed VPED strategy, two VPP systems are built by modifying the IEEE-34 bus test system and the IEEE-123 bus test system, respectively. In this work, the power error tolerance
ε in
Figure 1 is 0.05 kW and the iteration step
α is set to 0.002 s. For the convenience of simulation, the gain function
c[
k] is 0.5[1 + ln(
k + 1)]/(
k + 1) which can meet the conditions in the
Appendix A. The algebraic sum of power flowing through PCC is
Ps and the total loads are recorded as
PD. The purchase price
θ is 0.076$/kWh and the sale price
β is 0.072$/kWh. The parameters and the capacity limits of DERs are listed in
Table 1 and
Table 2, respectively.
The simulation implemented on the modified IEEE-34 bus test system is mainly designed to study the impact of communication time delays and channel noises on the distributed dispatch and the influence of changing Δ over the distributed dispatch algorithm. The modified IEEE-123 bus test system is aimed to investigate the adaptability of the distributed VPED algorithm under a large scale non-ideal communication network. It primarily discusses the time varying communication topology conditions arising from channel faults of communication links, DERs’ over-limit, and DERs’ plug-and-play.
4.1. The Modified IEEE-34 Bus Test System
As shown in
Figure 2, there are twenty schedulable DERs in the modified IEEE-34 bus test system. For the sake of making better use of renewable energies, PVs, and WGs will operate at their maximum power output. BEs can work in both charging and discharging modes and the MGGs may reduce their outputs to cut the fuel expenditure. BEs and MGGs are also able to adjust their outputs to deal with some unexpected events, which is aimed to maintain the system power balance. In comparison,
Table 3 offers the results optimized by using the centralized dispatch under the same operation condition and
Table 4 shows VPP’s average profits made by the two dispatch strategies. Three scheduling scenarios are provided as follows: (A) a distributed dispatch under the ideal communication network; (B) a distributed dispatch considering time delays and channel noises in communication network; and (C) a distributed dispatch with a different Δ.
(1) Scenario A: Distributed Dispatch under Ideal Communication
Figure 3 indicates the optimal scheduling results of each DER and
Figure 4 provides the variation of
Ps during the distributed optimization process. From
Figure 3 and
Table 3, we can find that the distributed dispatch proposed in this paper achieves the same scheduling scheme as the centralized dispatch does, which shows the effectiveness of the distributed dispatch strategy. From the viewpoint of profits, it is not difficult to find in
Table 4 that the distributed dispatch is the same with the centralized one.
Figure 4 illustrates that the VPP can sell electric energy to the main grid when its overall power is higher than load demands, but if the overall power is lower than the total loads, VPP will absorb power from the main grid to maintain the supply-demand balance.
(2) Scenario B: Distributed Dispatch Considering Communication Time Delays and Channel Noises
In practice, it is necessary to consider communication time delays and channel noises. When implementing the optimization, the delays are randomly distributed between 0 and 3; meanwhile, the noises are randomly distributed between 0 and 5 kW.
Figure 5 shows the optimization curves of this scenario.
Figure 6 provides the variation of
Ps and the power imbalance during the optimization. Since the transmission of the iteration information is postponed by time delays, curves for showing the variation of
Ps and the power imbalance will appear in cross-sections, such as M1 in
Figure 5c. Channel noises will cause the oscillation of power outputs; for example, M4 from the 16th to the 25th iterations. The more serious the delays and noises are, the rougher the curves will be. By the aid of the main grid, VPP can shrink the whole fluctuation and keep the system power balance (see
Figure 6). In the centralized scheduling, prediction of communication time delays and channel noises is needed and it will increase the scheduling burden. Based on the local communication mechanism, the proposed method can still reach the same result, but in a way of real-time scheduling, meaning that the proposed method is useful to improve the system noise immunity. The simulation shows the effectiveness of the distributed scheduling strategy in handling time delays and channel noises.
(3) Scenario C: Distributed Dispatch with a Different Δ
Changing the value of Δ means adjusting the consensus parameters in the distributed dispatch. Δ is set at 3 in scenario
a while Δ is set at 10 in this scenario. Contrasting
Figure 7 and
Figure 8 with
Figure 3 and
Figure 4, it is clear that the larger the value is, the faster the convergence speed, but the larger the oscillation that will be occurred in the optimization.
4.2. The Modified IEEE-123 Bus Test System
The modified IEEE-123 bus test system is shown in
Figure 9. In this example, forty DERs are dispersed in four areas in this test system. The operation parameters are the same with the previous test system. Three simulation scenarios are implemented as follows: (D) a distributed dispatch under the condition of the DERs’ over-limit; (E) a distributed dispatch under the condition of channel faults; and (F) a distributed dispatch under the condition of DERs’ play-and-plug.
(1) Scenario D: Distributed Dispatch under the Condition of DERs’ Over-Limit
In order to ensure the safe operation of the VPP, it is essential to consider the capacity limits of DERs. Suppose a few MGGs’ and BEs’ power outputs have reached the limits during the optimization.
Figure 10 shows that the over-limit DERs will run at the power limit and no longer iterate in the optimization, but continue to deliver data to their neighbors. Based on this local communication mechanism, the over-limit DERs may only affect the adjacent DERs rather than the whole.
Figure 11 indicates that, with regard to DERs’ over-limit events, the distributed method can still maintain the system power balance constraint.
(2) Scenario E: Distributed Dispatch under the Condition of Channel Faults
Channel faults will lead to the change of the communication topology. After the wrong channels are removed, the system recovers its power balance by reconstructing a new communication topology. The damaged channels are shown in
Figure 9 and the dispatch progress is displayed in
Figure 12 and
Figure 13. The channel faults can disturb DERs’ normal operation. Then, the system will build a new stable state by distributed VPED optimization.
(3) Scenario F: Distributed Dispatch with DERs’ Plug and Play
Compared with non-ideal communication conditions, DERs’ plug-and-play is most likely to occur under an actual large-scale VPP system. There are two DERs that temporarily plug-and-play during the distributed scheduling in this scenario. From
Figure 14, we can see that a PV plug off at about the 45th iteration for some reasons, but plug on at about the 50th iteration. However, by adjusting the power of MGGs, Bes, and
Ps, the VPP system immediately realizes a new supply-demand power balance (see
Figure 15). When this event happens again on a WG, the VPP system still restores its stability within a short time. Faced with the DER plug-and-play conditions, the system employing the proposed method in this paper shows a strong robustness.