Multi-Objective Optimization Operation of Multi-Agent Active Distribution Network Based on Analytical Target Cascading Method

Abstract

In the context of the green energy transition, the rapid expansion of flexible resources such as distributed renewable energy, electric vehicles (EVs), and energy storage has significantly impacted the operation of distribution networks. This paper proposes a multi-objective optimization approach for active distribution networks (ADNs) based on analytical target cascading (ATC). Firstly, a dynamic optimal power flow (DOPF) calculation method is developed using second-order conic relaxation (SOCR) to address power flow and voltage issues in the distribution network, incorporating active management (AM) elements. Secondly, this study focuses on aggregating the power of flexible resources within station areas connected to distribution network nodes and incorporating these resources into demand response (DR) programs. Finally, a two-layer model for collaborative multi-objective scheduling between station areas and the active distribution network is implemented using the ATC method. Case studies demonstrate the model’s effectiveness and validity, showing its potential for enhancing the operation of distribution networks amidst the increasing integration of flexible resources.

1. Introduction

Under the guidance of the “dual carbon” goal and driven by the construction of a new power system, the role of distribution networks as key energy infrastructure is becoming increasingly prominent. With the explosive growth and large-scale integration of distributed photovoltaics (DPV), electric vehicles (EVs), and new energy storage technologies, the distribution network is undergoing a profound transformation from passive to active and from one-way radiation to two-way interaction, facing unprecedented transformation pressure and innovation opportunities [1]. Against this backdrop, the active distribution network has emerged, tasked not only with ensuring the basic responsibility of power supply but also with enhancing carrying capacity, achieving intelligent operation and maintenance, promoting clean energy transition, and optimizing the efficient allocation of source-grid-load-storage resources.

The distributed nature, intermittency, and volatility of distributed photovoltaic power generation systems present challenges such as system peak regulation and may cause reverse power flow, leading to large fluctuations in grid voltage and power factors, thus reducing the stability of the power system [2,3,4,5,6]. Additionally, the high penetration of distributed photovoltaics can complicate the net load characteristics of the distribution network, affecting net load forecasting and source-load balance [7,8,9,10]. By effectively utilizing active management elements in the active distribution network, such as on-load tap changers (OLTCs), capacitor banks (CBs), static VAR compensators (SVCs), and energy storage systems (ESSs), reverse power flow and node voltage deviation can be reduced, optimizing power flow distribution. However, linear modeling of these elements poses a significant challenge.

Traditional distribution networks primarily rely on centralized control for power dispatch. However, with the large-scale integration of distributed resources, the number of points that need to be monitored and controlled has surged, making centralized control insufficient to handle the massive energy and information interactions at numerous nodes. Consequently, as distributed generation and diverse flexible loads grow explosively and are integrated on a large scale, directly controlling each distributed device is no longer feasible. Research has increasingly focused on analyzing the aggregation characteristics of various distributed resources and controlling them as a whole. The literature [11] analyzes the matching relationship between different types of power output and load based on the degree of inconsistency and complementarity between multiple power sources and load power changes, constructing an evaluation index for source-load cluster operation. The literature [12] analyzes the complementary characteristics of distributed power sources such as wind and solar, energy storage, and conventional loads, proposing a cluster partitioning method based on fuzzy clustering algorithms that consider the complementarity between nodes and electrical distance. Simulation results show that the integration of distributed resources helps reduce the reverse power flow problem. However, the study does not fully consider the intermittency and randomness of distributed power sources such as wind and solar, which present significant challenges to the security and stability of the grid. While the complementarity between power sources and loads is analyzed, the literature does not specify how to address these random and uncertain factors, especially under extreme weather conditions or significant load fluctuations, making the cluster partitioning and operational evaluation indicators potentially unable to accurately reflect the actual operating state.

At the regional power system level, particularly in the distribution network and feeder areas, managing the energy usage of controllable resources among different types of users in the feeder area is a key method to improve operational stability, economic efficiency, and the proportion of renewable energy integration. Many scholars, domestically and internationally, have studied the reasonable extraction and effective aggregation of controllable resources on the user side. The literature [13] aggregates air conditioner models, analyzing the relationship between aggregated air conditioning power and parameters such as indoor and outdoor environments. The literature [14,15,16] uses a 0–1 description of water heater operating modes to construct a water heater load aggregation model and analyze its aggregation effectiveness. The literature [17,18,19] establishes aggregated charging power models for electric vehicles based on the initial state of charge and initial charging time of different types of electric vehicles. However, these studies all focus on aggregating models for single, homogeneous loads, which are not suitable for the diverse load characteristics of feeder areas. For multiple types of controllable resources, the literature [20] introduces the concept of load aggregators to aggregate various user loads and optimize them based on economic cost. The literature [21] considers user energy satisfaction to establish physical dynamic models of controllable resources and develops multi-level aggregation algorithms. Currently, research on the aggregation and scheduling of diverse user-side resources in feeder areas is limited, and there is a lack of reasonable strategies for effective energy management in feeder areas.

To address this, this paper proposes a multi-objective optimization operation method for multi-agent active distribution networks based on the analytical target cascading (ATC) method. This method aggregates flexible resources such as distributed renewable energy, electric vehicles, and energy storage to participate in demand response, improving power flow and voltage in the distribution network. The paper first proposes a dynamic optimal power flow calculation method based on second-order cone relaxation, incorporating active management elements for control. Then, using the ATC method, a multi-objective collaborative optimization scheduling model for feeder areas and the distribution network is constructed, forming a two-level optimization model. Case studies validate the rationality and effectiveness of this model, demonstrating the advantages of the proposed method in improving photovoltaic absorption, reducing line losses, and optimizing the operation of the distribution network. This study establishes a demand response model for various flexible resources within the feeder area, achieving optimal economic efficiency and user comfort through coordinated scheduling.

This paper makes three main contributions:

(1)

For the impact of distributed photovoltaic integration on distribution network operation, considering the operational constraints of various active management elements, a multi-objective optimization model for dynamic optimal power flow based on mixed-integer second-order cone programming (MISOCP) is established with the goal of minimizing reverse power flow, node voltage deviation, line active power loss, and photovoltaic curtailment penalties, achieving optimized scheduling for the distribution network.

(2)

For various flexible resources within the feeder area, a power aggregation and demand response model is established, constructing a mixed-integer linear programming (MILP) model with the objective of optimizing the economic efficiency and energy comfort of load aggregators, achieving optimal scheduling for the feeder area.

(3)

A two-level hierarchical optimization scheduling model is established, with the MISOCP-DOPF model for the distribution network as the upper layer and the MILP model for the feeder area as the lower layer. The ATC method is used for model decoupling, achieving coordinated optimization of the distribution network and feeder area operations.

The rest of this paper is organized as follows. Section 1 presents the modeling of the active distribution network. Section 2 models the feeder area. Section 3 provides the solution method for the two-layer model. Section 4 presents a case study analysis. Section 5 concludes the paper.

2. Active Distribution Network Modeling

In the distribution network, AM elements include on-load tap changers (OLTCs) for voltage regulation, discrete reactive power compensation with capacitor banks (CBs), continuous reactive power adjustment with static VAR compensators (SVCs), active power regulation with energy storage systems (ESSs), and both active and reactive power regulation with photovoltaic inverters (PVIs).

2.1. Objective Function

To minimize reverse power flow and node voltage deviations while also considering operational efficiency, the objective is to minimize the sum of reverse power flow, node voltage deviations, active power losses in lines, and penalties for reducing photovoltaic active power. Normalization is performed using the ideal point method, resulting in a multi-objective optimization model aimed at the safe and economical operation of the distribution network. The model can be expressed as follows:

$$\begin{array}{c}\min f=w_1{\displaystyle \frac{f_1-f_1^{\min }}{f_1^{\max }-f_1^{\min }}}+w_2{\displaystyle \frac{f_2-f_2^{\min }}{f_2^{\max }-f_2^{\min }}}+\\ w_3{\displaystyle \frac{f_3-f_3^{\min }}{f_3^{\max }-f_3^{\min }}}+w_4{\displaystyle \frac{f_4-f_4^{\min }}{f_4^{\max }-f_4^{\min }}}\\ \left\{\begin{array}{l}{f}_1={\displaystyle \sum _{j\in B^{re}}{\displaystyle \sum _{t\in T^{re}}\sqrt{p_{j,t}^2+q_{j,t}^2}\mathsf{\Delta }t}}\\ f_2={\displaystyle \sum _{j\in B}{\displaystyle \sum _{t\in T}\left|V_{j,t}-V_{base}\right|\mathsf{\Delta }t}}\\ f_3={\displaystyle \sum _{ij\in E}{\displaystyle \sum _{t\in T}{I}_{ij}^2{r}_{ij}\mathsf{\Delta }t}}\\ f_4={\displaystyle \sum _{j\in B^{\mathrm{DG}}}{\displaystyle \sum _{t\in T}\left(P_{j,t}^{\mathrm{DG},\mathrm{PRE}}-P_{j,t}^{\mathrm{DG}}\right)\mathsf{\Delta }t}}\end{array}\right.\end{array}$$

