Optimal Configuration Model for Flexible Interconnected Distribution Transformer Areas Based on Load Aggregation

Abstract

The large-scale integration of new power loads, such as electric vehicles and energy storage devices, has led to challenges including insufficient regulation capacity and low resource coordination efficiency in low-voltage distribution transformer areas. To address these issues, this paper proposes an optimal configuration model for flexible interconnected distribution transformer areas based on load aggregation. First, a flexible interconnection architecture is constructed using multi-port power electronic conversion devices, enabling mutual power support and voltage stabilization between adjacent areas. Second, a load aggregator scheduling model is established to quantitatively assess the dispatchable potential of electric vehicle charging loads. On this basis, a multi-objective optimization configuration model is formulated with the objectives of minimizing the comprehensive cost of the system and minimizing the average peak-valley difference of substation transformer loads. Case study results demonstrate that the proposed model significantly improves both economic efficiency and operational reliability. Compared to the traditional independent operation mode, the coordinated optimization scheme reduces the comprehensive system cost by 29.6% and narrows the average load peak-valley difference by 50.8%. These findings verify the synergistic effectiveness of flexible interconnection and load aggregation technologies in enhancing equipment utilization, reducing distribution losses, and improving power supply resilience.

1. Introduction

With the sustained development of the socio-economy and accelerated transformation of the energy structure, power load profiles exhibit trends toward diversification and complexity. The large-scale integration of electric vehicle (EV) charging loads and the widespread deployment of energy storage devices pose severe challenges to the operational control and safety-carrying capacity of low-voltage distribution transformer areas [1,2]. As a critical link for direct interaction between the power system and end-users, the operational state of low-voltage distribution transformer areas directly determines power supply quality and reliability levels. However, traditional distribution transformer area architectures characterized by rigid interconnection and unidirectional power supply reveal deficiencies such as insufficient regulation capacity, poor power supply flexibility, and low resource coordination efficiency when confronted with highly volatile and strongly stochastic new loads. These limitations make it difficult to meet users’ demands for high-quality power supply [3,4].

In response to these challenges, various advanced management and coordination strategies have been proposed to enhance the operational performance of distribution networks. Among them, load aggregation has emerged as a promising approach. This technology aggregates dispersed, heterogeneous flexible loads into controllable virtual energy units through advanced metering and communication means, enabling active smoothing of load curves and centralized regulation of demand response [5]. It demonstrates significant advantages in enhancing system regulation margins, reducing peak-valley differences, and strengthening source-load interaction capabilities, thereby laying a technical foundation for refined management and control of distribution transformer areas [6]. Previous studies, such as those by Saleh et al. [5] and Jangid et al. [6], have demonstrated the potential of load aggregation in improving grid flexibility and efficiency.

Furthermore, the establishment of flexible interconnection in low-voltage distribution transformer areas has been recognized as a key pathway to break down regional resource barriers. By employing flexible interconnection devices such as AC/DC converters and soft open points (SOPs), power mutual support and voltage support between adjacent areas can be achieved. This effectively dismantles the traditional islanded operational mode of distribution transformer areas and substantially enhances the local consumption capacity of distributed energy resources (DERs) as well as power supply resilience [7,8]. Research by Xv et al. [1] and Tang et al. [7] has explored the implementation and benefits of flexible interconnection architectures, highlighting their role in improving system reliability and resource utilization.

Despite these advancements, existing research predominantly focuses on single-technology applications, either load aggregation or flexible interconnection, lacking systematic exploration of collaborative optimization configuration methods that integrate both technologies. For instance, while studies like [9] investigate flexible load aggregation modeling and [10] examine flexible interconnection key technologies, few address the synergistic potential of their combined application. This gap is significant because, on one hand, the flexibility resources unlocked by load aggregation rely on flexible interconnection channels for cross-area optimal allocation [9]. On the other hand, maximizing the effectiveness of flexible interconnection urgently requires the coordinated support of aggregated resources [10]. Their integration could significantly improve the economy, reliability, and adaptability of distribution transformer areas under complex operational scenarios.

