Distributed Dispatch of Distribution Network Operators, Distributed Energy Resource Aggregators, and Distributed Energy Resources: A Three-Level Conditional Value-at-Risk Optimization Model

Abstract

To enhance the participation enthusiasm of distributed energy resources (DERs) and DER aggregators in their demand response, this paper develops a three-level distributed scheduling model for the distribution network operators (DNO), DER aggregators, and DERs based on the conditional value-at-risk (CVaR) theory. First, a demand response model is established for the DNO, DER aggregators, and DERs. Next, we employ the analytical target cascading (ATC) method to construct a three-level distributed scheduling model, where incentive and compensation prices are shared as consensus variables across the model levels to amplify the influence of DER aggregators on incentive prices and DERs on compensation prices. Then, the photovoltaic output model is restructured using the CVaR theory to effectively measure the risk associated with photovoltaic output uncertainty. Finally, an analysis is conducted using the IEEE 33-node distribution network to validate the effectiveness of the proposed model.

1. Introduction

In China’s evolving electricity market, the diversity of trading entities and models is increasing, resulting in more complex interactions among market participants [1]. Relying exclusively on the power generation side poses challenges in maintaining the safety and stability of the power system. To enhance the response speed of the DNO, implementing a demand response mechanism on the DERs can effectively guide DERs to align with DNO regulations [2], thereby alleviating scheduling pressures on the DNO.

Given the vast number and wide distribution of DERs, they are difficult to directly regulate. Therefore, DER aggregatorsean consolidate DERs [3], creating a three-level response system of DNO–DER aggregators–DERs. However, the diversity of DERs, complex scheduling, and the influence of DERs operators’ psychology complicate resource management [4].

To achieve the “dual carbon” goal, the large-scale integration of DERs has become a norm in China’s power system [5]. These DERs, however, are highly volatile and unpredictable [6], requiring DNOs to maintain additional peak shaving capabilities and reserve capacity. Additionally, some DERs outputs peak during low-demand periods (like nighttime), leading to the underutilization of DERs.

To explore resource scheduling in demand responses, the authors of [7] developed a hierarchical game model and implemented a closed-loop inverse mechanism to enhance the tuning speed of the demand response strategy. The authors of [8], utilizing the theory of cross elasticity, established a multi-energy demand side model to accurately assess DERs behavior. Meanwhile, another study [9] introduced a dual incentive mechanism that offers differentiated incentives for various DERs types, effectively lowering incentive costs while meeting response objectives.

To predict the uncertainty of DERs, ref. [10] employed a graph deep learning-based retail dynamic price mechanism to accurately assess DERs energy consumption at varying prices, thereby effectively reducing demand response uncertainty. Some studies focus on pre-selecting output scenarios. For instance, ref. [11] developed a DERs energy management system to support demand-side participating in demand response, utilizing the Monte Carlo simulation to generate uncertain scenarios within the model. Additionally, refs. [12,13] pre-selected a set of typical output scenarios to account for the uncertainty in wind power output.

Given the wide distribution and large number of demand response entities, centralized algorithms necessitate frequent information transmission, which can raise privacy concerns. In contrast, distributed algorithms facilitate local computation for all participants, ensuring privacy protection and aligning with the trend toward decentralized technology. As a result, they are increasingly adopted. For example, the authors of [14] developed a demand response for virtual power plant participation and implemented distributed optimization using the ADMM algorithm, enhancing the security of scheduling information. Similarly, the authors of [15] established a master–slave game framework to address DERs interaction strategies, tackling demand response challenges through multi-agent distributed optimization.

However, existing studies, such as [7,8,9], primarily focus on the bidding strategies of higher-level institutions, ignoring the fact that DER aggregators and the DERs cannot decide the price to be passed onto them. The power of price setting is controlled by higher-level institutions, and it is easy to reduce the enthusiasm of the participants in demand response. As for the research on output uncertainty, most of the existing literature only considers risk measurement in the centralized algorithm, while the processing method of uncertainty in the distributed algorithm still needs further research.

In summary, based on the existing research, this paper first establishes the three-level demand response scheduling model of DNO, DER aggregators, and DERs. Then, in order to improve the enthusiasm of DER aggregators and DERs in participating in demand response, the ATC method is used to modify the model. The incentive price for DER aggregators and the compensation price for DERs were utilized as consensus variables between the levels, thereby increasing the decision-making power of DER aggregators and DERs regarding pricing. After that, to address the uncertainty of photovoltaic output on the DERs side, CVaR theory was applied to quantify risks, enabling DERs to select scheduling methods with varying risk levels based on their psychological expectations. Finally, the effectiveness of the proposed three-level model was validated through numerical analysis using IEEE 33 nodes.