(1)

where $f_1$ is the reverse power flow; $f_2$ is the node voltage deviations; $f_3$ is the active power losses in lines; $f_4$ is the penalties for reducing photovoltaic active power; $f_1^{\max }$, $f_2^{\max }$, $f_3^{\max }$, and $f_4^{\max }$ are the objective values before optimization; $f_1^{\min }$, $f_2^{\min }$, $f_3^{\min }$, and $f_4^{\min }$ are the ideal points for each objective after optimization; $w_1$, $w_2$, $w_3$, and $w_4$ are the weights of each objective, satisfying $w_1+w_2+w_3+w_4=1$; $p$ and $q$ are the active and reactive power injections at nodes, respectively; $i$ and $j$ are node identifiers; $B^{\mathrm{re}}$ is the set of nodes with reverse power flow; t is the time period; $T^{\mathrm{re}}$ is the set of periods with reverse power flow; $\mathsf{\Delta }t$ is the time interval; $T$ is the node voltage; $T$ is the reference voltage for the OLTC primary side, set to 1 p. u.; $T$ is the set of all nodes; $T$ is the total number of periods; $r$ is the set of all branches; ij identifies the branch; $r$ is the branch current; $r$ is the branch resistance; $B^{\mathrm{DG}}$ is the set of nodes containing distributed generators (DGs); $P_{j,t}^{\mathrm{DG}}$ is the active power output of DGs; and $P_{j,t}^{\mathrm{DG},\mathrm{PRE}}$ is the forecasted DG output.

2.2. Constraints

2.2.1. System Flow Constraints

The optimal power flow (OPF) model for the distribution network, based on the branch flow model (BFM), is a nonlinear model (see Appendix B for details). To facilitate computation, capacity constraints in the quadratic terms need to be linearized. Therefore, angle relaxation (AR) and second-order cone relaxation (SOCR) are used for convexification. By applying $I_{ij}^2=I_{ij}^{\prime }$,$V_j^2=V_j^{\prime }$ and extending single-period flow calculations to multiple periods, the basic dynamic optimal power flow (DOPF) model is constructed. The SOCR-DOPF model is as follows:

$$\begin{array}{c}\min f(p,q,P,Q,V^{\prime },I^{\prime })\\ \left\{\begin{array}{l}{p}_{i,t}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{k\in \delta (j)}{P}_{jk,t}}}-{\displaystyle \sum _{i\in \pi (j)}(P_{ij,t}-I_{ij,t}^{\prime }{r}_{ij})}+g_j{V}_{j,t}^{\prime },\\ p_i\in X_j^p,\forall j\in B,\forall t\in T\\ q_{i,t}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{k\in \delta (j)}{Q}_{jk,t}}}-{\displaystyle \sum _{i\in \pi (j)}(Q_{ij,t}-I_{ij,t}^{\prime }{x}_{ij})}+b_j{V}_{j,t}^{\prime },\\ q_i\in X_j^q,\forall j\in B,\forall t\in T\\ V_{j,t}^{\prime }=V_{i,t}^{\prime }-2(P_{ij,t}{r}_{ij}+Q_{ij,t}{x}_{ij})+I_{ij,t}^{\prime }(r_{ij}^2+x_{ij}^2),\forall ij\in E,\forall t\in T\\ {‖\begin{array}{c}2P_{ij,t}\\ 2Q_{ij,t}\\ I_{ij,t}^{\prime }-V_{i,t}^{\prime }\end{array}‖}_2\le I_{ij,t}^{\prime }+V_{i,t}^{\prime },\hspace{0.17em}\forall ij\in E,\forall t\in T\\ -S_{ij,\max }\le P_{ij,t}\le S_{ij,\max },\forall ij\in E,\forall t\in T\\ -S_{ij,\max }\le Q_{ij,t}\le S_{ij,\max },\forall ij\in E,\forall t\in T\\ -\sqrt{2}{S}_{ij,\max }\le P_{ij,t}+Q_{ij,t}\le \sqrt{2}{S}_{ij,\max },\forall ij\in E,\forall t\in T\\ -\sqrt{2}{S}_{ij,\max }\le P_{ij,t}-Q_{ij,t}\le \sqrt{2}{S}_{ij,\max },\forall ij\in E,\forall t\in T\\ I_{ij,\min }^2\le I_{ij,t}^{\prime }\le I_{ij,\max }^2,\forall ij\in E,\forall t\in T\\ V_{j,\min }^2\le V_{j,t}^{\prime }\le V_{j,\max }^2,\forall j\in B,\forall t\in T\\ -s_{j,\max }\le p_{j,t}\le s_{j,\max },\forall j\in B,\forall t\in T\\ -s_{j,\max }\le q_{j,t}\le s_{j,\max },\forall j\in B,\forall t\in T\\ -\sqrt{2}{s}_{j,\max }\le p_{j,t}+q_{j,t}\le \sqrt{2}{s}_{j,\max },\forall j\in B,\forall t\in T\\ -\sqrt{2}{s}_{j,\max }\le p_{j,t}-q_{j,t}\le \sqrt{2}{s}_{j,\max },\forall j\in B,\forall t\in T\end{array}\right.\end{array}$$

(2)

where $f(•)$ is the objective function; P and Q are the active and reactive power of the branches, respectively; x is the branch reactance; $X_j^p$ and $X_j^q$ are the constraints related to the active and reactive power injections at nodes, respectively; k is the node identifier; $\delta (j)$ is the set of end nodes for branches starting at node j; $\pi (j)$ is the set of starting nodes for branches ending at node j; $I_{ij,\max }$ and $I_{ij,\min }$ are the upper and lower limits of branch currents, respectively; $V_{j,\max }$ and $V_{j,\min }$ are the upper and lower limits of node voltages, respectively; $S_{ij,\max }$ is the branch capacity; $s_{j,\max }$ is the capacity of the node (or substation); and ${‖•‖}_2$ is the 2-norm.

2.2.2. OLTC Operating Constraints

$$\left\{\begin{array}{l}{V}_{j,t}^{\prime }={\left|V_{\mathrm{base}}\right|}^2{r}_{j,t},\forall j\in B^T,\forall t\\ r_j^{\min }\le r_{j,t}\le r_j^{\max },\forall j\in B^T,\forall t\\ r_{j,t}=r_j^{\min }+{\displaystyle \sum _s{r}_{j,s}}{\sigma }_{j,s,t}^T,\forall j\in B^T,\forall t\\ {\sigma }_{j,1,t}^T\ge {\sigma }_{j,2,t}^T\ge \dots \ge {\sigma }_{j,{\mathrm{SR}}_j,t}^T,\forall j\in B^T,\forall t\\ {\delta }_{j,t}^{\mathrm{UP}}+{\delta }_{j,t}^{\mathrm{DOWN}}\le 1,\forall j\in B^T,\forall t\\ {\displaystyle \sum _s{\sigma }_{j,s,t}^T}-{\displaystyle \sum _s{\sigma }_{j,s,t-1}^T}\ge {\delta }_{j,t}^{\mathrm{UP}}-{\delta }_{j,t}^{\mathrm{DOWN}}{\mathrm{SR}}_j,\forall j\in B^T,\forall t\\ {\displaystyle \sum _s{\sigma }_{j,s,t}^T}-{\displaystyle \sum _s{\sigma }_{j,s,t-1}^T}\le {\delta }_{j,t}^{\mathrm{UP}}{\mathrm{SR}}_j-{\delta }_{j,t}^{\mathrm{DOWN}},\forall j\in B^T,\forall t\\ {\displaystyle \sum _{t\in T}({\delta }_{j,t}^{\mathrm{UP}}+{\delta }_{j,t}^{\mathrm{DOWN}})}\le N_j^{T,\max },\forall j\in B^T\end{array}\right.$$

(3)

where $r_{j,t}$ is the square of the voltage ratio between the secondary and primary sides of the OLTC; $r_j^{\max }$ and $r_j^{\min }$ are the squares of the upper and lower limits of the adjustable tap ratio, respectively; $r_{j,s}$ is the squared difference between the tap ratios of position s + 1 and position s; ${\sigma }_{j,s,t}^T$ is a binary variable indicating whether there is an adjustment between adjacent OLTC tap positions; ${\delta }_{j,t}^{\mathrm{UP}}$ and ${\delta }_{j,t}^{\mathrm{DOWN}}$ are binary variables for adjacent time periods, respectively, where ${\delta }_{j,t}^{\mathrm{UP}}=1$ is an increase in the OLTC tap position from period t to period t − 1, and ${\delta }_{j,t}^{\mathrm{DOWN}}=1$ is a decrease; ${\mathrm{SR}}_j$ is the maximum allowable change in the OLTC tap position; and $N_j^{T,\max }$ is the maximum number of allowed tap adjustments within a period T.

