Energy Storage Configuration of Distribution Networks Considering Uncertainties of Generalized Demand-Side Resources and Renewable Energies

Abstract

With the growing proportion of advanced metering infrastructures and intelligent controllable equipment in power grids, demand response has been regarded as an effective and easily implemented approach to meet the demand–supply equilibrium. This paper innovatively proposes generalized demand-side resources combining the demand response with an energy storage system and constructs a configuration model to obtain scheduling plans. Firstly, this paper analyzes the characteristics of generalized demand-side resources and models the translational loads, reducible loads and energy storage system. Secondly, a deterministic energy storage configuration model aiming at achieving the lowest operation cost of distribution networks is established, from which the scheduling scheme of generalized demand-side resources can be obtained. Then, the fuzzy membership function and the probability density function are used to represent the uncertainty of the demand response, the prediction error of renewable energy output and the generalized demand-side resources that do not participate in the demand response. Therefore, this paper simulates daily operations to modify the capacity of energy storage. The problem is solved by using Monte Carlo simulation, fuzzy chance-constrained programming and mixed-integer programming. Finally, the effectiveness of this model is demonstrated with case studies in a 33-node distribution network. The results show that the uncertainty of this system is solved effectively. When only considering generalized demand-side resources, the total cost is reduced by 9.5%. After considering the uncertainty, the total cost is also decreased 0.3%. Simultaneously, the validity of the model is verified.

1. Introduction

Adhering to the direction of clean and low-carbon development, the penetration of renewable energy is continuously increasing in distribution networks [1]. However, the uncertain and intermittent power output makes the control of distribution networks more complex than ever. For the purposes of fully accommodating renewable energy and coping with the imbalance of supply and demand, existing studies have mostly focused on the effects of the demand response (DR) or the energy storage system (ESS) alone.

On the one hand, DR can improve the operation efficiency of distribution networks via the interaction between the grid side and the user side, and it can bring benefits to all participants. Different from traditional studies, in Ref. [2], a two-stage model for the day-ahead energy scheduling problem of DR was proposed. In the past few years, for different purposes, many models of DR have had different constraint conditions in various scenarios. In Ref. [3], a resource scheduling plan was formulated in the day-ahead stage on the basis of DR and unit constraints. Refs. [4,5] proposed that DR can decrease the peak-valley difference. Refs. [6,7] presented a model for locational marginal prices, including day-ahead, co-optimized, energy and spinning reserve markets. In Refs. [8,9], a bi-level robust optimization model with DR and thermal comfort was proposed for the capacity planning and operation problem. Above all, increasing DR provides substantial flexibility in renewable-based energy systems, but the deployment of DR is currently limited. Moreover, the actual electricity consumption situation changes unexpectedly because customers are temporarily unable to participate in the response plan. So, the uncertainty of DR can affect the dispatching operation of the distribution network. When only using DR, the optimization method will be affected by uncertainty.

On the other hand, ESS can smooth the fluctuations of renewable energies and ensure the safe and stable operation of power grids with a high proportion of renewable energy. Ref. [10] described a novel energy management strategy for hybrid energy storage systems, when used to supply urban electric vehicles. To minimize the total cost of a hybrid power system, a mathematical model for the configuration of battery energy storage systems was proposed in Ref. [11]. In Ref. [12], a unique energy storage method that combined wind, solar and gravity energy storage together was used to ensure the economy of the system. In order to minimize the total operating cost of microgrids, Ref. [13] established an optimized operation model considering the aging of the battery and the initial value of the state of charge. For the first time, Ref. [14] investigated the uncertain optimal allocation of ESS considering practical constraints, including prohibited zones, and the ramp rate. However, ESS is easily restricted by the geographical environment and high prices. Only relying on the configuration of ESS cannot give full play to the economy of the power system. Moreover, the installed ESS will be not fully utilized.

So, research on the coordination of DR and ESS is particularly important. Generalized demand-side resources (GDRs) considering the complementary characteristics of DR and ESS are thus proposed. GDRs are composed of distributed power source resources, load resources and energy storage resources. Nowadays, the influences of GDRs are becoming increasingly obvious. They can change the daily load at a much lower cost by using measures such as DR, V2G and ESS [15]. Moreover, configuration and scheduling schemes of GDRs can effectively reduce the dispatching costs of power grids. In Ref. [16], an overview of demand-side resource development is presented, from controllable loads to GDRs, including distributed generation (DG) and EES. However, to date, studies on the scheduling of GDRs have not taken into account uncertainty. Moreover, the capacity allocation for ESS while considering GDRs and renewable energy uncertainty has not been considered.

In order to further improve economic efficiency and enhance the system’s capacity of accepting renewable energy, it is essential to maximize the adjustable capacity of GDRs [17]. GDRs have great potential to participate in system scheduling. It has gradually become a common phenomenon to aggregate GDRs [18]. Ref. [19] proposed a detailed model for user-level planning, where cooling, heating, gas and electric DR were considered. Ref. [20] conducted a coordinated operation of energy storage and incentive-based GDRs. This can lower the impact of power grid failure for important loads and improve the line load rate. Ref. [21] achieved large-scale GDRs to participate in the “peak-shaving and valley-filling” of the power system. For the sake of controlling and coordinating a large number of GDRs, interaction control has been presented, and it is regarded as an innovative distributed approach [22,23]. Meanwhile, a distributed algorithm has been presented in order to aggregate a large number of households, with mixed-integer variables and intricate couplings between devices [24,25]. Within this context, the Stackelberg game theory has been applied to analyze the coordination between grid operators and demand-side participators [26,27]. In general, there are fewer studies on GDRs than on DR.