Therefore, this paper proposes an optimal configuration model for flexible interconnected distribution transformer areas based on load aggregation. The expected contributions of this research are summarized as follows:

(1)

It establishes a quantifiable dispatch potential model for EV charging loads under the management of a load aggregator, accurately capturing the adjustable capacity of decentralized flexible resources.

(2)

It develops a multi-objective optimal configuration model that simultaneously minimizes the comprehensive cost of the system and the average peak-valley difference of transformer loads, addressing both economic and operational reliability concerns.

(3)

It demonstrates, through comparative case studies, the significant synergistic benefits achieved by the deep integration of flexible interconnection and load aggregation technologies, including improved equipment utilization, reduced distribution losses, and enhanced power supply resilience and economy.

2. Flexible Interconnection System for Distribution Transformer Areas

Aiming to address the problems of insufficient regulation capacity and low resource coordination efficiency in traditional low-voltage distribution transformer areas operating under an islanded mode when confronted with highly volatile and strongly stochastic new loads, constructing an efficient and flexible interconnected architecture is key to achieving resource optimization and enhancing resilience [11]. The structure of the flexible interconnection system for distribution transformer areas proposed in this paper is shown in Figure 1.

This structure centers around a multi-port power electronic conversion device. Through AC/DC converters, the AC busbars of adjacent substation areas are connected to a common DC busbar system. Necessary DC/DC conversion interfaces are integrated onto the DC busbar to connect local DC source-load resources such as energy storage or to provide additional DC load access points. Furthermore, adjacent substation areas can be flexibly interconnected via substation area DC tie lines, enabling effective regulation of mutual power support between areas and facilitating efficient interaction among sources, loads, and storage across different substation areas. This flexible interconnected architecture, characterized by rich control degrees of freedom, creates indispensable physical conditions for the coordinated scheduling and optimal configuration of load aggregation resources over a broader scope. It enables the integration and mutual support of flexibility resources across substation areas, thereby supporting the maximization of the overall operational efficiency of the interconnected system.

3. Analysis of Electric Vehicle Charging Load Dispatch Potential

This section aims to quantitatively evaluate the dispatch potential of EV charging loads managed by a load aggregator (LA). It first elaborates on the role of the LA as a key entity integrating dispersed flexible loads and its value in optimizing charging schedules and smoothing load peaks. Subsequently, a refined dispatch potential model is constructed based on typical EV charging/discharging characteristics and user travel patterns. Vehicle states are categorized, with state of charge (SOC) thresholds introduced to ensure user travel demands are met. Finally, by aggregating the individual potential of all EVs within the region, a formula for the total dispatch potential characterizing the LA’s centralized control capability is derived, laying the foundation for subsequent optimal configuration.

3.1. Definition and Characteristics of Load Aggregator

As a crucial participant in the new electricity market, LA refers to a specialized entity that effectively integrates geographically dispersed, heterogeneous small-scale flexible loads into a virtual energy unit with observable, measurable, and controllable characteristics through advanced metering infrastructure, communication technology, and intelligent control strategies [12]. Its core value lies in overcoming the inherent fragmented distribution characteristics and rigid operational constraints of traditional load resources, transforming massive dispersed flexibility resources on the user side into system-level regulation capabilities characterized by scale and dispatchable responsiveness.

In the context of EV charging load dispatch, the involvement of LAs demonstrates significant advantages. By centrally managing the charging behavior of large-scale EV clusters, it can effectively smooth load peaks caused by uncontrolled charging, mitigating the overload risk of transformers [13]. Specifically, based on the overall load curve of the distribution transformer area, real-time electricity price signals, or system security constraints, the LA can intelligently adjust the charging schedule and power of EVs within its aggregation. For instance, through price-based demand response or direct load control commands, the LA can shift a portion of the charging load from peak periods to off-peak hours, or even modulate charging rates in real time, thereby achieving peak shaving and valley filling. This not only directly reduces the peak-valley difference of the area’s load curve, optimizes transformer capacity utilization, and reduces equipment capacity expansion needs and network losses caused by excessive peak loads but also enhances the alignment between load demand and renewable generation patterns. More significantly, the flexibility resources unlocked by the aggregator provide valuable regulation margins for the distribution network. When combined with a flexible interconnection system, the LA can further transcend the physical boundaries of a single distribution transformer area, coordinating the dispatch of EV charging loads across a wider region, thereby significantly enhancing the power supply reliability and resilience of the entire interconnected system.