2. Three-Level Demand Response Scheduling Model

2.1. Upper-Level DNO Scheduling Optimization Model

2.1.1. DNO Model Objective Function

The upper-level DNO model needs to perform the demand response while maintaining network security and stability. Its objective function is minimum operating cost, given as follows:

$$f^G=\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t\left[{\lambda }_t^{Pb}{P}_{i,s,t}^{Dbuy}-{\lambda }_t^{Ps}{P}_{i,s,t}^{Dsell}-{\delta }_t^G{P}_{i,t}^G-{\kappa }_P{P}_{i,t}^{Ex}-{\kappa }_Q{Q}_{i,t}^{Ex}\right]$$

(1)

Among them, $f^G$ is the cost of DNO; ${\rho }_s$ is the probability of scenario $s$ appearing; $\mathsf{\Delta }t$ is the unit time intervals; ${\lambda }_t^{Pb}$ and ${\lambda }_t^{Ps}$ are the prices for purchasing and selling active energy from the electricity market; $P_{i,s,t}^{Dbuy}$ and $P_{i,s,t}^{Dsell}$ are the quantities of active energy purchased and sold by DNO from node $i$ to the electricity market during time period $t$; ${\delta }_t^G$ is the incentive price issued by DNO to DER aggregators; and $P_{i,t}^G$ is the electricity required by DNO to DER aggregators on node $i$. When there are no DERs participating in the response on node $i$, the value of the electricity required by DNO to DER aggregators is 0; ${\kappa }_P$ and ${\kappa }_Q$ are the excess penalty terms for the active and reactive power of DNO, respectively; $P_{i,t}^{Ex}$ and $Q_{i,t}^{Ex}$ are the excess active and reactive power of DNO, respectively; $N$ is the set of DNO nodes; $i$ is the node number; $S$ is the set of photovoltaic output scenarios; $s$ is the photovoltaic output scenario; $T$ is the set of runtime periods; and $t$ represents the runtime periods.

2.1.2. DNO Model Constraints

(1)

Network constraints

$$P_{i,s,t}^{Dsell}-P_{i,s,t}^{Dbuy}+P_{i,t}^{Base}-P_{i,t}^G=\sum _{l|i\in F(i)}{P}_{l,t}^{flow}-\sum _{l|i\in T(i)}\left(P_{l,t}^{flow}-I_{l,t}^2{r}_l\right)+P_{i,t}^{Ex}$$

(2)

$$Q_{i,s,t}^{Dsell}-Q_{i,s,t}^{Dbuy}+Q_{i,t}^{Base}=\sum _{l|i\in F(i)}{Q}_{l,t}^{flow}-\sum _{l|i\in T(i)}\left(Q_{l,t}^{flow}-I_{l,t}^2{x}_l\right)+Q_{i,t}^{Ex}$$

(3)

$$V_{j,t}=V_{i,t}-2\left(P_{l,t}^{flow}{r}_l+Q_{l,t}^{flow}{x}_l\right)+I_{l,t}^2(r_l^2+x_l^2)$$

(4)

$$I_{l,t}^2{V}_{i,t}\ge (P_{l,t}^{flow}{)}^2+(Q_{l,t}^{flow}{)}^2$$

(5)

$$V_i^{min}\le V_{i,t}\le V_i^{max}$$

(6)

$$(P_{l,t}^{flow}{)}^2+(Q_{l,t}^{flow}{)}^2\le {\left(S_l^{max}\right)}^2$$

(7)

Among them, $P_{i,t}^{Base}$ is the basic active load of node $i$ during time period $t$; $P_{l,t}^{flow}$ is the active power of branch $l$ during time period $t$; $I_{l,t}^2$ is the square value of branch current of branch $l$ during time period $t$; $r_l$ is the resistance of branch $l$; $Q_{i,s,t}^{Dbuy}$ and $Q_{i,s,t}^{Dsell}$ are the quantities of reactive energy purchased and sold by DNO from node $i$ to the electricity market during time period $t$, respectively; $Q_{l,t}^{flow}$ is the reactive power of branch $l$ during time period $t$; $x_l$ is the branch reactance of branch $l$; $Q_{i,t}^{Base}$ is the basic reactive load of node $i$ during time period $t$; $V_{i,t}$ is the square of voltage amplitude at node $i$ during time period $t$; $V_i^{max}$ and $V_i^{min}$ are the maximum and minimum square of voltage amplitude at node $i$, respectively; $S_l^{max}$ is the maximum capacity of branch $l$; $l$ is the branch Number; $F\left(i\right)$ is the set of branch head nodes; and $T\left(i\right)$ is the set of branch end nodes.

2.2. Middle-Level DER Aggregators Quotation Optimization Model

DER Aggregators Model Objective Function

The revenue of the middle-level DER aggregators model comes from the difference between the incentive price of the upper-level DNO and the compensation price of the lower-level DERs. Its objective function is maximum economic benefit, given as follows:

$$f^A=\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t({\delta }_t^{G^{\prime }}-{\delta }_t^A)P_{i,t}^G$$

(8)

Among them, $f^A$ is the revenue of DER aggregators; ${\delta }_t^{G^{\prime }}$ is the incentive price uploaded from DER aggregators to DNO; and ${\delta }_t^A$ is the compensation price issued to DERs by DER aggregators.