However, in distribution networks, studies of ESSs’ configurations considering the uncertainty of GDRs, wind power and photovoltaic power prediction are rarely carried out. Therefore, in this paper, an ESS configuration model is established, which considers the uncertainty of GDRs and renewable energies. The contribution of this paper can be summarized as follows:

• A proposal of GDRs that combine the complementary characteristics of DR with ESS and apply them to the optimal operation of distribution network.
• A provision of a new approach to contract price formulation among load aggregator (LA), distribution networks and main network by considering GDRs.
• A solution for user response uncertainty and renewable energy uncertainty by establishing a GDR configuration model.

The remaining sections of this paper are organized as follows: Section 2 analyzes the characteristics of the GDRs and builds models for them. Section 3 establishes a configuration model of ESS. This involves the participation of GDRs in distribution networks. The ESS’s configuration plans and GDRs’ response plans are obtained using this deterministic model. Section 4 takes the uncertainty of the system into account and then proposes approaches to solve the uncertainty. Section V demonstrates the effectiveness and validity of the model. Section 6 draws conclusions and provides prospects for future works.

2. GDR Model

In distribution networks, there are many kinds of GDRs. Due to the development of electronic equipment, GDRs are also easily regulated. At present, GDRs are composed of distributed power source resources, load resources and an energy storage system. The load resources are divided into non-translational loads, translational loads and reducible loads.

However, GDRs’ schedulable capabilities have great limitations. Meanwhile, due to the constraints of distribution networks, the characterization of their flexibility is complicated. Therefore, GDRs should be integrated and quantified in multiple dimensions of space and time [28]. Aggregating GDRs is one of the effective means used to solve the hidden problems caused by large-scale distributed energy access to distribution networks.

The modeling analyses of TLs, reducible loads and ESS are presented below.

2.1. Transferable Loads

TLs refer to electrical equipment whose working power cannot be changed but whose working time can be shifted forward or backward. The power distribution vector of TLs is determined by translational time and the number of electrical appliances.

In general, the change in daily load curves caused by the translation of TLs is numerically equal to the power of transferred TLs minus the power of pre-load participating in DR.

$$\begin{array}{l}{p}_{TL,i}^{k,t}=p_{k,e}({\displaystyle \sum _{m=1}^M{\lambda }_{i,k}^{m,t}\times n_{TL,i,k}^m}-{\gamma }_{i,k}^t{\displaystyle \sum _{m=1}^M{n}_{{}_{TL,i,k}}^m})\\ p_{TL,i}^t={\displaystyle \sum _{k=1}^K{p}_{TL,i}^{k,t}}\\ {\gamma }_{i,k}^t=\{\begin{array}{l}1t\in \left[t_{i,k}^b,t_{i,k}^e\right]\\ 0t\in \mathrm{else}\end{array}\end{array}$$

(1)

The power of TLs on a single node is the sum of all types of TLs on the node. M is the number of shifting modes set in this paper. ${\lambda }_k^{m,t}$ is a binary variable; if its value is 1, there is a TL of type k at this time according to translational mode m.

TLs are affected by various constraints when they work. Simply setting the above variables makes the processing of continuity constraints and time constraints more complex. Therefore, this paper introduces intermediate variables and a time distribution matrix containing all translational possibilities to describe constraints.

There are many types of TLs, and different types have different rated powers and working times. Taking k as an example: Its normal working hours is [$t_{begin,k}$, $t_{end,k}$]. It is allowed to move forward for $t_{forw,k}$ hours and to move backward for $t_{backw,k}$ hours.

$$\{\begin{array}{l}{A}_{i,k}^{h,t}=1 t\in [t_{i,k}^{begin},t_{i,k}^{end}]+h-t_{forw,i,k}-1 \\ A_{i,k}^{h,t}=0 t\in else, h\in [1,H]\\ \mathrm{H}=t_{forw,k}+t_{backw,k}+1\end{array}$$

(2)

H means that translation has H possibilities. This paper selects M translational models from H possibilities to dispatch TLs.$A_k$ represents the distribution of time among all possibilities.

$$\{\begin{array}{l}{\displaystyle \sum _{h=1}^H{\delta }_{i,k}^{m,h}}=1 m\in [1,\mathrm{M}]\\ {\lambda }_{i,k}^{m,t}={\delta }_{i,k}^{m,h}\times A_{i,k}^{h,t} \end{array}$$

(3)

$$0\le n_{TL,i,k}^m\le n_{TL,i,k}^{\max }$$

(4)

${\delta }_{i,k}^{m,h}$ is a binary variable. If its value is 1, then it indicates that translation strategy m corresponds to translation possibility h among all possibilities. Formula (4) represents the upper and the lower limits of the number of TLs.