3.2. Load Aggregator Charging Load Dispatch Potential Model

Currently, most charging piles adopt a constant-current constant-voltage (CC-CV) two-stage charging mode. That is, the constant-current charging mode is used in the initial charging stage. When the voltage rises to the maximum voltage of the lithium battery, it switches to the constant-voltage charging mode. During constant-voltage charging, the current gradually decreases until it reaches zero, at which point charging is considered complete [14]. During this process, the charging/discharging equations for the electric vehicle are as follows:

$$\left\{\begin{array}{l}U(t)=E_0+{\displaystyle \frac{KCi}{0.1\mathrm{C}+h(t)}}-{\displaystyle \frac{KCh(t)}{\mathrm{C}-h(t)}}+{\mathrm{Ae}}^{-\mathrm{Bh}(t)}+Ri\\ h(t)=[1-\mathrm{SOC}(t_0)]\mathrm{C}-{\displaystyle {\int }_{t_0}^{t}idt}\end{array}\right.$$

(1)

In (1), $U(t)$ is the voltage during charging; $E_0$ is the rated voltage; $K$ is the polarization voltage of the battery; $C$ is the rated capacity of the battery; $i$ is the battery current; $A$ and $B$ are constants for the battery charging/discharging process; $R$ is the battery internal resistance; $\mathrm{SOC}(t)$ represents the battery’s SOC at time $t$.

According to the charging/discharging characteristics of EVs during different time periods within a day, their states can be categorized into off-grid state during the morning commuting period, grid-connected charging state in the morning period, off-grid state during the evening commuting period, and grid-connected charging state in the evening period. Whether an EV user’s trip is affected by the aggregator’s dispatch is closely related to its SOC. Therefore, the SOC threshold for each EV trip is defined as:

$$SO{C}_{th}={\displaystyle \frac{l}{\mathrm{L}}}$$

(2)

In (2), $SO{C}_{th}$ is the SOC threshold for the EV trip; $l$ is the distance of the trip; $\mathrm{L}$ is the EV’s maximum design mileage. The SOC at the start of each trip must not be lower than this travel state threshold.

The SOC of the EV under different charging/discharging states in various periods is:

$$SOC(t)=\left\{\begin{array}{ll}SOC(t_{sl})-SO{C}_{th}\cdot {\displaystyle \frac{\mathsf{\Delta }t}{T}}& t_{sl}\le t\le t_{fa}\\ SOC(t_{fa})+{\displaystyle \frac{\alpha }{C}}{\displaystyle {\int }_{t_{fa}}^{t_{fl}}U(t)\mathrm{I}dt}& t_{fa}<t\le t_{fl}\\ SOC(t_{fl})-SO{C}_{th}\cdot {\displaystyle \frac{\mathsf{\Delta }t}{T}}& t_{fl}<t\le t_{sa}\\ SOC(t_{sa})+{\displaystyle \frac{\alpha }{C}}{\displaystyle {\int }_{t_{sa}}^{t_{sl}}U(t)\mathrm{I}dt}& t_{sa}<t\le t_{sl}\end{array}\right.$$

(3)

In (3), $SOC(t)$ is the SOC at time $t$; $t_{fa}$ is the grid-connection time in the morning period; $t_{fl}$ is the off-grid time in the evening period; $t_{sa}$ is the grid-connection time in the evening period; $t_{sl}$ is the off-grid time the next morning; $\alpha$ is the charging efficiency.