2.2.3. CB and SVC Operating Constraints

The constraints for CBs are established as follows:

$$\left\{\begin{array}{l}{Q}_{j,t}^{\mathrm{CB}}=y_{j,t}^{\mathrm{CB}}{Q}_{j,t}^{\mathrm{CB},\mathrm{one}},\forall j\in B^{\mathrm{CB}},\forall t\\ y_{j,t}^{CB}\le Y_j^{\max },\forall j\in B^{CB},\forall t\\ y_{j,t}^{\mathrm{CB}}={\displaystyle \sum _s{\sigma }_{j,s,t}^{\mathrm{CB}},\forall j\in B^{\mathrm{CB}},\forall t}\\ {\sigma }_{j,1,t}^{\mathrm{CB}}\ge {\sigma }_{j,2,t}^{\mathrm{CB}}\ge \cdots \ge {\sigma }_{j,Y_j^{\mathrm{CB},\max },t}^{\mathrm{CB}},\forall j\in B^{\mathrm{CB}},\forall t\\ {\delta }_{j,t}^{\mathrm{CB},\mathrm{IN}}+{\delta }_{j,t}^{\mathrm{CB},\mathrm{DE}}\le 1,\forall j\in B^{\mathrm{CB}},\forall t\\ y_{j,t}^{\mathrm{CB}}-y_{j,t-1}^{\mathrm{CB}}\ge {\delta }_{j,t}^{\mathrm{CB},\mathrm{IN}}-{\delta }_{j,t}^{\mathrm{CB},\mathrm{DE}}{Y}_j^{\mathrm{CB},\max },\forall j\in B^{\mathrm{CB}},\forall t\\ y_{j,t}^{\mathrm{CB}}-y_{j,t-1}^{\mathrm{CB}}\le {\delta }_{j,t}^{\mathrm{CB},\mathrm{IN}}{Y}_j^{\mathrm{CB},\max }-{\delta }_{j,t}^{\mathrm{CB},\mathrm{DE}},\forall j\in B^{\mathrm{CB}},\forall t\\ {\displaystyle \sum _{t\in T}({\delta }_{j,t}^{\mathrm{CB},\mathrm{IN}}+{\delta }_{j,t}^{\mathrm{CB},\mathrm{DE}})}\le N_j^{CB,\max },\forall t,\forall j\in B^{CB}\end{array}\right.$$

(4)

where $B^{\mathrm{CB}}$ is the set of nodes containing CBs; $Q_{j,t}^{\mathrm{CB}}$ is the reactive power compensation provided by the CBs; $Q_{j,t}^{\mathrm{CB},\mathrm{one}}$ is the reactive power compensation of a single CB unit; $y_{j,t}^{\mathrm{CB}}$ is the number of CB units in operation; $Y_j^{\max }$ is the maximum allowable number of CB units in operation; ${\sigma }_{j,s,t}^{\mathrm{CB}}$ is a binary variable indicating whether a CB unit is in operation; ${\delta }_{j,t}^{\mathrm{CB},\mathrm{IN}}$ and ${\delta }_{j,t}^{\mathrm{CB},\mathrm{DE}}$ are binary variables for adjacent time periods, respectively, where ${\delta }_{j,t}^{\mathrm{CB},\mathrm{IN}}=1$ is a decrease in the number of CB units in operation from period t − 1 to period t; and $N_j^{CB,\max }$ is the maximum allowable number of CBs switching operations within period T.

The model for SVCs is established as follows:

$$Q_j^{\mathrm{SVC},\min }\le Q_{j,t}^{\mathrm{SVC}}\le Q_j^{\mathrm{SVC},\max },\forall j\in B^{\mathrm{SVC}},\forall t$$

(5)

where $B^{\mathrm{SVC}}$ is the set of nodes containing SVCs; $Q_{j,t}^{\mathrm{SVC}}$ is the reactive power compensation provided by the SVCs; and $Q_j^{\mathrm{SVC},\max }$ and $Q_j^{\mathrm{SVC},\min }$ are the lower and upper limits of the SVC’s reactive power compensation, respectively.

2.2.4. Constraints for DG Operation

The model for DG is established as follows:

$$\left\{\begin{array}{l}{F}_j^{\mathrm{DG}}{P}_{j,t}^{\mathrm{DG},\mathrm{PRE}}\le P_{j,t}^{\mathrm{DG}}\le P_{j,t}^{\mathrm{DG},\mathrm{PRE}},\forall j\in B^{\mathrm{DG}},\forall t\\ -P_{j,t}^{\mathrm{DG}}\tan ({\phi }_j^{\mathrm{DG}})\le Q_{j,t}^{\mathrm{DG}}\le P_{j,t}^{\mathrm{DG}}\tan ({\phi }_j^{\mathrm{DG}}),\forall j\in B^{\mathrm{DG}},\forall t\end{array}\right.$$

(6)

where $P_{j,t}^{\mathrm{DG}}$ is the real power output of the DG; $P_{j,t}^{\mathrm{DG},\mathrm{PRE}}$ is the predicted power output; $F_j^{\mathrm{DG}}$ is the ratio of the DG’s power output to its maximum capacity; $Q_{j,t}^{\mathrm{DG}}$ stands for the reactive power output; and ${\phi }_j^{\mathrm{DG}}$ corresponds to the impedance angle associated with the DG’s minimum power factor.

3. Modeling of Distribution Feeder Areas

3.1. System Architecture

Figure 1 illustrates the control structure of flexibility resources in the distribution network’s substation areas. In certain substations, there are significant flexibility resources, such as electric vehicles, temperature-controlled loads, energy storage, and distributed photovoltaics. Due to the high willingness of users to participate in demand response, these flexibility resources are aggregated through a substation aggregation center. Building on this, the substation management center interacts with the distribution network via tie-line power and adjusts user electricity consumption habits through demand response. This approach balances economic efficiency and user comfort within the substation area, achieving power balance and increasing the photovoltaic absorption rate. In the distribution network, nodes have decentralized photovoltaic integration. Given the impact of renewable energy integration on the distribution network, AM elements are additionally integrated at the nodes to actively adjust voltage and power flow. The distribution network control center can adjust these AM elements to optimize the technical parameters of the network.

Figure 1. Schematic diagram of hierarchical regulation and control of “distribution network-station area-flexible resources”.

3.2. Objective Function

To effectively encourage user participation in demand response and support photovoltaic integration while considering user comfort, this section develops an objective function based on both economic efficiency and user temperature satisfaction. This forms a multi-objective optimization scheduling model for the distribution feeder area. Because the objective functions have different units, normalization is required. The process is as follows:

$$C_{\mathrm{econ}}^{\prime }={\displaystyle \frac{C_{\mathrm{econ}}-C_{\mathrm{econ},\min }}{C_{\mathrm{econ},\max }-C_{\mathrm{econ},\min }}}$$

(7)

where $C_{\mathrm{econ}}^{\prime }$ is the normalized value of the economic indicator; $C_{\mathrm{econ}}$ is the actual cost value before normalization; $C_{\mathrm{econ},\max }$ is the maximum possible value of the economic indicator before normalization; and $C_{\mathrm{econ},\min }$ is the minimum possible value before normalization. The user temperature satisfaction indicator is expressed as a percentage, so it does not require additional normalization. Therefore, the overall objective function can be expressed as follows:

$$C={\omega }_1{C}_{\mathrm{econ}}^{\prime }+{\omega }_2{C}_{\mathrm{temp}}$$

(8)

where ${\omega }_1$ and ${\omega }_2$ are the weight coefficients for the two objective functions, respectively (${\omega }_1+{\omega }_2=1$); and $C_{\mathrm{temp}}$ is the user satisfaction indicator.

3.2.1. Economic Indicators

The economic indicator is viewed from the perspective of the district management center and considers the control costs of various flexible resources over period T. Its expression is as follows:

$$C_{\mathrm{econ}}=\min {\displaystyle \sum _{t=1}^T(C_t^{\mathrm{DW}}+C_t^{\mathrm{DR}}}+C_t^{\mathrm{PV}}+C_t^{\mathrm{YW}})$$

(9)

where $C_t^{\mathrm{DW}}$ is the fee paid by the district management center to the distribution network at time t; $C_t^{\mathrm{DR}}$ is the demand response cost at time t; $C_t^{\mathrm{PV}}$ refers to the penalty for curtailed solar power at time t; and $C_t^{\mathrm{YW}}$ is the operation and maintenance cost of energy storage within the district at time t.

$$C_t^{\mathrm{DW}}=X_{\mathrm{b}}{P}_{\mathrm{buy},t}^{\mathrm{grid}}-X_{\mathrm{s}}{P}_{\mathrm{sell},t}^{\mathrm{grid}}$$

(10)

where $P_{\mathrm{buy},t}^{\mathrm{grid}}$ and $P_{\mathrm{sell},t}^{\mathrm{grid}}$ are the power purchased and sold by the district to the distribution network at time t, respectively; and $X_{\mathrm{b}}$ and $X_{\mathrm{s}}$ are the unit prices for purchasing and selling electricity from the district management center to the distribution network, respectively.

$$C_t^{\mathrm{DR}}={\displaystyle \sum _{m=1}^{N_{\mathrm{m}}}({\lambda }_m^{\mathrm{curt}}}{P}_{m,t}^{\mathrm{curt}})+{\displaystyle \sum _{t=1}^T\left|P_t^{\mathrm{ac}}-P_t^{\mathrm{ac}(0)}\right|}{\lambda }_{\mathrm{ac}}+{\displaystyle \sum _{t=1}^T\left|P_t^{\mathrm{ev}}-P_t^{\mathrm{ev}(0)}\right|}{\lambda }_{\mathrm{ev}}$$