Formula (1) contains the product of binary variables and integer variables. By using the Big-M method, Formula (1) can be transformed into Formula (5):

$$\begin{array}{l}-V\cdot {\lambda }_{i,k}^{m,t}\le p_{TL,i}^{k,t}+p_{k,e}{\displaystyle \sum _{m=1}^M{\gamma }_{i,k}^t{n}_{{}_{TL,i,k}}^m}\\ \le V\cdot {\lambda }_{i,k}^{m,t}-(1-{\lambda }_{i,k}^{m,t})V+n_{{}_{TL,i,k}}^m{p}_{k,e}\\ \le p_{TL,i}^{k,t}+p_{k,e}{\displaystyle \sum _{m=1}^M{\gamma }_{i,k}^t{n}_{{}_{TL,i,k}}^m}\\ \le (1-{\lambda }_{i,k}^{m,t})V+n_{{}_{TL,i,k}}^m{p}_{k,e}\end{array}$$

(5)

V is a large positive number.

2.2. Reducible Loads

Based on the amount of electricity that can be reduced, reducible loads have been classified into two types: interruptible loads (ILs) and curtailable flexible loads (CFLs) [29]. ILs mean that loads can be immediately reduced to zero. However, this type of electrical equipment is not usually used. CFLs mean that loads can be reduced from rated power to a certain value [30]. Simply, this certain value is not set too small considering the comfort of users.

$$\{\begin{array}{ll}{P}_{IL,i}^t=p_{IL,e}{n}_{IL,i}& t\in [t_{IL}^{on},t_{IL}^{off}]\\ P_{IL,i}^t=0& t\in else\end{array}$$

(6)

$$0\le n_{IL,i}\le n_{IL,i}^{\max }{n}_{IL,i} is \mathrm{integer}$$

(7)

$$0\le P_{CFL,i}^t\le \varpi P_{CFL,i}^{t,e}$$

(8)

Formulas (6) and (7) describe the interrupted power of ILs. Formula (8) describes the reduced power of CFLs. The final power obtained by the above loads can be expressed using Formula (9):

$$P_{DR,i,t}=-P_{CFL,i}^t-P_{IL,i}^t+P_{TL,i}^t$$

(9)

2.3. Energy Storage System

ESS can give full play to the advantages of DR. On the one hand, they can purchase and sell electricity by taking advantage of the price difference at different times and bring profits to distribution networks. On the other hand, they can solve the grid scheduling problems caused by the uncertainties of renewable energies output and users’ electricity consumption. Therefore, ESS models should be established. Formula (10) considers the constraints, such as charge and discharge power and the SOC state.

$$\{\begin{array}{l}-P_{ess,i,\max }\le P_{ess,i}^t\le P_{ess,i,\max }\\ e_{ess,i}^{t+\Delta t}=e_{ess,i}^t+P_{ess,i}^t\Delta t\\ SO{C}_{ess,\min }<={\displaystyle \raisebox{1ex}{$e_{ess,i}^t$}\!\left/ \!\raisebox{-1ex}{$e_{ess,i,\max }$}\right.}<=SO{C}_{ess,\max }\\ {\displaystyle \sum _{t=1}^T{P}_{ess,i,t}^t=0i=1,2,\cdots ,\mathrm{N}}\end{array}$$

(10)

After establishing the models of GDRs, it is essential to perform a simple analysis on the contract pricing and establish the configuration model of GDRs.

3. GDR Configuration Model

In the previous sections, the GDRs’ models were built. From the perspective of distribution networks, there are transactions between distribution networks, LA and the main network. Distribution networks need to conduct resources regulation, but the specific behaviors of specific electricity users participating in DR are controlled by LA. Distribution networks and LA represent the relationship between the whole and the details. Distribution networks need to determine their approximate electricity purchase amount from the main network a few days in advance. This can facilitate the main network to prepare startup plans. According to the difficulty of LA, pricing contracts should be designed for different prices and sizes in order to regulate different resources.

3.1. Formulation of Contract Electricity Price

When it comes to the contract pricing strategy, lots of information needs to be considered: the time of use tariff (TOU) determined by the main network, the willingness of users to participate in DR and the benefits that LA can receive from distribution networks. In this paper, the following interaction processes are considered: TOU is determined using load curves, which are submitted by the distribution networks. During peak hours, in order to reduce the amount of electricity purchase from the main network, GDRs are encouraged to participate in DR. Therefore, the contract price should be similar to TOU. That is, the contract should have different prices at different times. The translational time and the continuous working time determine the difficulty of TLs to participate in DR. To ensure the interest of the users who participate in DR, contract prices should be set according to the difficulty. In most cases, the prices should be set higher than CFLs. After integrating GDRs according to the wishes of the users, LA will distribute the determined power consumption plans to the corresponding users. The specific pricing strategy is shown in Figure 1. From the perspective of the distribution networks, it is necessary to protect the interest of LA and the users in contract pricing and to improve the enthusiasm of GDRs to participate in DR.