2.3. Lower-Level DERs Optimization Model

2.3.1. DERs Model Objective Function

Lower-level DERs need to participate in demand response while meeting their own basic load. Their objective function is maximum economic benefit, given as follows:

$$f^{LA}=\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t\left({\delta }_t^{A^{\prime }}{P}_{i,t}^G-f_{i,s,t}^{PV}-f_{i,s,t}^{ESS}-f_{i,s,t}^{CDG}-f_{i,s,t}^{FL}-f_{i,s,t}^{DNO}\right)$$

(9)

Among them, $f^{LA}$ is the total revenue of all DERs; ${\delta }_t^{A^{\prime }}$ is the compensation price uploaded by the DERs to DER aggregators; $f_{i,s,t}^{PV}$ is the photovoltaic output cost of DERs on node $i$ during time period $t$ in the photovoltaic output scenario $s$. For the sake of simplicity, the subscript $s$ will not be explained in the following text. In addition, $f_{i,s,t}^{ESS}$, $f_{i,s,t}^{CDG}$, $f_{i,s,t}^{FL}$, $f_{i,s,t}^{DNO}$ are the cost of energy storage system, gas turbine, flexible load, and purchasing electricity in the electricity market for DERs on node $i$ during time period $t$, respectively.

2.3.2. DERs Model Constraints

This article establishes four types of DERs, photovoltaics, energy storage systems, gas turbines, and flexible loads.

(1)

Photovoltaic output constraints

$$f_{i,s,t}^{PV}={\sigma }_{PV}{P}_{i,s,t}^{PV}$$

(10)

$$0\le P_{i,s,t}^{PV}\le P_{i,t}^{PV,max}$$

(11)

Among them, ${\sigma }_{PV}$ is the unit cost of photovoltaic power output; $P_{i,s,t}^{PV}$ is the photovoltaic power output of node $i$ during time period $t$; and $P_{i,t}^{PV,max}$ is the maximum power output of photovoltaic on node $i$ during time period $t$.

(2)

Energy storage system constraints

$$f_{i,s,t}^{ESS}={\sigma }_c(P_{i,s,t}^{ESS,c}{)}^2+{\sigma }_d(P_{i,s,t}^{ESS,d}{)}^2$$

(12)

$$0\le P_{i,s,t}^{ESS,c}\le P_{i,t}^{ESS,c,max}$$

(13)

$$0\le P_{i,s,t}^{ESS,d}\le P_{i,t}^{ESS,d,max}$$

(14)

$$e_{i,s,t}={\beta }_i{e}_{i,s,t-1}+P_{i,s,t}^{ESS,c}-P_{i,s,t}^{ESS,d}$$

(15)

$$E_i^{min}\le e_{i,s,t}\le E_i^{max}$$

(16)

Among them, $P_{i,s,t}^{ESS,c}$ and $P_{i,s,t}^{ESS,d}$ are charging and discharging power of energy storage system for node $i$ during time period $t$; ${\sigma }_c$ and ${\sigma }_d$ are unit charging and discharging costs; $P_{i,t}^{ESS,c,max}$ and $P_{i,t}^{ESS,d,max}$ are the maximum charging and discharging power of node $i$ during time period $t$; $e_{i,s,t}$ is the current energy storage level of the energy storage system at node $i$ during time period $t$; and ${\beta }_i$ is the power drop rate of the energy storage system at node $i$. It can reflect the proportion of stored electricity loss over time; $E_i^{max}$ and $E_i^{min}$ are the maximum and minimum energy storage level on node $i,\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}.$

(3)

Gas turbine constraints

$$f_{i,s,t}^{CDG}={\sigma }_{C1}(P_{i,s,t}^{CDG}{)}^2+{\sigma }_{C2}{P}_{i,s,t}^{CDG}+d$$

(17)

$$P_i^{CDG,min}\le P_{i,s,t}^{CDG}\le P_i^{CDG,max}$$

(18)

$$-\mathsf{\Delta }{P}_i^{CDG,max}\le P_{i,s,t}^{CDG}-P_{i,s,t-1}^{CDG}\le \mathsf{\Delta }{P}_i^{CDG,max}$$

(19)

Among them, ${\sigma }_{C1}$, ${\sigma }_{C2}$, $d$ are cost coefficients; $P_{i,s,t}^{CDG}$ is the power output of the generators at node $i$ during time period $t$; $P_i^{CDG,max}$ and $P_i^{CDG,min}$ are the maximum and minimum generators power output for node $i$; and $\mathsf{\Delta }{P}_i^{CDG,max}$ is the maximum climbing rate of the generators at node $i$.

(4)

Flexible load constraints

$$f_{i,s,t}^{FL}=\tau (P_{i,s,t}^{FL}-P_{i,t}^{FL,ba}{)}^2$$

(20)

$$P_{i,t}^{FL,min}\le P_{i,s,t}^{FL}\le P_{i,t}^{FL,max}$$

(21)

$$\sum _{t=1}^T{P}_{i,s,t}^{FL}=\sum _{t=1}^T{P}_{i,t}^{FL,ba}$$

(22)

Among them, $\tau$ is the flexible load transfer cost coefficient; $P_{i,s,t}^{FL}$ is the actual power consumption of flexible load at node $i$ during time period $t$; $P_{i,t}^{FL,ba}$ is the flexible load basic value of node $i$ during time period $t$, and is the flexible load that originally needs to be consumed at this time; and $P_{i,t}^{FL,max}$ and $P_{i,t}^{FL,min}$ are the maximum and minimum actual power consumption of the flexible load of node $i$ during time period $t$.