(11)

where $N_{\mathrm{m}}$ is the total number of interruption levels; ${\lambda }_m^{curt}$ is the compensation price for interruptible load at level m; $P_{m,t}^{\mathrm{curt}}$ is the amount of interruptible load at level m at time t; ${\lambda }_{\mathrm{ac}}$ is the compensation price for air conditioning load; ${\lambda }_{\mathrm{ev}}$ is the compensation price for EV charging load; $P_t^{\mathrm{ac}\left(0\right)}$ is the air conditioning load before response at time t; $P_t^{\mathrm{ac}}$ is the air conditioning load after response at time t; $P_t^{\mathrm{ev}}$ is the electric vehicle charging load after response at time t; and $P_t^{\mathrm{ev}\left(0\right)}$ is the electric vehicle charging load before response at time t.

$$C_t^{\mathrm{PV}}={\displaystyle \sum _{t=1}^T{c}_{\mathrm{pv}}{P}_t^{\mathrm{pv},\mathrm{loss}}}$$

(12)

where $c_{\mathrm{pv}}$ is the penalty cost per unit of curtailed solar power, and $P_t^{\mathrm{pv},\mathrm{loss}}$ is the amount of curtailed solar power in the area at time t.

$$C_t^{\mathrm{YW}}={\displaystyle \sum _{t=1}^T(P_{\mathrm{out},t}^{\mathrm{ess}}}+P_{\mathrm{in},t}^{\mathrm{ess}})y_{\mathrm{ess}}$$

(13)

where $P_{\mathrm{out},t}^{\mathrm{ess}}$ and $P_{\mathrm{in},t}^{\mathrm{ess}}$ are the discharge and charge power of the energy storage system at time t, respectively; and $y_{\mathrm{ess}}$ is the maintenance cost per unit of discharge and charge power for the energy storage system.

3.2.2. User Temperature Satisfaction Indicator

During demand response, air conditioners in the distribution area adjust the temperature set by users, leading to a difference between the actual indoor temperature and the users’ expectations. This discrepancy can impact users’ daily lives and may lead to dissatisfaction. Therefore, it is important to model user temperature satisfaction. First, calculate the percentage deviation between the indoor temperature after response and the ideal temperature:

$$B_{\mathrm{temp},t}=\left|{\displaystyle \frac{T_{in,t}-0.5(T_{\mathrm{in},\max }+T_{\mathrm{in},\min })}{0.5(T_{\mathrm{in},\max }-T_{\mathrm{in},\min })}}\right|\times 100\%$$

(14)

where $T_{\mathrm{in},t}$ is the indoor temperature at time t; and $T_{\mathrm{in},\max }$ and $T_{\mathrm{in},\min }$ are the maximum and minimum indoor temperatures, respectively.

User temperature satisfaction can then be expressed as follows:

$$C_{\mathrm{temp}}=\min \left({\displaystyle \frac{{\displaystyle \sum _{t=1}^{24}}{S}_t\frac{{\mathrm{e}}^{B_{\mathrm{temp},t}}-1}{\mathrm{e}-1}}{{\displaystyle \sum _{t=1}^{24}}{S}_t}}\right)$$

(15)

where $S_t$ is the state of the air conditioning cluster at time t, which is a binary variable where a value of 0 indicates that the air conditioning cluster is in a standby state, and a value of 1 signifies that the cluster is operating at its rated capacity.

3.3. Constraints

3.3.1. EV Constraints

Unlike traditional loads, EV charging loads exhibit significant uncertainty. To accurately predict changes in EV load, it is necessary to model its key characteristics and describe its dynamic behavior. Based on traditional vehicle travel patterns, the following describes the charging and behavior characteristics of EVs.

Research typically assumes that the daily driving distance of an EV follows a log-normal distribution, with its probability density function expressed as follows:

$$f_D(x)={\displaystyle \frac{1}{x{\delta }_D\sqrt{2\pi }}}\exp [-{\displaystyle \frac{{(\ln x-{\mu }_D)}^2}{2{{\delta }_D}^2}}]$$

(16)

where ${\mu }_D$ and ${\delta }_D$ are the mean and standard deviation of the $\ln x$, respectively.

For a one-day period, the initial state of charge of an EV before travel follows a uniform distribution, with its probability density function expressed as follows:

$$f_S(S_{\mathrm{begin}}^{\mathrm{ev}})={\displaystyle \frac{1}{b-a}}$$

(17)

where a and b are the lower and upper limits of $S_{\mathrm{begin}}^{\mathrm{ev}}$, respectively.

Based on Equations (16) and (17), the state of charge of the EV from the end of the trip to the charging station can be determined as follows:

$$S_{\mathrm{arrive}}^{\mathrm{ev}}=S_{\mathrm{begin}}^{\mathrm{ev}}-{\displaystyle \frac{x{E}_{\mathrm{b}}}{100E_{\mathrm{h}}}}$$

(18)

where $E_{\mathrm{h}}$ is the battery capacity of the EV; and $E_{\mathrm{b}}$ is the energy consumption per 100 km.

The initial charging time of the EV when it connects to the grid follows a normal distribution, with the probability density function expressed as follows:

$$f_{\mathrm{s}}(t_{\mathrm{begin}})={\displaystyle \frac{1}{{\delta }_{\mathrm{s}}\sqrt{2\mathsf{\pi }}}}\exp [-{\displaystyle \frac{{(t_{\mathrm{begin}}-{\mu }_{\mathrm{s}})}^2}{2{{\delta }_{\mathrm{s}}}^2}}]$$

(19)

where ${\mu }_{\mathrm{s}}$ and ${\delta }_{\mathrm{s}}$ are the mean and standard deviation of $t_{\mathrm{begin}}$, respectively.

Based on Equations (18) and (19), the time required for the EV to complete charging can be calculated as follows:

$$t_{\mathrm{c}}={\displaystyle \frac{(1-S_{\mathrm{arrive}}^{\mathrm{ev}})E_{\mathrm{h}}}{{\eta }_{\mathrm{ev}}{P}_{\mathrm{ev}}}}$$

(20)

where ${\eta }_{\mathrm{ev}}$ is the EV charging efficiency; and $P_{\mathrm{ev}}$ is the EV’s rated changing power.

Therefore, the end time of EV charging is given by the following:

$$t_{\mathrm{over}}=t_{\mathrm{c}}+t_{\mathrm{begin}}$$

(21)

The end time of EV charging is processed, and the charging load power for a single EV is represented by time intervals as follows:

$$H_t=\left\{\begin{array}{l}{\eta }_{\mathrm{ev}}{P}_{\mathrm{ev}},\left\{\begin{array}{c}\begin{array}{l}\mathrm{if} 0<t_{\mathrm{over}}\le 24,\mathrm{then}\\ t\in \left[t_{\mathrm{begin}},t_{\mathrm{over}}\right]\end{array}\\ \begin{array}{l}\mathrm{if} t_{\mathrm{over}}>24,\mathrm{then} \\ t\in \left[t_{\mathrm{begin}},24\right]\cup \left[1,t_{\mathrm{over}}-24\right]\end{array}\end{array}\right.\hfill \\ 0,\mathrm{other} \mathrm{cases}\hfill \end{array}\right.$$

(22)

After conducting $M_{\mathrm{c}}$ number of simulations, the average is computed to determine the original charging load power for the $N_{\mathrm{c}}$ EVs:

$$P_t^{\mathrm{ev}\left(0\right)}={\displaystyle \frac{1}{M_{\mathrm{c}}}}\sum _{i=1}^{M_{\mathrm{c}}}\sum _{j=1}^{N_{\mathrm{c}}}{H}_{i,j,t}$$

(23)

where $H_{i,j,t}$ is the charging load power of the j EV at time t during the i simulation. This describes how the EV’s charging load varies over time.

According to psychology and economics theories, adjusting electricity prices can influence user consumption behavior. By implementing a time-of-use pricing strategy, the scheduling department can maximize the potential of flexible resources and guide the basic flexible resource loads to align with system scheduling requirements.

The time-of-use pricing periods for a typical day are determined using K-means clustering on the net load within the distribution area. Peak, shoulder, and off-peak periods are then matched with their respective electricity prices. The net load model for the distribution area is as follows:

$$P_t^{\mathrm{LOAD}}=P_t^{\mathrm{load}}+P_t^{\mathrm{ev}\left(0\right)}+P_t^{\mathrm{ac}\left(0\right)}-P_{\mathrm{all},t}^{\mathrm{pv}}$$

(24)

where $P_t^{\mathrm{LOAD}}$ is the net load of the distribution area at time t; and $P_t^{\mathrm{load}}$ is the forecasted load at time t.

The impact of electricity prices on user consumption behavior is quantified using the price elasticity of demand. The expression is as follows:

$$\kappa ={\displaystyle \frac{\mathsf{\Delta }{p}_t/p_t^{(0)}}{\mathsf{\Delta }{c}_t/c_t^{(0)}}}$$

(25)

where $\mathsf{\Delta }{p}_t$ and $\mathsf{\Delta }{c}_t$ are the changes in EV charging load and electricity price, respectively; $p_t^{(0)}$ and $c_t^{(0)}$ is the original EV charging load and original price, respectively. The equation, thus, is the ratio of load transfer rate to price transfer rate at time t.