3.2. Establishment of GDR Configuration Model

From an economic perspective of distribution networks, a configuration model for GDRs is established. This model takes into account the ESS’s configuration cost, GDRs’ assignments, the network loss and power purchase cost. Its optimal goal is to minimize the sum of all costs.

$$\min F=F_{ess}+F_{DR}+F_{loss}+F_{grid}$$

(11)

The calculation methods of costs in the above formula are shown in Equations (12)–(15):

$$\{\begin{array}{l}{F}_{ess}=F_{ess,E}+F_{ess,p}\\ F_{ess,\mathrm{E}}=\frac{T}{8760}{\displaystyle \sum _{i=1}^N({\partial }_e}{\mathrm{C}}_E^{inv}+{\mathrm{C}}_E^{fix})e_{ess,i,\max }\\ F_{ess,\mathrm{P}}=\frac{T}{8760}{\displaystyle \sum _{i=1}^N(}{\mathrm{C}}_p^{fix}{P}_{ess,i}^t)\\ {\partial }_e=\frac{r_{ess}}{1-{(1+r_{ess})}^{-h_{ess}}}\end{array}$$

(12)

$$F_{DR}={\displaystyle \sum _{i=1}^N({\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=1}^M{n}_{TL,i,k}^m{d}_{TL,i}^{k,m}+}}}{n}_{IL,i}{d}_{IL,i}+{\displaystyle \sum _{t=1}^T{P}_{CFL,i}^t{d}_{CFL,i}^t})$$

(13)

$$F_{loss}={\displaystyle \sum _{t=1}^T[{\delta }^{loss}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j\in c(i)}{U}_{ij,t}^2/r_{ij}}}]}$$

(14)

$$F_{grid}={\displaystyle \sum _{t=1}^T[{\eta }_t{P}_{grid,t}]}$$

(15)

The objection function (11) describes the expected minimum sum of the ESS’s investment and operational cost (12), the DR cost (13), the line loss cost (14) and the main network electricity purchasing cost (15).

Among them, the ESS’s cost consists of the energy storage configuration cost and the energy storage operation cost. Then, the DR cost almost comes from the DR contract; the line loss cost and the main network electricity purchasing cost are related to factors such as the power loss price and the active power of the line, respectively.

Constraints (16)–(17) represent the internal constraints of the distribution networks:

$$\begin{array}{l}-P_{ess,i,t}+P_{WT,i,t}+P_{PV,i,t}+P_{grid,i,t}-P_{load,i,t}-P_{DR,i,t}=P_{i,t}\\ P_{i,t}={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j\in c(i)}(}}{U}_{i,t}^2{g}_{ij}-U_{i,t}{U}_{j,t}(g_{ij}\cos {\theta }_{ij,t}+b_{ij}\sin {\theta }_{ij,t}))\end{array}$$

(16)

$$U_i^{\min }\le U_{i,t}\le U_i^{\max }$$

(17)

3.3. Simplification of Model

By linearly approximating the alternating current power flow Equation (16) in the model, the original nonlinear configuration model is transformed into a quadratic configuration model.

$$\{\begin{array}{ll}\mathrm{Re}\left\{U_{j,t}\right\}=1+{\displaystyle \sum _{j_1\in N^{\prime }}R(j,j_1)P_{j_1}+X(j,j_1)Q_{j_1}}, & \forall j\in N^{\prime }\\ \mathrm{Im}\left\{U_{j,t}\right\}={\displaystyle \sum _{j_1\in N^{\prime }}X(j,j_1)P_{j_1}-R(j,j_1)Q_{j_1}}, & \forall j\in N^{\prime }\end{array}$$

(18)

Equation (18) is the linearized expression of the power flow equation. Equation (14) can be transformed into Equation (19) according to Equation (18).

$$\{\begin{array}{l}{P}_{loss,t}={\displaystyle \sum _{(j,j_1)\in \epsilon }\mathrm{Re}\left\{y_{j,j_1}^{\ast }\right\}}[{(\mathrm{Re}\left\{U_{j,t}\right\}-\mathrm{Re}\left\{U_{j_1,t}\right\})}^2+\\ {(\mathrm{Im}\left\{U_{j,t}\right\}-\mathrm{Im}\left\{U_{j_1,t}\right\})}^2]\\ F_{loss}={\displaystyle \sum _{t=1}^T[{\delta }^{loss}{P}_{loss,t}]}\end{array}$$

(19)

The purchased power of the main network can be expressed as the total network loss subtracting the sum of the injected power of all the nodes, except for those connected to the main distribution networks. So, $P_{grid,t}$ can be expressed as (20):

$$P_{grid,t}=-{\displaystyle \sum _{j\in N^{\prime }}{P}_{j,t}+P_{loss,t}}$$

(20)

In this paper, the solution process is divided into two stages. The above model is the first-stage model; it is a mixed-integer quadratic configuration model and also a deterministic model.

In the first stage, this deterministic model is used to determine the installation nodes and the energy storage capacity. Due to the introduction of 0–1 variables, the constraints are relatively complex. Moreover, because the optimal problem contains multiple variables and constraints, the calculation becomes difficult. Therefore, a genetic algorithm (GA) is used to transform the above problem into several small optimal problems. So, GA and a quadratic programming algorithm are combined to solve this model.

A flowchart is shown in Figure 2.