The dispatch potential modeling for EVs under different periods and charging/discharging states is detailed below.

(a) If the EV starts responding to discharge dispatch immediately after connecting to the grid, and begins charging at the end of the dispatch period until it disconnects. Define the SOC at the off-grid time as $SO{C}_l$, expressed as:

$$SO{C}_l=SOC(t_a)-{\displaystyle \frac{\alpha }{C}}{\displaystyle {\int }_{t_a}^{t_{end}}U(t)\mathrm{Id}t}+{\displaystyle \frac{\alpha }{C}}{\displaystyle {\int }_{t_{end}}^{t_l}U(t)\mathrm{I}dt}$$

(4)

In (4), $t_a$ is the start time of the EV’s response to charging/discharging dispatch; $t_{end}$ is the end time of the EV’s response to charging/discharging dispatch. When $SO{C}_l\ge SO{C}_{th}$, the EV at that time is considered to have discharge dispatch potential.

(b) If the EV can still complete the charging dispatch task assigned by the aggregator after connecting to the grid, it is considered to have charging dispatch potential at that time. Define the SOC after charging dispatch as $SO{C}_{end}$, expressed as:

$$SO{C}_{end}=SOC(t_a)+{\displaystyle \frac{\alpha }{C}}{\displaystyle {\int }_{t_a}^{t_{end}}U(t)\mathrm{I}dt}$$

(5)

(c) The dispatch potential in the off-grid state includes the dispatch potential during the morning commute and the dispatch potential during the evening commute. Its expression is:

$$\left\{\begin{array}{c}{P}_{\mathrm{disch}}=0\\ P_{\mathrm{ch}}=0\end{array}\right.$$

(6)

In (6), $P_{\mathrm{ch}}$ is the charging dispatch potential in the off-grid state, and $P_{\mathrm{disch}}$ is the discharging dispatch potential in the off-grid state.

(d) The dispatch potential in the grid-connected state includes the dispatch potential during the daytime grid-connected period and the evening grid-connected period. An EV is considered to have dispatch potential if its SOC meets the minimum state threshold requirement; otherwise, it does not. The expressions are as follows:

$$P_{\mathrm{disch}}=\left\{\begin{array}{ll}-U(t)\mathrm{I}& SO{C}_l\ge SO{C}_{th}\\ 0& SO{C}_l\le SO{C}_{th}\end{array}\right.$$

(7)

$$P_{\mathrm{ch}}=\left\{\begin{array}{cc}U(t)\mathrm{I}\hfill & SO{C}_l\le 1\\ 0\hfill & SO{C}_l\ge 1\end{array}\right.$$

(8)

The total dispatch potential of all charging loads within the aggregator’s aggregation region is:

$$\left\{\begin{array}{l}{P}_{\mathrm{ev}\_\mathrm{ch}}(t)={\displaystyle \sum _{i=1}^M}{P}_{\mathrm{ch}}(t)\\ P_{{\mathrm{ev}}_{-}\mathrm{disch}}(t)={\displaystyle \sum _{i=1}^M}{P}_{\mathrm{disch}}(t)\end{array}\right.$$

(9)

In (9), $P_{\mathrm{ev}\_\mathrm{ch}}$ is the total charging dispatch potential of the region at time $t$; $P_{{\mathrm{ev}}_{-}\mathrm{disch}}$ is the total discharging dispatch potential of the region at time $t$; $M$ is the total number of EVs within the aggregation region.

4. Optimal Configuration Model for Flexible Interconnected Distribution Transformer Areas Based on Load Aggregation

This section establishes an optimal configuration model for flexible interconnected distribution transformer areas based on load aggregation. The model adopts a dual-objective optimization framework: the economic objective minimizes the system’s comprehensive cost; the reliability objective minimizes the average peak-to-valley difference of the transformer loads within all areas of the interconnected system. The proposed model is built upon several key hypotheses to ensure both practicality and solvability:

(a)

The system operates under strict physical boundaries including AC/DC power balance within each area;

(b)

Key equipment such as transformers, converters, and energy storage units are subject to capacity and operational status limits;

(c)

The load aggregator’s scheduling capability follows the dispatch potential model established in Section 3 and is constrained within its feasible region;

(d)

Given the short distances between interconnected areas, power transmission losses of tie-lines are neglected;

(e)

The multi-objective optimization model considers both economic and reliability objectives, and the trade-off is resolved objectively via Pareto optimality and knee-point identification without subjective weight assignment.