(5)

Power purchase constraints

$$f_{i,s,t}^{DNO}={\omega }^b{P}_{i,s,t}^{buy}-{\omega }^s{P}_{i,s,t}^{sell}$$

(23)

Among them, ${\omega }^b$ and ${\omega }^s$ are the unit electricity prices for purchasing and selling electricity from the electricity market for DERs, respectively; $P_{i,s,t}^{buy}$ and $P_{i,s,t}^{sell}$ are the amount of electricity purchased and sold by DERs on node $i$ to the electricity market during time period $t$.

(6)

Balance constraints

$$P_{i,s,t}^{sell}-P_{i,s,t}^{buy}-P_{i,t}^G=P_{i,s,t}^{PV}-P_{i,s,t}^{ESS,c}+P_{i,s,t}^{ESS,d}+P_{i,s,t}^{CDG}-P_{i,s,t}^{FL}-P_{i,t}^{Base}$$

(24)

(7)

Demond response quotation constraints

$${\delta }_t^{G,min}\le {\delta }_t^G\le {\delta }_t^{G,max}$$

(25)

$${\delta }_t^{G^{\prime },min}\le {\delta }_t^{G^{\prime }}\le {\delta }_t^{G^{\prime },max}$$

(26)

$${\delta }_t^{A,min}\le {\delta }_t^A\le {\delta }_t^{A,max}$$

(27)

$${\delta }_t^{A^{\prime },min}\le {\delta }_t^{A^{\prime }}\le {\delta }_t^{A^{\prime },max}$$

(28)

Among them, ${\delta }_t^{G,max}$ and ${\delta }_t^{G,min}$ are the maximum and minimum incentive prices issued by DNO to DER aggregators, respectively; ${\delta }_t^{G^{\prime },max}$ and ${\delta }_t^{G^{\prime },min}$ are the maximum and minimum incentive price uploaded by DER aggregators to DNO; ${\delta }_t^{A,max}$ and ${\delta }_t^{A,min}$ are the maximum and minimum compensation prices issued by DER aggregators to DERs; and ${\delta }_t^{A^{\prime },max}$ and ${\delta }_t^{A^{\prime },min}$ are the maximum and minimum compensation prices uploaded by DERs to DER aggregators.

3. ATC Method

3.1. ATC Method of Multi-Level Model

The ATC method is a distributed algorithm designed to address coordination problems in decentralized, multi-level models. It enables each element at every level to make autonomous decisions, while parent consensus variables coordinate and optimize the choices of child consensus variables to achieve an overall optimal solution [16]. Compared to other optimization methods, the ATC method offers advantages, such as parallel optimization, flexibility in structure, and rigorous convergence proofs, making it a popular choice for solving system optimization problems in multi-level models [17]. In this section, we first discuss the two-level model, and then we extend it to the multi-level model.

To modify the original two-level model through the ATC method, it is essential to identify a consensus variable that can be transferred between the upper and lower levels. The objective is to minimize the deviation between the target and response while adhering to the specified constraints.

$$\begin{gathered} \underset{x}{min}‖T_A-R_A‖ \\ \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} R_A=r\left(x\right) \end{gathered}$$

(29)

$$g_a\left(x\right)\le 0 a=1,\dots ,m_a$$

(30)

$$h_b\left(x\right)=0 b=1,\dots ,m_b$$

(31)

Among them, $T_A$ is the goal passed from the upper level to the lower level; $R_A$ is the response passed from the lower level to the upper level; $x$ represents the variables in the model; $r(\cdot )$ is the function related to $x$ in the lower-level model; $g_a(\cdot )$ is the inequality constraints in two-level model; $h_b\left(x\right)$ is the equality constraints in the two-level model; $a$ and $b$ are the number of iterations for the inner and outer levels of the current ATC method; and $m_a$ and $m_b$ are the total number of iterations for the inner and outer levels of the ATC method.

The two-level model shown in Equations (29)–(31) can be extended to a general multi-level model, as follows:

$$\begin{gathered} \mathit{min} ‖R_{ab}-R_{ab}^U‖+‖y_{ab}-y_{ab}^U‖+{\epsilon }_R+{\epsilon }_y \\ \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} R_{ab}=r_{ab}(R_{(a+1)b},\widetilde{x_{ab}},y_{ab}) \end{gathered}$$

(32)

$$\sum _{k\in C_{ab}}‖R_{(a+1)k}-R_{(b+1)k}^U‖\le {\epsilon }_R$$

(33)

$$\sum _{k\in C_{ab}}‖y_{(a+1)k}-y_{(a+1)k}^U‖\le {\epsilon }_y$$

(34)

$$g_{ab}\left(R_{ab},\widetilde{x_{ab}},y_{ab}\right)\le 0$$

(35)

$$h_{ab}\left(R_{ab},\widetilde{x_{ab}},y_{ab}\right)=0$$

(36)

Among them, $R_{ab}$ is the response of the current level to the upper level; $R_{ab}^U$ is the upper-level target for the current level; $y_{ab}$ and $y_{ab}^U$ are the consensus variables of the current level and the upper level, respectively; ${\epsilon }_R$ and ${\epsilon }_y$ are the tolerance errors of the current level and the upper level, respectively; $x_{ab}$ and $y_{ab}$ are the variables in the current level and upper level; $k$ is the current total number of iterations; $\mathrm{a}\mathrm{n}\mathrm{d} C_{ab}$ is the total maximum number of iterations. Equations (33) and (34) represent the convergence conditions of the iteration. When the difference between the consensus variables in the current level and the upper level is less than the tolerance error, the inner and outer iterations of the ATC method enter the next round.