First, the initial population is generated. Each individual in the population represents a different location plan for energy storage in the distribution networks. The load curves, renewable energy power output curves, GDRs and time-sharing electricity price information are known in the model. Each individual can obtain a corresponding energy storage capacity configuration and a final cost of system. The objective function is obtained by screening small individuals. Through crossover and mutation, a new population is generated and then used to solve the model. After several calculations, the individual whose objective function value is the smallest is finally determined, thus obtaining the energy storage configuration scheme.

The initial energy storage configuration scheme and the scheduling of GDRs can be obtained using this model. However, the uncertainty of the system is ignored in the actual operation process. It is necessary to clear the sources of uncertainty by using the analyses below and by modifying the energy storage capacity to solve the uncertainty.

4. Uncertainty Analysis

Without considering the uncertainty, the above model is a deterministic model. In the actual operation process, although electricity users receive the dispatching instructions from LA, they may be affected by unexpected conditions the next day. The actual response situation will fluctuate. A configurable scheme for ESS and a scheduling scheme for GDRs can be obtained by using the model in Section 3. Concurrently, there are uncertainties in the renewable energy power output and DR. In this paper, the fuzzy parameter of the trigonometric membership property is used to express the actual response resources [31]. Sometimes, the scheduling plan of GDRs does not use all GDRs, so extra GDRs can solve the uncertainty. $P_{DR,i,t}$ in Equation (16) is replaced by ${\overline{P}}_{DR,i,t}$ in Equation (21):

$${\overline{P}}_{DR,i,t}={\overline{P}}_{TL,i}^t-P_{CFL,i}^t-{\overline{P}}_{IL,i}^t$$

(21)

4.1. Uncertainty of TLs

According to the results of the deterministic model, the scheduling plan of TLs is determined. Although TLs’ translational plans are formulated, a situation exists where users do not participate in DR. Therefore, this paper discusses the uncertainty of TLs in moving forward and backward situations.

(1)

The uncertainty of TLs moving forward:

The triangle membership function is used to describe the uncertainty of TLs moving forward.

$${\overline{n}}_{TL,i,s1}^{forw}=(\xi r_1{n}_{TL,i,s1}^{forw},\xi r_2{n}_{TL,i,s1}^{forw},n_{TL,i,s1}^{forw})$$

(22)

When additional TLs are not considered to solve the uncertainty, $r_1$ and $r_2$ are determined according to the agreement between LA and the distribution networks. Due to the uncertainty, the distribution networks need to increase the energy storage capacity. In this paper, TLs that are not involved in DR are used to solve the uncertainty during actual operation. If some TLs are not translated, other TLs not participating in the DR plan can be arranged to make up for it. The value of $\xi$ depends on the number of additional pieces of electricity equipment. The value of $\xi$ is bigger than 1. The greater the addition of TLs, the greater the value of $\xi$. To some extent, the more the resources aggregated by LA, the less the uncertainty of the power system.

(2)

The uncertainty of TLs moving backward:

$${\overline{n}}_{TL,i,s2}^{backw}=(r_1{n}_{TL,i,s2}^{backw},r_2{n}_{TL,i,s2}^{backw},n_{TL,i,s2}^{backw})$$

(23)

If TLs are not shifted, the uncertainty is compensated for by the configuration of ESS. When the translational time is determined, the number of TLs is actually uncertain, which is expressed by the fuzzy membership function.

4.2. Uncertainty of CFLs and ILs

CFLs and ILs do not take into account uncertainty because of the precise control of advanced power electronic equipment.

4.3. Uncertainty of Renewable Energy

After considering the wind power output error and the photovoltaic power output error, $P_{WT,i,t}$ in Equation (16) is replaced by ${\widehat{P}}_{WT,i,t}$ in Equation (24), and ${\widehat{P}}_{WT,i,t}^{loss}$ satisfies Equation (25).$P_{PV,i,t}$ in Equation (16) is replaced by ${\widehat{P}}_{PV,i,t}$ in Equation (27), and ${\widehat{P}}_{PV,i,t}^{loss}$ satisfies Equation (28).

$${\widehat{P}}_{WT,i,t}=P_{WT,i,t}+{\widehat{P}}_{WT,i,t}^{loss}$$

(24)

$$f({\widehat{P}}_{WT,i,t}^{loss})=\frac{1}{{\sigma }_1\sqrt{2\pi }}{e}^{-\frac{{({\widehat{P}}_{WT,i,t}^{loss}-{\mu }_1)}^2}{2{\sigma }_1^2}}$$

(25)

The probability density of Formula (25) is shown in Figure 3. By analyzing the probability density curves, it can be seen that, in the right part of the vertical coordinate, the actual output is greater than the predicted output. At this time, it is useless to reduce CFLs. In the left part, the actual output is less than the predicted output. Some wind power output errors can be compensated for by dispatching CFLs. The final error is determined by the amount of load reduction on the node.