The demand elasticity matrix is composed of demand elasticity coefficients and is expressed as follows:

$$\mathit{K}=\left[\begin{array}{ccc}{\kappa }_{\mathrm{ff}}& {\kappa }_{\mathrm{fp}}& {\kappa }_{\mathrm{fg}}\\ {\kappa }_{\mathrm{pf}}& {\kappa }_{\mathrm{pp}}& {\kappa }_{\mathrm{pg}}\\ {\kappa }_{\mathrm{gf}}& {\kappa }_{\mathrm{gp}}& {\kappa }_{\mathrm{gg}}\end{array}\right]$$

(26)

By guiding changes in user electricity consumption through adjustments in electricity prices, the goal of “peak shaving and valley filling” can be achieved. The expression for this time-of-use pricing response model in the context of EV applications is as follows:

$$\left[\begin{array}{l}{\widehat{p}}_t^{\mathrm{f}}\hfill \\ {\widehat{p}}_t^{\mathrm{p}}\hfill \\ {\widehat{p}}_t^{\mathrm{g}}\hfill \end{array}\right]=\left[\begin{array}{ccc}{p}_t^{\mathrm{f}\left(0\right)}& 0& 0\\ 0& p_t^{\mathrm{p}\left(0\right)}& 0\\ 0& 0& p_t^{\mathrm{g}\left(0\right)}\end{array}\right]\mathit{K}\left[\begin{array}{c}{\mathsf{\Delta }\mathrm{c}}_t^{\mathrm{f}}/c_t^{\mathrm{f}\left(0\right)}\\ {\mathsf{\Delta }\mathrm{c}}_t^{\mathrm{p}}/c_t^{\mathrm{p}\left(0\right)}\\ {\mathsf{\Delta }\mathrm{c}}_t^{\mathrm{g}}/c_t^{\mathrm{g}\left(0\right)}\end{array}\right]+\left[\begin{array}{c}{p}_t^{\mathrm{f}\left(0\right)}\\ p_t^{\mathrm{p}\left(0\right)}\\ p_t^{\mathrm{g}\left(0\right)}\end{array}\right]$$

(27)

where ${\widehat{p}}_t^{\mathrm{f}}$, ${\widehat{p}}_t^{\mathrm{p}}$, and ${\widehat{p}}_t^{\mathrm{g}}$ are the user electricity loads for the peak, flat, and off-peak periods after the response, respectively; $p_t^{\mathrm{f}\left(0\right)}$, $p_t^{\mathrm{p}\left(0\right)}$, and $p_t^{\mathrm{g}\left(0\right)}$ is the original user electricity loads for the peak, flat, and off-peak periods before the response, respectively; and $c_t^{\mathrm{f}\left(0\right)}$, $c_t^{\mathrm{p}\left(0\right)}$, and $c_t^{\mathrm{g}\left(0\right)}$ and ${\mathsf{\Delta }\mathrm{c}}_t^{\mathrm{f}}$, ${\mathsf{\Delta }\mathrm{c}}_t^{\mathrm{p}}$, and ${\mathsf{\Delta }\mathrm{c}}_t^{\mathrm{g}}$ are the original electricity prices and their changes for the peak, flat, and off-peak periods.

$$\left\{\begin{array}{l}{\displaystyle \sum _{t=1}^{24}{P}_t^{\mathrm{ev}}\mathrm{\Delta }t}={\displaystyle \sum _{t=1}^{24}{P}_t^{\mathrm{ev}\left(0\right)}\mathrm{\Delta }t}\\ {\displaystyle \sum _{t=1}^{24}\left[P_t^{\mathrm{ev}}\left(c_t^{(0)}+\mathrm{\Delta }{c}_t\right)\right]\mathrm{\Delta }t}\le {\displaystyle \sum _{t=1}^{24}\left[P_t^{\mathrm{ev}\left(0\right)}{c}_t^{(0)}\right]\mathrm{\Delta }t}\\ P_t^{\mathrm{ev}\left(0\right)}\left(1-\sigma \right)\le P_t^{\mathrm{ev}}\le P_t^{\mathrm{ev}\left(0\right)}\left(1+\sigma \right)\end{array}\right.$$

(28)

where $\sigma$ is the proportion of EV charging load participating in the demand response. This describes the model for price-based demand response participation by EVs.

3.3.2. Air Conditioning Constraints

With the ongoing development of smart appliances, temperature-controlled loads are becoming increasingly prevalent within distribution networks. The scheduling department can manage these temperature-controlled loads as a collective group. Given that most of these loads are air conditioners, they are used as a representative example. Building thermal resistance and air thermal capacity are modeled as electrical resistance and capacitance, respectively. Light and ambient temperature are represented as electrical sources. An equivalent thermal parameter model is used to simulate the heat exchange process of air conditioners, thereby establishing a power aggregation model.

Air conditioners are electrical devices that depend on ambient temperature. To describe the indoor temperature controlled by air conditioners, an equivalent thermal parameter model (ETP) is often used. The expression for this model is as follows:

$$T_{\mathrm{in},t}=T_{\mathrm{out},\mathrm{t}}-P_t^{\mathrm{ac}}R-(T_{\mathrm{out},\mathrm{t}}-P_t^{\mathrm{ac}}R-T_{\mathrm{in},t-1})e^{-{\displaystyle \frac{\mathrm{\Delta }t}{RC}}}$$

(29)

where $T_{\mathrm{out},t}$ is the outdoor temperature at time t; $P_t^{\mathrm{ac}}$ is the actual operating power of the air conditioner at time t; R is the thermal resistance of the air conditioner; and C is the thermal capacitance of the air conditioner.

The energy consumption of an air conditioner is closely related to its operating mode. The logic for determining the operating state of the air conditioner is expressed as follows:

$$S_t=\left\{\begin{array}{c}1,T_{\mathrm{in},t-1}\ge T_{\mathrm{set},t}+\epsilon \\ 0,T_{\mathrm{in},t-1}\le T_{\mathrm{set},t}-\epsilon \\ S_{t-\delta },else\end{array}\right.$$

(30)

where $\epsilon$ is the temperature hysteresis range for air conditioner state transitions; and $\delta$ is the simulation period. When the actual indoor temperature exceeds the sum of the air conditioner’s set temperature and the hysteresis temperature, the air conditioner must operate at its rated capacity. Conversely, when the actual indoor temperature falls below the difference between the set temperature and the hysteresis temperature, the air conditioner stops operating. In all other scenarios, the air conditioner maintains the same operational state as in the previous time step. Consequently, the actual operating power of the air conditioner can be expressed as follows:

$$P_t^{\mathrm{ac}\left(0\right)}={\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}{P}_{t,i}^{\mathrm{ac}}}={\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}{P}_{\mathrm{e}}}{S}_t$$

(31)

where $P_{t,i}^{\mathrm{ac}}$ is the operating power of the i air conditioner within the area; $P_{\mathrm{e}}$ is the rated operating power of the air conditioner; and $N_{\mathrm{ac}}$ is the total number of air conditioners in the area.

Although different air conditioning units share the same operating mechanism for temperature control, their equivalent thermal resistance, thermal capacity, and energy efficiency ratios vary. Under a common temperature control target, all air conditioning units in the area can be connected in parallel to create an aggregated equivalent thermal parameter model. By increasing the average set temperature of the air conditioners, the total power of the temperature control load cluster can be reduced, enabling incentive-based demand response. The model is expressed as follows:

$$\left\{\begin{array}{l}{R}_{\mathrm{eq}}={\displaystyle \frac{1}{{\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}\frac{1}{R_i}}}}\\ C_{\mathrm{eq}}={\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}{C}_i}\\ k_{\mathrm{eq}}={\displaystyle \frac{N_{\mathrm{ac}}}{{\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}\frac{1}{k_i}}}}\end{array}\right.$$

(32)

where $R_{\mathrm{eq}}$ is the equivalent thermal resistance after aggregating the air conditioners; $R_i$ is the thermal resistance of the i air conditioner; $C_{\mathrm{eq}}$ signifies the equivalent thermal capacity after aggregation; $C_i$ is the thermal capacity of the i air conditioner; $k_{\mathrm{eq}}$ is the equivalent energy efficiency ratio after aggregation; and $k_i$ is the energy efficiency ratio of the i air conditioner.

$$T_{\mathrm{in},\mathrm{eq},t}=T_{\mathrm{out},\mathrm{eq},t}-P_t^{\mathrm{cold}}{R}_{\mathrm{eq}}-(T_{\mathrm{out},\mathrm{eq},\mathrm{t}}-P_t^{\mathrm{cold}}{R}_{\mathrm{eq}}-T_{\mathrm{in},\mathrm{eq},t-1})e^{-{\displaystyle \frac{\mathrm{\Delta }t}{R_{\mathrm{eq}}{C}_{\mathrm{eq}}}}}$$

(33)

where $T_{\mathrm{in},\mathrm{eq},t}$ and $T_{\mathrm{out},\mathrm{eq},t}$ are the equivalent indoor and outdoor temperatures at time t, respectively; and $P_t^{\mathrm{cold}}$ is the equivalent cooling power of the aggregated air conditioners at time t.