3.2. DNO–DER Aggregators–DERs Three-Level Distributed Scheduling Model

To enhance the decision-making power of DERs and DER aggregators regarding the prices issued from the upper level, the incentive price issued from the DNO to the DER aggregators and the compensation price issued from the DER aggregators to the DERs are utilized as ATC consensus variables. Therefore, the DNO–DER aggregators–DERs three-level distributed scheduling model is constructed.

The upper level is the DNO optimization scheduling model, where the DNO first conducts power flow calculations to decide demand response and incentive prices. The middle level is the DER aggregators quotation optimization model, in which the DER aggregators accept incentive and compensation prices from both the upper and lower levels to optimize their bidding strategy. The lower level is the DERs optimization model. DERs optimize their own resource output according to the demand response while ensuring their own needs are met. The framework of the DNO–DER aggregators–DERs three-level model is illustrated in Figure 1.

3.3. ATC Method for Model Modification

3.3.1. ATC Method-Revised DNO Model

$$f^{G,ATC}=\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t\left[{\lambda }_t^{Pb}{P}_{i,s,t}^{Dbuy}-{\lambda }_t^{Ps}{P}_{i,s,t}^{Dsell}-{\delta }_t^G{P}_{i,t}^G-{\kappa }_P{P}_{i,t}^{Ex}-{\kappa }_Q{Q}_{i,t}^{Ex}-{\kappa }_t^{GA}{f}^{GLA,change}\right]$$

(37)

$$f^{GLA,change}={\left({\delta }_t^G-{\delta }_t^{G^{\prime },k_1,k_2}\right)}^2$$

(38)

Among them, $f^{G,ATC}$ represents the optimizing objectives for the ATC method of DNO; ${\kappa }_t^{GA}$ is the ATC penalty coefficient for DNO-DER aggregators; $f^{GLA,change}$ is the ATC penalty term; ${\delta }_t^{G^{\prime },k_1,k_2}$ is the middle-level response uploaded from DER aggregators to DNO; $k_1$ and $k_2$ are number of iterations for the outer and inner levels of the ATC method for DNO–DER aggregators, respectively. When the ATC method converges, the values of ${\delta }_t^G$ and ${\delta }_t^{G^{\prime },k_1,k_2}$ tend to be consistent, making the penalty term close to 0 without interfering with the optimization results. The ATC penalty term in the formula will gradually make the incentive price between DNO and DER aggregators converge during iteration, which is the bargaining process.

3.3.2. ATC Method-Revised DER Aggregators Model

$$f^{A,ATC}=\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t\left[({\delta }_t^{G^{\prime }}-{\delta }_t^A)P_{i,t}^G-{\kappa }_t^{AG}{f}^{LAG,change}-{\kappa }_t^{ALA}{f}^{ALA,change}\right]$$

(39)

$$f^{LAG,change}={\left({\delta }_t^{G,k_1,k_2}-{\delta }_t^{G^{\prime }}\right)}^2$$

(40)

$$f^{ALA,change}={\left({\delta }_t^A-{\delta }_t^{A^{\prime },k_3,k_4}\right)}^2$$

(41)

Among them, $f^{A,ATC}$ represents the optimizing objectives for the ATC method of DER aggregators; ${\kappa }_t^{AG}$ and ${\kappa }_t^{ALA}$ are the ATC penalty coefficients for DER aggregators–DNO and DER aggregators–DERs; $f^{LAG,change}$ and $f^{ALA,change}$ are the ATC penalty terms; ${\delta }_t^{G,k_1,k_2}$ is the upper-level target issued from DNO to DER aggregators; ${\delta }_t^{A^{\prime },k_3,k_4}$ is the lower-level response given by the DERs to DER aggregators; and $k_3$ and $k_4$ are the number of iterations for the outer and inner levels of the ATC method for DER aggregators–DNO.

3.3.3. ATC Method Revised DERs Model

$$f^{LA,ATC}=\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t\left[{\delta }_t^{A^{\prime }}{P}_{i,t}^G-f_{i,s,t}^{PV}-f_{i,s,t}^{ESS}-f_{i,s,t}^{CDG}-f_{i,s,t}^{FL}-f_{i,s,t}^{DNO}-{\kappa }_t^{LA}{f}^{LAA,change}\right]$$

(42)

$$f^{LAA,change}={\left({\delta }_t^{A,k_3,k_4}-{\delta }_t^{A^{\prime }}\right)}^2$$

(43)

Among them, $f^{LA,ATC}$ represents the optimizing objectives for the ATC method of DERs; ${\kappa }_t^{LA}$ is the ATC penalty coefficients for DERs–DER aggregators; $f^{LAA,change}$ is the ATC penalty term; and ${\delta }_t^{A,k_3,k_4}$ is the middle-level goal issued to DERs by DER aggregators.