In the real-time dispatch process, CFLs that do not participate in DR can be used to deal with the uncertainty of renewable energy power prediction errors. So, ${\widehat{P}}_{WT,i,t}^{loss}$ can be combined with $\Delta p_{CFL,i,t}$ to represent ${\widehat{P}}_{WT,i,t}^{newloss}$.

$$\begin{array}{l}\{\begin{array}{ll}{\widehat{P}}_{WT,i,t}^{newloss}=\min (\Delta p_{CFL,i,t}+{\widehat{P}}_{WT,i,t}^{loss},0)& {\widehat{P}}_{WT,i,t}^{loss}\le 0\\ {\widehat{P}}_{WT,i,t}^{newloss}={\widehat{P}}_{WT,i,t}^{loss}& {\widehat{P}}_{WT,i,t}^{loss}>0\end{array}\\ {\widehat{P}}_{WT,i,t}=P_{WT,i,t}+{\widehat{P}}_{WT,i,t}^{newloss}\end{array}$$

(26)

The probability distribution function of ${\widehat{P}}_{WT,t}^{new}$ is shown in Figure 3. It clearly illustrates Equation (26). The probability that ${\widehat{P}}_{WT,t}^{new}$ is equal to zero increases. The probability of ${\widehat{P}}_{WT,t}^{new}$ being less than 0 is a, and the probability of ${\widehat{P}}_{WT,t}^{new}$ being greater than 0 is b, so the probability of ${\widehat{P}}_{WT,t}^{new}$ being equal to zero is 1-a-b.

In the following function, the prediction probability distribution of the photovoltaic output under different weather types can be obtained by fitting the photovoltaic power generation prediction error.

$${\widehat{P}}_{PV,i,t}=P_{PV,i,t}+{\widehat{P}}_{PV,i,t}^{loss}$$

(27)

$$f({\widehat{P}}_{PV,i,t}^{loss})=\frac{\Gamma (\frac{v_t+1}{2})}{{\sigma }_2\sqrt{v_t\pi }\Gamma (\frac{v_t}{2})}{\left[\frac{v_t+(\frac{{\widehat{P}}_{PV,i,t}^{loss}-{\mu }_2}{{\sigma }_2})}{v_t}\right]}^{-(\frac{v_t+1}{2})}$$

(28)

To obtain a reliable result for the ESS’s configuration, it is necessary to consider the renewable energy output prediction in various scenarios. Many scenarios of renewable energy output can be simulated using Monte Carlo simulation. The specific steps are as follows:

• Determine the number of simulations;
• Extract the simulations’ results using the functions (24)–(28);
• Substitute the simulations’ results into the model for calculation;
• Accumulate the results for each simulation;
• Output the results and judge the simulation number;
• End.

Because of the fuzzy parameters, the constraints cannot be given the certain feasible sets, so the opportunity constraint model is adopted, that is, ensuring the probability of the constraints is not less than a certain confidence level.

$$\begin{array}{l}{C}_{\tau }\{-{\overline{P}}_{TL,i}^t-{\widehat{P}}_{WT,i,t}-{\widehat{P}}_{PV,i,t}=P_{i,t}+P_{ess,i,t}+P_{load,i,t}-P_{CFL,i}^t-P_{IL,i}^t\}\ge \alpha \\ {\overline{P}}_{TL,i}^t={\displaystyle \sum _{s1=1}^{S1}({\overline{n}}_{TL,i,s1}^{forw}{\lambda }_{i,s1}^{t,forw}-{\overline{n}}_{TL,i,s}^{forw}{\gamma }_i^t)}+{\displaystyle \sum _{s2=1}^{S2}({\overline{n}}_{TL,i,s2}^{backw}{\lambda }_{i,s2}^{t,back}-{\overline{n}}_{TL,i,s2}^{backw}{\gamma }_i^t)}\end{array}$$

(29)

$\alpha$ describes the system’s imbalance risk caused by the uncertainty of renewable energy. The maximum is determined by the system’s risk-bearing capacity. Moreover, $1-\alpha$ reflects the reliability of the normal operation.

When solving the fuzzy opportunity constraint programming, the credibility opportunity constraints can be transformed into the corresponding clear equivalent class according to the uncertainty configuration theory. As $\alpha$ increases, the total cost decreases; that is, the greater risk, the better economy. So, when the confidence coefficient $\alpha$ > 1/2 [32], the power balance constraint formula can be converted into a clear equivalence class:

$$\begin{array}{l}(2-2\alpha )(-\xi r_2{\displaystyle \sum _{s1=1}^{S1}{n}_{TL,i,s1}^{forw}{\lambda }_{i,s1}^{t,forw}}-r_2{\displaystyle \sum _{s2=1}^{S2}{n}_{TL,i,s2}^{backw}{\lambda }_{i,s2}^{t,back}})\\ +(2\alpha -1)(-\xi r_1{\displaystyle \sum _{s1=1}^{S1}{n}_{TL,i,s1}^{forw}{\lambda }_{i,s1}^{t,forw}}-r_1{\displaystyle \sum _{s2=1}^{S2}{n}_{TL,i,s2}^{backw}{\lambda }_{i,s2}^{t,back}})\\ =P_{i,t}+P_{ess,i,t}+P_{load,i,t}-P_{CFL,i}^t-P_{IL,i}^t-{\widehat{P}}_{WT,i,t}-{\widehat{P}}_{PV,i,t}\end{array}$$

(30)

4.4. Solution of GDR Configuration Model

The second stage is based on the results of the energy storage configuration at the first stage. Under the initial energy storage schemes, the daily operation of the system is simulated to obtain schemes for GDRs’ participation in DR. Then, the ESS’s capacity is reconfigured while considering the uncertainty of the system. The calculation flow of the second stage is shown in Figure 4.