$$\left\{\begin{array}{l}{P}_{\mathrm{e},t}^{\mathrm{cold}}={\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}{k}_i}{P}_{\mathrm{e}}\\ T_{\mathrm{ac},\mathrm{eq}}={\displaystyle \frac{1}{N_{\mathrm{ac}}}}{\displaystyle \sum _{i=1}^{N_{\mathrm{ac}}}{T}_{\mathrm{ac},i}}\\ T_{\mathrm{set},\mathrm{eq}}=T_{\mathrm{ac},\mathrm{eq}}+G\\ T_{\mathrm{in}}^{\min }\le T_{\mathrm{in},\mathrm{eq},t}\le T_{\mathrm{in}}^{\max }\\ 0\le G\le G_{\max }\\ 0\le P_t^{\mathrm{cold}}\le P_{\mathrm{e}}{S}_{\mathrm{eq},t}\\ P_t^{\mathrm{ac}}={\displaystyle \frac{T_{\mathrm{out},\mathrm{eq},t}-T_{\mathrm{in},\mathrm{eq},t}}{k_{\mathrm{eq}}{R}_{\mathrm{eq}}}}\end{array}\right.$$

(34)

where $P_{\mathrm{e},t}^{\mathrm{cold}}$ is the equivalent rated cooling power of the aggregated air conditioners; $T_{\mathrm{ac},\mathrm{eq}}$ is the equivalent original set temperature after aggregation; $T_{\mathrm{ac},i}$ is the original set temperature of the i air conditioner; G is the air conditioner’s temperature adjustment level; $G_{\max }$ is the maximum adjustment level; and $T_{\mathrm{in}}^{\max }$ and $T_{\mathrm{in}}^{\min }$ are the maximum and minimum indoor temperatures, respectively. The following is an improved version of Equation (30):

$$S_{\mathrm{eq},t}=\left\{\begin{array}{c}1,T_{\mathrm{in},\mathrm{eq},t-1}\ge T_{\mathrm{set},\mathrm{eq},t}+\epsilon \\ 0,T_{\mathrm{in},\mathrm{eq},t-1}\le T_{\mathrm{set},\mathrm{eq},t}-\epsilon \\ S_{\mathrm{eq},t-\delta },\mathrm{else}\end{array}\right.$$

(35)

where $T_{\mathrm{set},\mathrm{eq}}$ is the set temperature of the air conditioners after participating in demand response; and $S_{\mathrm{eq},t}$ is the real-time operating status of the air conditioners in the aggregated model at time t.

3.3.3. Constraints on Interruptible Loads

Based on the impact of load interruptions on users’ daily lives and production activities, interruptible loads are typically categorized into three types.

$$P_t^{\mathrm{curt}}={\displaystyle \sum _{m=1}^{N_{\mathrm{m}}}{P}_{m,t}^{\mathrm{curt}}}$$

(36)

where $P_t^{\mathrm{curt}}$ is the total amount of interruptible load for the entire area at time t.

Different interruption ratios and compensation prices are set according to the interruptible load level. The higher the interruption level, the lower the ratio coefficient and the higher the compensation price. The interruptible load demand response model is established as follows:

$$\begin{array}{c}{P}_{m,t}^{\mathrm{curt}}+P_{m,t-1}^{\mathrm{curt}}\le P_{m,t,\max }^{\mathrm{cl}}=k_{\mathrm{c},m}{P}_t^{\mathrm{load}1}\\ 0\le P_{m,t}^{\mathrm{curt}}\le k_{\mathrm{curt},m}{P}_t^{\mathrm{load}1}\end{array}$$

(37)

where $P_{m,t,\max }^{\mathrm{cl}}$ is the maximum amount of level m interruptible load over two consecutive periods; $k_{\mathrm{c},m}$ is the upper limit of the interruptible load ratio for level $m$ within a continuous period; and $k_{\mathrm{curt},m}$ is the upper limit of the interruptible load ratio for level $m$ at time t.

3.3.4. Constraints for Energy Storage and Distributed Photovoltaics

For user-side energy storage and distributed photovoltaics, we assume that the parameters of each device follow a uniform distribution within a certain range. We use the Monte Carlo method to approximate and generate data by analyzing the power characteristics of individual devices and then aggregating the power of all devices to develop a model for the power aggregation of the cluster. The energy storage system model is described as follows:

$$\left\{\begin{array}{l}{S}_{\max }^{\mathrm{ess}}={\displaystyle \sum _{d=1}^{N_{\mathrm{d}}}{S}_{d,\max }^{\mathrm{ess}}},S_{\min }^{\mathrm{ess}}={\displaystyle \sum _{d=1}^{N_{\mathrm{d}}}{S}_{d,\min }^{\mathrm{ess}}}\\ P_{\mathrm{i}\mathrm{n},t}^{\mathrm{ess}}={\displaystyle \sum _{d=1}^{N_{\mathrm{d}}}{P}_{\mathrm{in},d,t}^{\mathrm{ess}}{\eta }_{\mathrm{in}}}\\ P_{\mathrm{out},t}^{\mathrm{ess}}={\displaystyle \sum _{d=1}^{N_{\mathrm{d}}}{P}_{\mathrm{out},d,t}^{\mathrm{ess}}/{\eta }_{\mathrm{out}}}\\ S_{\min }^{\mathrm{ess}}\le S_t^{\mathrm{ess}}\le S_{\max }^{\mathrm{ess}}\\ {\alpha }_{\mathrm{out}}+{\alpha }_{\mathrm{in}}\le 1\\ 0\le P_{\mathrm{out},t}^{\mathrm{ess}}\le {\alpha }_{\mathrm{out}}{P}_{\mathrm{out},\max }^{\mathrm{ess}}\\ 0\le P_{\mathrm{i}\mathrm{n},t}^{\mathrm{ess}}\le {\alpha }_{\mathrm{in}}{P}_{\mathrm{i}\mathrm{n},\max }^{\mathrm{ess}}\\ S_t^{\mathrm{ess}}=S_{t-1}^{\mathrm{ess}}+P_{\mathrm{in},t}^{\mathrm{ess}}{\eta }_{\mathrm{in}}\mathrm{\Delta }t-P_{\mathrm{out},t}^{\mathrm{ess}}/{\eta }_{\mathrm{out}}\mathrm{\Delta }t\end{array}\right.$$

(38)

where $S_{\max }^{\mathrm{ess}}$ and $S_{\min }^{\mathrm{ess}}$ are the maximum and minimum storage capacities of the energy storage system, respectively; $S_{d,\max }^{\mathrm{ess}}$ and $S_{d,\min }^{\mathrm{ess}}$ are the maximum and minimum storage capacities of the d storage unit within the distribution area, respectively; $P_{\mathrm{out},t}^{\mathrm{ess}}$ and $P_{\mathrm{i}\mathrm{n},t}^{\mathrm{ess}}$ are the discharge and charge powers of the energy storage system at time t, respectively; $P_{\mathrm{in},d,t}^{\mathrm{ess}}$ and $P_{\mathrm{out},d,t}^{\mathrm{ess}}$ refer to the discharge and charge powers of the d storage unit at time t, respectively; ${\eta }_{\mathrm{out}}$ and ${\eta }_{\mathrm{in}}$ are the discharge and charge efficiencies of the energy storage system at time t, respectively; $S_t^{\mathrm{ess}}$ is the storage amount of the energy storage system at time t; $P_{\mathrm{out},\max }^{\mathrm{ess}}$ and $P_{\mathrm{in},\max }^{\mathrm{ess}}$ are the maximum discharge and charge powers of the energy storage system, respectively; ${\alpha }_{\mathrm{out}}$ and ${\alpha }_{\mathrm{in}}$ are the discharge and charge states of the energy storage system, respectively; and $\mathrm{\Delta }t$ is the time step.

To ensure that the actual consumption of PV energy does not exceed the generated PV energy, the following constraint applies to the consumption of rooftop PV within the distribution area:

$$\left\{\begin{array}{l}{P}_{\mathrm{all},t}^{\mathrm{pv}}={\displaystyle \sum _{d=1}^{N_{\mathrm{pv}}}{P}_{\mathrm{all},d,t}^{\mathrm{pv}}}\\ 0\le P_t^{\mathrm{pv}}\le P_{\mathrm{all},t}^{\mathrm{pv}}\end{array}\right.$$

(39)

where $P_t^{\mathrm{pv}}$ is the actual PV output power at time t; $P_{\mathrm{all},t}^{\mathrm{pv}}$ is the total PV generation within the distribution area at time t; $P_{\mathrm{all},d,t}^{\mathrm{pv}}$ is the total generation of the d distributed PV unit within the distribution area at time t; and $N_{\mathrm{pv}}$ is the total number of distributed PV units in the distribution area.