3.3.4. ATC Convergence Conditions

The ATC consensus variable between DNO and DER aggregators is the incentive price transmitted between each other, and its inner convergence condition is given as follows:

$$\left|{\delta }_t^{G,k_1,k_2}-{\delta }_t^{G,k_1,k_2-1}\right|\le {\epsilon }_{in}^G$$

(44)

$$\left|{\delta }_t^{G^{\prime },k_1,k_2}-{\delta }_t^{G^{\prime },k_1,k_2-1}\right|\le {\epsilon }_{in}^G$$

(45)

The outer convergence condition is given as follows:

$$\left|{\delta }_t^{G^{\prime },k_1,k_2}-{\delta }_t^{G,k_1,k_2}\right|\le {\epsilon }_{out}^G$$

(46)

Among them, ${\epsilon }_{in}^G$ and ${\epsilon }_{out}^G$ are the tolerance errors for the inner and outer levels of the ATC method between DNO and DER aggregators and are usually set to be small to ensure the accuracy of the results.

The ATC consensus variable between DER aggregators and DERs is the compensation price communicated between each other, and its inner convergence condition is given as follows:

$$\left|{\delta }_t^{A,k_3,k_4}-{\delta }_t^{A,k_3,k_4-1}\right|\le {\epsilon }_{in}^A$$

(47)

$$\left|{\delta }_t^{A^{\prime },k_3,k_4}-{\delta }_t^{A^{\prime },k_3,k_4-1}\right|\le {\epsilon }_{in}^A$$

(48)

The outer convergence condition is given as follows:

$$\left|{\delta }_t^{A^{\prime },k_3,k_4}-{\delta }_t^{A,k_3,k_4}\right|\le {\epsilon }_{out}^A$$

(49)

Among them, ${\epsilon }_{in}^A$ and ${\epsilon }_{out}^A$ are the tolerance errors for the inner and outer levels of the ATC method between DER aggregators and DERs, respectively.

4. Uncertainty of Photovoltaic Output Measured Using CVaR Theory

4.1. Introduction to CVaR Theory

Value-at-risk (VaR) and CVaR theories were initially applied in the financial field to quantify potential risks [18]. VaR represents the maximum expected loss of an asset over a specified future period at a given confidence level, calculated as follows:

$$\vartheta (x_v,\alpha )={\int }_{f(x_v,y_v)}\sigma \left(y_v\right)d{y}_v$$

(50)

$$C_v=\mathit{min}\left\{\alpha \in R:\vartheta (x_v,\alpha )\ge \theta \right\}$$

(51)

Among them, $x_v$ represents the decision variables; $y_v$ represents the random variables; $\vartheta (x_v,\alpha )$ is the distribution function of loss function $f_v(x_v,y_v)$ and is no greater than the boundary value $\alpha$; $\sigma \left(y_v\right)$ is the probability density function of the random variable $y_v$; and $C_v$ is the VaR value under confidence level $\theta$.

Because VaR cannot calculate the risk information beyond the sub-point of confidence level $\theta$, ref. [19] proposed a CVaR risk measurement method, with the following calculation formula:

$$C_{va}={\displaystyle \frac{1}{1-\theta }}{\int }_{f(x_v,y_v)\ge C_v}f(x_v,y_v)\sigma \left(y_v\right)d{y}_v$$

(52)

Among them, $C_{va}$ is the CVaR value under confidence level $\theta$ and represents the average loss value of financial assets exceeding the VaR.

The transformation function derived from Formula (52) $F_{\theta }(x_v,\alpha )$ is given as follows:

$$F_{\theta }(x_v,\alpha )=\alpha +{\displaystyle \frac{1}{1-\theta }}\int {\left(f(x_v,y_v)-\alpha \right)}^{+}\sigma \left(y_v\right)d{y}_v$$

(53)

$${\left(f(x_v,y_v)-\alpha \right)}^{+}=\mathit{max}\left\{f(x_v,y_v)-\alpha ,0\right\}$$

(54)

At this point, $\alpha$ is the VaR value.

For ease of solution, discretizing Equation (53) yields the following:

$$F_{\theta ,d}(x_v,\alpha )=\alpha +{\displaystyle \frac{1}{q_a(1-\theta )}}\sum _{q=1}^{q_a}{\left(f(x_v,y_{v,q})-\alpha \right)}^{+}$$

(55)

$$C_{va}=\mathit{min}{F}_{\theta ,d}(x_v,\alpha )$$

(56)

Among them, $y_{v,q}$ is the $q$-th sample of random variable $y_v$; $q_{\alpha }$ is the total number of samples.

4.2. Using CVaR Theory to Deal with Uncertainty in Photovoltaic Output

To quantify the risk arising from the uncertainty of photovoltaic output in the DERs model, CVaR theory is employed to assess DERs risk. For DERs at node $i$, the CVaR value, which takes into account the uncertainty of photovoltaic output, is expressed as follows:

$$C_{aR}=C_R+{\displaystyle \frac{1}{1-\theta }}\sum _{s=1}^S{\rho }_s{C}_{ex}$$

(57)

Among them, $C_{aR}$ is the CVaR value of DERs cost on node $i$; $C_R$ is the VaR value of DERs cost on node $i$; and $C_{ex}$ is the DERs cost on node $i$ exceeds the VaR value.