The daily operation of the system with the initial energy storage schemes is simulated, and then the schemes of GDRs participating in DR are obtained. GDRs that do not participate in DR can change the uncertainty of the system. The revised uncertainty can be represented by changes in the probability density function and the fuzzy membership function. New renewable energy power output curves are obtained by using Monte Carlo simulation. New energy storage configuration results are obtained by keeping the nodes of the energy storage configuration from changing and by taking Equation (11) as the objective function. Compared with the initial energy storage configuration results, the energy storage capacity is modified.

5. Case Study

Note that all studies are carried out on a 64-bit 2.7 GHz computer with 8 GB physical memory (RAM) and eight physical core processors.

5.1. Description

The example analysis is carried out using a 33-node distribution network. Its topology is shown in Figure 5.

In this system, nodes 5, 8, 12, 20, 25 and 32 are all connected wind turbines with an installed capacity of 500 kW. The reference voltage is 12.66 kV, and the upper and lower voltages of the other nodes are 1.1 pu and 0.9 pu, respectively. The power loss price is 0.1 USD/kWh. A lithium battery is used for the energy storage configuration. The specific parameters of the ESS adopted in this paper are listed in

Table 1. Parameters of battery energy storage unit.

Parameter	Value
Rated power/kWh	1.0
Maximal charge and discharge power/kW	0.5
Max state of charge	0.9
Min state of charge	0.1
Initial state of charge	0.5
Investment costs/USD	200
Operation and maintenance costs/(USD/year)	2.94
Life cycle/years	20
Discount rate	0.1
Unit power operating cost/(USD/kW)	0.002

, and their maximum continuous charging or discharging time is set as two hours.

A typical daily load curve and typical wind power output curves are shown in Figure 6 and Figure 7, respectively.

The total amount of TLs and ILs aggregated by LA is shown in

Table 2. Number of GDRs.

GDRs	Number-Rated Power/kW
ILs	100-5
ILs	100-5

.

Table 3. Contract pricing between LA and distribution network.

Types	Unit Price/(USD/kWh)
CFLs	0.5 × TOU
ILs	0.2
	Translational time (h)	Unit price (USD/kWh)
TLs	6	0.053
	5	0.044
	4	0.035
	3	0.026
	2	0.018
	1	0.009
	0	0

shows the contract price between LA and the distribution networks. CFLs on the node connected to the wind turbines are used for practical scheduling in order to solve the uncertainty of the system. Except for the nodes that are connected to the wind turbines, CFLs on the other nodes are set to 5% of the load on each node, and they are used as the constraints of CFLs in the deterministic model.

Later, the configuration results of a 33-node distribution network and the scheduling plans of GDRs considering uncertainty are analyzed using simulations.

5.2. Analysis of Configuration Results

5.2.1. Analysis of ESS’s Configuration and Economic Factors

After calculation, the ESS’s configuration results considering the uncertainty of GDRs and renewable energies are obtained, and they are shown in

Table 4. ESS’s configuration results considering uncertainty of GDRs and renewable energies.

	Node5	Node16	Node28
Energy storage configuration (MWh)	1.24	7.45	1.50

.

On the basis of ESS’s configuration results in Table 4, the daily operating costs of the distribution network are obtained as shown in

Table 5. Costs of ESS’s configuration.

Daily Operating Cost of ESS (USD)	Network Loss Cost (USD)	Electricity Purchase Cost (USD)	DR Cost (USD)	Total Cost (USD)
86	62	5126	200	5474

.

For further comparative analyses of the configuration data, a 33-node distribution network operating in the five different situations described below is used. The main differences between these situations are whether ESS is configured and whether DR and uncertainty are considered.

(1)

ESS, DR and the uncertainty of this system are ignored;

(2)

DR and the uncertainty of this system are not considered; the model is used to reduce costs under the typical scenario;

(3)

DR is not considered, but the uncertainty of this system is considered. Aimed at reducing the total cost of the distribution networks, the ESS’s configuration under the typical scenario is determined;

(4)

Considering DR but ignoring the uncertainty of this system, the costs and initial energy storage configuration are determined;

(5)

Based on content (4), a daily operation simulation is carried out to update the energy storage configuration considering uncertainty.

The configuration results for the above five situations are analyzed using simulations. The comparison analyses are shown in

Table 6. Analysis of configuration under different contents.

Content	Energy Storage Configuration (MWh)	Total Cost of ESS Operation and Maintenance (USD)	Total Cost of ESS Investment (×106 USD)
1	-	-	-
2	node5: 12.94	1068.08	2.928
	node16: 0.56
	node28: 1.14
3	node5: 13.85	1144.73	3.138
	node16: 0.59
	node28: 1.25
4	node12:1.23	720.45	1.978
	node25:7.19
	node31:1.47
5	node12:1.24	742.35	2.038
	node25:7.45
	node31:1.50

and

Table 7. Cost comparison of different contents.

Content	Network Loss Cost (USD)	Electricity Purchase Cost (USD)	Daily Operating Cost of ESS (USD)	DR Cost (USD)	Total Cost (USD)
1	850	7721	-	-	8571
2	384	5610	110	-	6049
3	380	5542	114	-	6036
4	63	5145	84	200	5492
5	62	5126	86	200	5474

.