3.3.5. Overall Constraints of the Distribution Area

In addition to the flexible resources within the distribution area, the primary power supply for the area is also sourced from power purchased by load aggregators from the distribution network. The internal power balance model for the distribution area is described as follows:

$$P_t^{\mathrm{load}0}+P_t^{\mathrm{ev}}+P_t^{\mathrm{ac}}+P_t^{\mathrm{load}1}-{\displaystyle \sum _{m=1}^{N_{\mathrm{m}}}{P}_{m,t}^{\mathrm{curt}}}+P_{\mathrm{sell},t}^{\mathrm{grid}}+P_{\mathrm{in},t}^{\mathrm{ess}}=P_t^{\mathrm{pv}}+P_{\mathrm{buy},t}^{\mathrm{grid}}+P_{\mathrm{out},t}^{\mathrm{ess}}$$

(40)

where $P_t^{\mathrm{load}0}$ and $P_t^{\mathrm{load}1}$ are the original fixed load power and interruptible load power at time t, respectively; and $P_{\mathrm{buy},t}^{\mathrm{grid}}$ and $P_{\mathrm{sell},t}^{\mathrm{grid}}$ are the power purchased from and sold to the distribution network at time t.

The constraints on power purchases and sales within the distribution area are as follows:

$$\left\{\begin{array}{l}0\le P_{\mathrm{buy},t}^{\mathrm{grid}}\le u_{\mathrm{buy},t}{P}_{\mathrm{buy},\max }^{\mathrm{grid}}\\ 0\le P_{\mathrm{sell},t}^{\mathrm{grid}}\le u_{\mathrm{sell},t}{P}_{\mathrm{sell},\max }^{\mathrm{grid}}\\ 0\le u_{\mathrm{buy},t}+u_{\mathrm{sell},t}\le 1\end{array}\right.$$

(41)

where $u_{\mathrm{buy},t}$ and $u_{\mathrm{sell},t}$ are binary variables (0–1) indicating the status of power purchase and sale to the distribution network at time t, respectively; and $P_{\mathrm{buy},\max }^{\mathrm{grid}}$ and $P_{\mathrm{sell},\max }^{\mathrm{grid}}$ are the maximum power that can be purchased from or sold to the distribution network at time t, respectively, as determined by the capacity of the interconnection line between the distribution area and the grid.

4. Solution Methodology for the Bi-Level Model

4.1. Model Preprocessing

As described earlier, the bi-level model for the coordinated scheduling of the distribution network and the distribution area consists of two layers. The upper layer focuses on the active distribution network, which includes AM (active management) elements. This layer establishes constraints related to power flow and the operation of AM elements, evaluating aspects such as reverse power flow, node voltage, line active power loss, and curtailed solar energy. The lower layer focuses on the distribution area, which includes various flexible resources. It sets operational constraints for these resources and overall constraints for the area, assessing the economic performance of the distribution area’s management center and the temperature satisfaction of its users. The bi-level model for coordinated scheduling between the distribution network and the distribution area is illustrated in Figure 2.

Figure 2. Two-layer model of distribution network-station area cooperative dispatching.

4.2. Solving the Bi-Level Model Using ATC

Based on the established objective functions and constraints for both the upper and lower layers, the global optimization model is formulated as follows:

$$\begin{array}{l}\min ({\delta }_{\mathrm{tq}}{\displaystyle \frac{C}{H_{\mathrm{tq}}}}+{\delta }_{\mathrm{grid}}{\displaystyle \frac{f}{H_{grid}}})\\ \left\{\begin{array}{c}(2)–(6),(28),(34),(35),(37)–(41)\\ {\delta }_{\mathrm{tq}}+{\delta }_{\mathrm{grid}}=1\end{array}\right.\end{array}$$

(42)

where ${\delta }_{\mathrm{tq}}$ and ${\delta }_{\mathrm{grid}}$ are the weighting coefficients for the optimization objectives of the lower-level distribution area and the upper-level distribution network, respectively; and $H_{tq}$ and $H_{grid}$ are the normalization factors corresponding to these objectives.

To simplify the resolution of the complex coupled model involving multiple stakeholders, the ATC algorithm is used to decouple the aforementioned global optimization model. The power transactions between the distribution area and the distribution network are treated as linkage power, and the augmented Lagrangian function is employed as the penalty function. This transforms the global optimization model in Equation (42) into the following bi-level optimization model.

Upper-level:

$$\begin{array}{c}\min \{{\delta }_{\mathrm{grid}}{\displaystyle \frac{f}{H_{\mathrm{grid}}}}+{\displaystyle \sum _{t\in T}{\displaystyle \sum _i[{\pi }_{i,t}^{\mathrm{p},1}\mathrm{\Delta }{P}_{i,t}^{\mathrm{up}}}}+{\pi }_{i,t}^{\mathrm{p},2}{(\mathrm{\Delta }{P}_{i,t}^{\mathrm{up}})}^2]\}\\ \left\{\begin{array}{c}\mathrm{\Delta }{P}_{i,t}^{\mathrm{up}}=P_{i,t}^{\mathrm{up}}-{\stackrel{⌢}{P}}_{i,t}^{\mathrm{low}}\\ (2)–(6)\end{array}\right.\end{array}$$

(43)

Lower-level:

$$\begin{array}{c}\min \{{\delta }_{\mathrm{tq}}{\displaystyle \frac{C}{H_{\mathrm{tq}}}}+{\displaystyle \sum _{t\in T}{\displaystyle \sum _i[{\pi }_{i,t}^{\mathrm{p},1}\mathrm{\Delta }{P}_{i,t}^{\mathrm{low}}}}+{\pi }_{i,t}^{\mathrm{p},2}{(\mathrm{\Delta }{P}_{i,t}^{\mathrm{low}})}^2]\}\\ \left\{\begin{array}{c}\mathrm{\Delta }{P}_{i,t}^{\mathrm{low}}={\stackrel{⌢}{P}}_{i,t}^{\mathrm{up}}–P_{i,t}^{\mathrm{low}}\\ (28),(34),(35),(37)-(41)\end{array}\right.\end{array}$$

(44)

where ${\pi }_{i,t}^{\mathrm{p},1}$ and ${\pi }_{i,t}^{\mathrm{p},2}$ are the first- and second-order Lagrange multipliers corresponding to the linkage power of the i transmission line at time t, respectively; $\mathrm{\Delta }{P}_{i,t}^{\mathrm{up}}$ is the difference between the upper-level linkage power and the lower-level optimized linkage power at time t on the i transmission line; ${\stackrel{⌢}{P}}_{i,t}^{\mathrm{low}}$ is the value that needs to be optimized at the lower level and is communicated to the upper level; $\mathrm{\Delta }{P}_{i,t}^{\mathrm{low}}$ is the difference between the upper-level optimized linkage power and the lower-level linkage power; ${\stackrel{⌢}{P}}_{i,t}^{\mathrm{up}}$ is the value that needs to be optimized at the upper level, and this value is passed from the upper to the lower level.

The convergence criteria for the iterative solution of the bi-level optimization model are as follows:

$$\left\{\begin{array}{l}\left|P_{i,t,k}^{\mathrm{up}}-P_{i,t,k}^{\mathrm{low}}\right|\le {\xi }_1\\ \left|{\displaystyle \frac{(C_k+{\displaystyle \sum _i{f}_{i,k}})-(C_{k-1}+{\displaystyle \sum _i{f}_{i,k-1}})}{(C_k+{\displaystyle \sum _i{f}_{i,k}})}}\right|\le {\xi }_2\end{array}\right.$$

(45)

where $P_{i,t,k}^{\mathrm{up}}$ and $P_{i,t,k}^{\mathrm{low}}$ are the linkage power on the i transmission line at time t for the upper and lower levels during the k iteration, respectively; $C_k$ and $f_{i,k}$ are the objective function values for the distribution network and the i district in the k iteration, respectively; and ${\xi }_1$ and ${\xi }_2$ are the convergence tolerances, respectively.

The Lagrange multiplier update criterion is as follows:

$$\left\{\begin{array}{l}{\pi }_{i,t,k}^{\mathrm{p},1}={\pi }_{i,t,k-1}^{\mathrm{p},1}+2{({\pi }_{i,t,k-1}^{\mathrm{p},2})}^2({\stackrel{⌢}{P}}_{i,t,k-1}^{\mathrm{up}}-{\stackrel{⌢}{P}}_{i,t,k-1}^{\mathrm{low}})\\ {\pi }_{i,t,k}^{\mathrm{p},2}=\beta {\pi }_{i,t,k-1}^{\mathrm{p},2}\end{array}\right.$$

(46)

The relevant solving process is shown in Algorithm 1.

Algorithm 1: Analysis target cascading.

Step 1: Initialize parameters, $k=1, {\pi }_{i,t,1}^{\mathrm{p},1}={\pi }_{i,t,1}^{\mathrm{p},2}=0.5, {{\displaystyle \stackrel{⌢}{P}}}_{i,t,1}^{\mathrm{low}}={{\displaystyle \stackrel{⌢}{Q}}}_{i,t,1}^{\mathrm{low}}=0.5, \beta =2$.