For ease of use, relaxing the above equation yields the following:

$$C_{ex}\ge 0$$

(58)

$$C_{ex}\ge \sum _{t=1}^T\left[{\delta }_t^{A^{\prime }}{P}_{i,t}^G-f_{i,s,t}^{PV}-f_{i,s,t}^{ESS}-f_{i,s,t}^{CDG}-f_{i,s,t}^{FL}-f_{i,s,t}^{DNO}-{\kappa }_t^{LA}{f}^{LAA,change}\right]-C_R$$

(59)

The DERs model based on CVaR value is given as follows:

$$\mathit{min}{f}^{LA,C}=(1-\chi )\sum _{i=1}^N\sum _{s=1}^S\sum _{t=1}^T{\rho }_s\mathsf{\Delta }t\left[{\delta }_t^{A^{\prime }}{P}_{i,t}^G-f_{i,s,t}^{PV}-f_{i,s,t}^{ESS}-f_{i,s,t}^{CDG}-f_{i,s,t}^{FL}-f_{i,s,t}^{DNO}-{\kappa }_t^{LA}{f}^{LAA,change}\right]+\sum _{i=1}^N\chi C_{aR}$$

(60)

Among them, $f^{LA,C}$ is the cost of DERs based on CVaR; and $\chi$ is the risk preference coefficient, indicating the level of risk aversion, with a value range of [0, 1]. The closer $\chi$ is to 1, the more likely DERs are to make low-risk decisions.

5. Model Solving

Solution Steps

The solution flowchart for the three-level scheduling model proposed in this article is illustrated in Figure 2, with the specific steps as follows:

Step 1: Initialize model parameters.

Step 2: The upper-level DNO and lower-level DERs optimize their respective strategies and transmit incentive and compensation prices to the middle-level DER aggregators.

Step 3: The middle-level DER aggregators optimize their strategy based on the received prices, determine their own incentive and compensation prices, and re-transmit them to the DNO and DERs.

Step 4: If the incentive price transmitted between the DNO and DER aggregators and the compensation price transmitted between the DER aggregators and DERs both meet the convergence conditions of the ATC method, output the incentive and compensation prices. Otherwise, repeat steps 2–3.

6. Case Analysis

6.1. Parameter Settings

This paper uses an improved IEEE33 node distribution network [20] for simulation calculations. The topology structure is shown in Figure 3, with three DERs in nodes 25, 31, and 13, respectively. Among them, DERs1 includes photovoltaic power plants, energy storage system, and gas turbines, while DERs2 and DERs3 include photovoltaic power plants, energy storage system, gas turbines, and flexible loads. Due to the small scale of the distribution network, only one DER aggregator is set up to participate in the response. The compensation prices for the three DERs are consistent, and they can independently trade electricity in the electricity market. Setting the scenario that the required power is 1.124 MWh in 24 h, and performing demand response can be used to obtain the required power. We set the incentive price ${\delta }_t^G$ range issued by DNO to 200–3000 CNY/MWh, and the compensation price ${\delta }_t^A$ range issued by DER aggregators to 150–2800 CNY/MWh. The parameters of the photovoltaic plant, energy storage system, gas turbine, and flexible load are shown in

Table 1. DERs model parameters.

Type	Parameter	Value
Photovoltaic	${\sigma }_{PV}$	0.0852 (CNY/kW·h)
Energy storage system	${\sigma }_c$$, {\sigma }_d$	0.00018 (CNY/kW·h)
	$P_{i,t}^{ESS,c,max}$$, P_{i,t}^{ESS,d,max}$	60 (kW)
	${\beta }_i$	0.98
	$E_i^{max}$$, E_i^{min}$	200 (kW), 30 (kW)
Gas turbine	$P_i^{CDG,max}$$, P_i^{CDG,min}$	120 (kW), 15 (kW)
	$\mathsf{\Delta }{P}_i^{CDG,max}$	50 (kW)
Flexible load	$\tau$	0.035 (CNY/kW·h)
	$P_{i,t}^{FL,max}$$, P_{i,t}^{FL,min}$	250 (kW), 100 (kW)

. Three scenarios of photovoltaic output data were generated using the Latin hypercube sampling method, as shown in Figure 4. The rated power of the photovoltaic power stations for DERs 1, 2, and 3 are 120 kW, 130 kW, and 170 kW, respectively. The basic load of the DERs is shown in Figure 5, and other parameters are referenced in [17]. The algorithm used is modeled and calculated using Gams software version 24.4.6 and runs on a WINDOWS 11 PC (3.20 GHz, 16 G).

To validate the efficacy of the model presented in this article, we conducted a comparative analysis involving four distinct schemes:

Scheme 1: This scheme entirely excludes the ATC method, not utilizing it among the DNO, DER aggregators, and DERs models.

Scheme 2: The ATC method is used between DNO and DER aggregators models.

Scheme 3: The ATC method is used between DER aggregators and DERs models.

Scheme 4: The ATC method is used among the DNO, DER aggregators, and DERs models.

6.2. Analysis of Scheduling Results

6.2.1. Analysis of Optimization Results

Table 2. Calculation results of 4 scenarios.