Both contents 2 and 3 consider DR, but content 3 also considers the uncertainty of the system. It can be seen in Table 5 that the energy storage configuration of each node in content 3 is larger than that of each node in content 2. Similarly, on the basis of content 4, content 5, which takes uncertainty factors into account, is also configured with more energy storage at each node. So, it can be seen from the configuration results in Table 5 that the system considering uncertainty needs to provide more energy storage and that DR can effectively reduce the energy storage capacity.

In order to analyze the economy of the system under several conditions, the scenario with the largest cost in the typical scenario is selected. In this scenario, the costs of each content are calculated.

The following can be seen in Table 6: Considering DR can effectively reduce the network loss and the power purchase ability of the main network. Although the system increases the cost of DR, the total cost is significantly reduced. Compared with the results of content (4) and content (5) in Table 6 and Table 7, the difference in the electricity purchase cost is small due to the small difference in the energy storage capacity.

5.2.2. Analysis of GDRs

After the energy storage configuration plans are determined, the participation of GDRs in DR is analyzed. Based on the ESS configuration schemes obtained using content (5), in the minimum output scenario of wind power and with the goal of minimizing the cost of the network loss and the cost of the power purchase in the main network, this paper determines the scheduling of GDRs.

In this paper, GDRs are divided into two categories: one is used to reduce the cost of the system, and the other is used to solve the uncertainty in actual scheduling. Figure 8 shows a situation where TLs participate in DR. Loads at high prices are shifted to the rest time periods according to different shifting plans.

A situation where CFLs participate in DR is shown in Figure 9.

Figure 10 shows the corresponding charging and discharging powers at each moment and the energy storage capacity under content (4) and content (5). The main difference between them lies in the 5–7 period and the 14–16 period. During these periods, the electricity prices change greatly. Content (5) has a lower charging power than content (4), because the energy storage capacity of content (5) is larger than that of content (4), and the electricity price at time 7 is higher than at time 6. The charging power at time 7 and the system cost are reduced.

Figure 11 shows the corresponding power purchase conditions of the main network under content (3) and content (5). During some high price periods, the main reason for the large amount of electricity purchased is the influence of the network structure. The power purchases in the main network are similar in both cases. DR can significantly reduce the network loss, thereby reducing the power of the main network’s purchases.

5.2.3. Comparison and verification of simulation results

In Section 3, the nonlinear problem (NLP) is converted into a quadratic programming problem (QP). So, the results and efficiency obtained by using the two methods in this paper need to be compared.

QP is a special case of NLP. When the objective function is quadratic and the constraints are linear, this kind of NLP can become QP.

Table 8. Comparison of results obtained using NLP and QP models.

Model	Objective Function Value (USD)	Energy Storage (MW/MWh)	Calculating Time (s)
QP	5603	7.23/14.46	7.04
NLP	5609	7.25/14.50	38.8

lists the calculation results and time of the nonlinear programming model and the quadratic programming model. It can be seen that the calculation results are very similar. The objective function value of QP is only 6 USD less than the function value of NLP, and the energy storage configuration values are also similar. Furthermore, the quadratic programming model requires less calculation time than the nonlinear programming model.

Therefore, this transformation can effectively shorten the calculation time without affecting the configuration results. This conversion is reliable and reasonable for the linearization of the power flow equation. The confidence coefficient α in Formula (28) is equal to 95%.

Considering the above simulation data, the ESS’s configuration results are compared to verify the validity of the model.

As can be seen in Figure 12, all costs are reduced considering the GDRs and uncertainty of this system. When only considering GDRs, the electricity purchase cost is reduced by 8.6%, the daily operating cost of ESS is reduced by 21.8%, and the total cost is reduced by 9.5%. Moreover, when considering the uncertainty of this system, the expenses do not increase. The network loss cost is reduced by 1.6%, the daily operating cost of ESS is reduced by 2.4%, and the total cost is also decreased by 0.3%.

To sum up, the ESS’s configuration results considering the uncertainty of GDRs and renewable energies can effectively reduce the system cost while reducing the power of the main network purchases (Figure 11). The accuracy and validity of the model are demonstrated.

6. Conclusions

Considering the uncertainty of users’ responses and the prediction errors of renewable energy, a GDR configuration model for distribution networks is proposed in this paper. Through the analyses of calculation examples, the accuracy and validity of the model are demonstrated.

So, not only can the results obtained in this paper provide a new approach for the formulation of incentive contracts, but they can also play an applicable role in the actual planning and operation of distribution networks in the future. The objective function is aimed at the expenses of the distribution networks, and the comfort of consumers is reflected by the constraints of time and contract prices. Through the regulation of GDRs, the uncertainty problems of users’ responses and renewable energies can be solved. Compared with only using DR or simply utilizing energy storage, the overall cost can be reduced by 29.6% and by 35.9%, respectively.

This paper obtains the scheduling scheme of GDRs and the configuration of ESS considering the uncertainty. In the future, real-time scheduling strategies for various resources from the perspective of LA should be determined, and a detailed analysis of the contract should be carried out.