Step 2: Given the expected values of tie-line power ${{\displaystyle \stackrel{⌢}{P}}}_{i,t,k-1}^{\mathrm{low}} \mathrm{and} {{\displaystyle \stackrel{⌢}{Q}}}_{i,t,k-1}^{\mathrm{low}}$ from the lower-level model, solve the upper-level model and record the resulting values of tie-line power ${{\displaystyle P}}_{i,t,k}^{\mathrm{up}} \mathrm{and} {{\displaystyle Q}}_{i,t,k}^{\mathrm{up}}$, denoted as ${{\displaystyle \stackrel{⌢}{P}}}_{i,t,k}^{\mathrm{up}} \mathrm{and} {{\displaystyle \stackrel{⌢}{Q}}}_{i,t,k}^{\mathrm{up}}$, respectively.

Step 3: Given the expected values of tie-line power ${{\displaystyle \stackrel{⌢}{P}}}_{i,t,k}^{\mathrm{up}}\mathrm{ }\mathrm{and}\mathrm{ }{{\displaystyle \stackrel{⌢}{Q}}}_{i,t,k}^{\mathrm{up}}$ from the upper-level model, solve the lower-level model and record the resulting values of tie-line power ${{\displaystyle P}}_{i,t,k}^{\mathrm{low}} \mathrm{and} {{\displaystyle Q}}_{i,t,k}^{\mathrm{low}}$, denoted as ${{\displaystyle \stackrel{⌢}{P}}}_{i,t,k}^{\mathrm{low}}\mathrm{ }\mathrm{and}\mathrm{ }{{\displaystyle \stackrel{⌢}{Q}}}_{i,t,k}^{\mathrm{low}}$, respectively.

Step 4: Determine convergence based on the convergence criterion given in Equation (45). If convergence is achieved, terminate the iteration.

Step 5: $k=k+1$, update the Lagrange multipliers according to the update rule in Equation (46). Return to Step 2.

5. Case Studies

This study uses the IEEE 33-node model as the research subject to validate the model’s effectiveness. Some nodes in the distribution network are integrated with AM elements, with specific parameters detailed in Appendix A, Table A1 [22]. The power generation and load power of photovoltaic systems are predicted using the K-means++ algorithm, as shown in Appendix B, Figure A1. The topology connecting the substation area, AM elements, and the active distribution network is illustrated in Figure 3. The model is coded with the YALMIP toolbox 20230622 in MATLAB R2024a for Windows environment and solved by commercial optimization software GUROBI 11.0.

Figure 3. Topology diagram of station area, AM element, and distribution network connection.

5.1. Analysis of Active Distribution Network Optimization Dispatch Results

According to relevant industry standards, a reverse power flow exceeding 80% of the transformer capacity is considered a reverse heavy overload, and a node voltage exceeding 107% of the reference voltage is classified as node overvoltage. The dispatch results of various AM elements within the active distribution network are shown in Figure 4, Figure 5, Figure 6 and Figure 7. During midday, when photovoltaic generation is high and the load is low, power back feed and node overvoltage frequently occur in the distribution network. Around 12:00 to 13:00, both the SVC and ESS absorb a significant amount of active power, and the OLTC tap ratio decreases, which helps reduce the “source” power in the network and effectively increases the “load” power.

Figure 4. OLTC scheduling results.

Figure 5. CB scheduling results.

Figure 6. SVC scheduling results.

Figure 7. ESS scheduling results.

Table 1 presents a comparison of the simulation results for various scenarios in the active distribution network. The comparison confirms that large-scale photovoltaic (PV) integration causes reverse power flow and increases node voltage deviation but reduces active power losses in the lines. However, the integration of AM elements can effectively mitigate the reverse power flow and the increased node voltage deviation caused by PV integration while further reducing active power losses in the lines. Figure 8 illustrates the voltage levels at various nodes in the distribution network: (a) without PV integration, (b) after the integration of high-percentage PV, and (c) after the integration of AM elements into a network with high penetration of distributed PV. A comparison between Figure 8a and Figure 8b clearly shows that significant distributed PV generation can lead to overvoltage for distribution network users. A comparison with Figure 8c demonstrates that the integration of AM elements results in a flatter voltage distribution, significantly improving system voltage quality and effectively mitigating overvoltage at distribution network nodes. Analysis of Figure 9 reveals that some nodes still have an abandoned light at certain times, and the photovoltaic consumption rate is still not 100%, indicating that the photovoltaic carrying capacity of these nodes with an abandoned light has reached its limit. This observation can be used to evaluate the large-scale distributed photovoltaic consumption capacity of different nodes.

Table 1. Comparison of simulation results.

Case Scenario	Comprehensive Objective/p.u.	Reverse Power Flow in Distribution Network/p.u.	Node Voltage Deviation/p.u.	Active Power Loss of Lines/p.u.	PV Active Power Curtailment/p.u.	PV Consumption Rate/%
Without photovoltaic integration	\	0	16.3738	0.2589	\	\
With photovoltaic integration (penetration rate of 80%)	\	1.4561	19.2383	0.1676	\	\
AM	0.1624	0.6783	11.1845	0.1429	0.1974	93.5597

Figure 8. Node voltage in the distribution network.

Figure 9. Proportion of distributed PV consumption in the distribution network.

5.2. Analysis of Distribution Substation Optimization Dispatch Results

The parameters of various flexible resources integrated into the substation area are listed in Appendix A, Table A2, Table A3, Table A4, Table A5 and Table A6 [23]. Before dispatch, the peak, flat, and valley periods are determined based on netload, and the corresponding electricity prices are matched, as shown in Appendix B, Figure A3. The overall dispatch results for the substation area are presented in Figure 10, and the dispatch results for each flexible resource are shown in Figure 11, Figure 12, Figure 13 and Figure 14. Analysis of Figure 10 indicates that the substation area primarily relies on flexible resources for dispatch without selling electricity back to the distribution network. The energy storage dispatch results are shown in Figure 11. During valley price periods, energy storage charges as much as possible to build reserves, and during flat and peak price periods, it discharges to meet the substation area’s load demand, thereby reducing electricity costs. The stored energy increases during charging and decreases during discharging, which aligns with its operational requirements.

Figure 10. Station scheduling results.

Figure 11. Station ESS scheduling results.

Figure 12. Station EV scheduling results.

Figure 13. Station air conditioning scheduling results.

Figure 14. Station interruptible load scheduling results.

The EV dispatch results for the substation area are shown in Figure 12. During peak price periods, the EV charging load decreases and shifts to valley price periods, while it remains unchanged during flat price periods, meeting the requirements of price-based demand response. The air conditioning dispatch results for the substation area are shown in Figure 13. It is evident that the air conditioning load decreases throughout the 24 h period. The control center increases the air conditioning set temperature to reduce the electric load while ensuring that the maximum indoor temperature requirement is not exceeded, thus minimizing electricity costs. The dispatch results for interruptible loads are shown in Figure 14. During peak price periods, the control center first interrupts the first category of interruptible loads to minimize user impact. However, to ensure that the interruption time for the same category of loads does not exceed constraints within continuous periods, the center does not continuously interrupt the first category. Instead, it switches to interrupting the second and third categories of loads. No loads are interrupted during flat and valley price periods. The results of the objective functions after dispatch are shown in Table 2. The total electricity cost for the substation area is 19,817 RMB, with 18,330 RMB spent on purchasing electricity from the distribution network, representing the primary expense. The curtailment penalty is 0 RMB, indicating that all the PV energy is fully utilized. The energy comfort index is 0, meaning that after dispatch, the indoor temperature in the substation area is as close as possible to the most comfortable level, which is the average of the maximum and minimum indoor temperature standards.

Table 2. Objective function results.

Name	$C^{\mathrm{DW}}$/¥	$C^{\mathrm{DR}}$/¥	$C^{\mathrm{PV}}$/¥	$C^{\mathrm{YW}}$/¥	$C_{\mathrm{econ}}$/¥	$C_{\mathrm{temp}}$
Calculated value	18,330	1391	0	96	19,817	0

5.3. Analysis of Algorithm Solving Performance

Figure 15 shows the convergence error values corresponding to the two convergence conditions of Equation (45).

Figure 15. Convergence error values.

As can be seen from Figure 15, the error value 2 satisfies the convergence accuracy ${\xi }_2$ after three iterations, and the error value 1 satisfies the convergence accuracy ${\xi }_1$ after 7 iterations. The algorithm exhibits fast convergence speed and high computational accuracy. The calculation time for the collaborative optimization scheduling of the distribution transformer area is around 40 s, indicating high computational efficiency that can meet the operational requirements of the system.

6. Conclusions

This paper presents a multi-objective dispatch optimization model for an active distribution network involving multiple entities. The upper layer focuses on the active distribution network, while the lower layer targets the distribution substation area. The power exchanged between these two layers serves as the linking power for iterative, coordinated optimization, ultimately leading to convergence. This study aggregates the power of various flexible resources within the substation area, enabling unified control of each type of load and collective participation in demand response. This approach ensures efficient and unified multi-objective dispatch, balancing the interests of both the substation management center and the users while maintaining operational flexibility. Simulation results demonstrate that the two-layer optimization model effectively addresses the coordinated optimization of the active distribution network and the substation area, improving PV utilization and enhancing user-side dispatch flexibility. The limitation of this paper lies in the fact that the proposed method is only applicable to a certain scale of distribution systems. For future research, we plan to extend this method to larger-scale distribution systems to further validate its engineering application value.