	Scheme 1	Scheme 2	Scheme 3	Scheme 4
DNO cost (CNY)	4662.1	4707.7	4693.6	4737.0
DER aggregators revenue (CNY)	987.2	1132.6	886.2	1042.0
Total DERs revenue (CNY)	870.6	819.8	1098.2	1106.7
Total cost to participants(CNY)	2804.3	2755.3	2709.2	2588.3

presents the calculation results for the four schemes. Given that the parts of schemes 1, 2, and 3 that do not use the ATC method, they are solved using the hierarchical game model in reference [7]. This model is often used to solve demand response problems. Comparing Scheme 1 and Scheme 2 reveals that adopting the ATC method increases costs for the DNO and revenues for the DER aggregators, while DERs revenues decrease. This occurs because the ATC method enhances the DER aggregators’ decision-making power over their own incentive prices, yet the DER aggregators independently set compensation prices for DERs, resulting in only DER aggregators profit.

When comparing Scheme 1 and Scheme 3, it becomes evident that by using the ATC method to enhance DERs’ bargaining power, DERs secure higher compensation prices and increased profits. However, DER aggregators’ revenue, derived from the price difference between the DNO and DERs declines, as it cannot influence the DNO’s price.

The comparison between Scheme 1 and Scheme 4 indicates that employing the ATC method across the DNO, DER aggregators, and DERs models empowers both the DER aggregators and DERs to negotiate prices, ultimately enhancing profits for both parties. Overall, using the ATC method in the three-level DNO–DER aggregators–DERs model increases the benefits for demand response participants, thereby enhancing their motivation.

Comparing the total costs of the four schemes reveals that the total cost of Scheme 4 has decreased by 7.7% compared to Scheme 1. This reduction is attributed to the ATC method, which grants DERs bargaining power, enabling them to set prices based on resource output. This flexibility in resource scheduling lowers the costs associated with distributed resource management and ultimately reduces the model’s total cost. These findings demonstrate that the three-level scheduling model constructed using the ATC method effectively reduces DERs resource scheduling costs and enhances the overall economic benefits of demand response.

6.2.2. Analysis of CVaR Results

Using Scheme 4 as an example, we analyze the impact of the CVaR model on the total cost of DERs photovoltaic output, with the effective frontier curve illustrated in Figure 6. The figure indicates that at high $\chi$ values, DERs adopt a more conservative approach, opting to lower the CVaR value to mitigate potential risks from the uncertainty of photovoltaic output. Conversely, at low $\chi$ values, DERs are more aggressive, seeking to reduce total costs and increase the CVaR value. Overall, as $\chi$ rise, the total cost increases while the CVaR value decreases. Figure 6 illustrates the relationship between the total cost of photovoltaic output and the CVaR value, providing risk estimation for DERs.

Table 3. Comparison of benefits between DERs deterministic model and CVaR model.

Type	Deterministic Model			CVaR Model
	Day-Ahead Revenue (CNY)	Mid-Day Revenue (CNY)	Total Revenue (CNY)	Day-Ahead Revenue (CNY)	Mid-Day Revenue (CNY)	Total Revenue (CNY)
DERs1	446.9	−55.3	391.6	435.2	−77.8	357.4
DERs2	207.1	−98.7	108.4	198.8	48.5	247.3
DERs3	598.3	−115.7	482.6	577.2	−75.2	502.0
Total revenue	1252.3	−269.7	982.6	1211.2	−104.5	1106.7

compares the benefits of the DERs deterministic model and the CVaR model. In the day-ahead stage, the deterministic model, which does not account for the uncertainty of photovoltaic output, directly uses typical output scenarios. As a result, all three DERs have higher revenues in the day-ahead stage compared to the CVaR model. However, because the deterministic model overlooks the uncertainty of photovoltaic output, DERs may face lower-than-expected output and may need to purchase additional power at a high price from the electricity market. This leads to lower mid-day revenue compared to the CVaR model. The CVaR model, by considering the uncertainty of photovoltaic output in the day-ahead stage, enables an increase in mid-day revenue, ultimately resulting in higher total revenue compared to the deterministic model. This indicates that the CVaR theory has excellent economic viability.

7. Conclusions

This article proposes a three-level distributed scheduling model for DNO–DER aggregators–DERs, employing the ATC method to coordinate between levels and enhance the bargaining power of DER aggregators and DERs. In addition, CVaR theory is used to quantify the uncertainty of photovoltaic output to assist users in making decisions. Through the analysis of the case, the following conclusions are obtained:

(1)

The proposed three-level model increases the bargaining power between DER aggregators and DERs, improving their profits and enhancing their enthusiasm for participating in demand response. Additionally, the model grants DERs greater scheduling freedom, allowing them to set prices based on their actual output, thereby boosting the overall economic benefits of the demand response.

(2)

The use of CVaR theory enables an evaluation of the relationship between the total cost of photovoltaics and the associated risk level, helping DERs select risk preferences that align with their psychological expectations and make informed decisions. Moreover, it has been shown that applying the CVaR theory can enhance the economic efficiency of photovoltaics by quantifying output uncertainty.

It is important to note that this model currently only focuses on electricity scheduling and does not account for carbon emissions. And it ignores factors such as the regional environmental policy for renewable energy and their competitive relationship with other energy sources. Future research can integrate the carbon footprint of renewable energy sources and consider more different factors for a comprehensive analysis.