These assumptions provide a realistic and structured foundation for evaluating the synergistic benefits of flexible interconnection and load aggregation technologies.

4.1. Objective Function

This paper adopts a multi-objective function that considers both economic efficiency and system reliability, expressed as follows:

$$\min f=(f_1,f_2)$$

(10)

In (10), $f_1$ is the system’s comprehensive cost; $f_2$ is the average peak-to-valley difference of the transformer loads across all areas within the interconnected system.

From an economic perspective, the objective is to minimize the comprehensive cost of the interconnected system, composed of fixed-asset investment cost, equipment operating cost, and load aggregation resource scheduling cost, expressed as:

$$f_1=f_{\mathrm{int}}+f_{\mathrm{ope}}+f_{agg}$$

(11)

In (11), $f_{\mathrm{int}}$ is the fixed-asset investment cost; $f_{\mathrm{ope}}$ is the equipment operating cost; $f_{agg}$ is the load aggregation resource scheduling cost.

The fixed-asset investment cost consists of the converter investment cost $f_{\mathrm{VSC}.\mathrm{int}}$ and the energy storage unit investment cost $f_{es.\mathrm{int}}$, where the energy storage unit investment cost includes the investment cost of the energy storage unit DC/DC interface and the investment cost of the energy storage battery, namely:

$$f_{\mathrm{int}}=\xi (f_{\mathrm{VSC}.\mathrm{int}}+f_{es.\mathrm{int}})$$

(12)

$$f_{\mathrm{VSC}.\mathrm{int}}=c_{\mathrm{VSC}.\mathrm{int}}\sum _{i=1}^n{P}_{i.\max }^{\mathrm{VSC}}$$

(13)

$$f_{es.\mathrm{int}}=c_{\mathrm{DC}/\mathrm{DC}.\mathrm{int}}\sum _{i=1}^n{P}_{i.\max }^{es}+c_{es.\mathrm{int}}\sum _{i=1}^n{E}_{i.\max }^{es}$$

(14)

In (12)–(14), $\xi$ is the equivalent annual value conversion coefficient; $c_{\mathrm{VSC}.\mathrm{int}}$ is the unit capacity investment cost of the converter; $c_{\mathrm{DC}/\mathrm{DC}.\mathrm{int}}$ is the unit capacity investment cost of the energy storage unit DC/DC interface; $c_{es.\mathrm{int}}$ is the unit capacity investment cost of the energy storage battery; $n$ is the number of areas within the interconnected system; $P_{i.\max }^{\mathrm{VSC}}$ is the rated capacity of the converter; $P_{i.\max }^{es}$ is the rated capacity of the energy storage unit DC/DC interface; $E_{i.\max }^{es}$ is the energy storage battery capacity.

The rated capacity of the converter is determined by the maximum anticipated power exchange required for both local AC/DC conversion and cross-area flexible power support. Its value is optimized to handle peak loads while ensuring operation within the safe operating area (SOA). The SOA is defined by the manufacturer’s specifications to prevent thermal overstress and electrical overloading of semiconductor devices, ensuring operational reliability and longevity.

The equipment operating cost consists of the transformer operating cost $f_{\mathrm{tr}.ope}$, the converter operating cost $f_{VSC.ope}$, and the energy storage unit operating cost $f_{es.ope}$. In this paper, the interconnected areas are in close proximity with short tie-line lengths; therefore, the power transmission losses of the tie-lines are neglected. The expression for the equipment operating cost is as follows:

$$f_{\mathrm{ope}}=f_{\mathrm{tr}.\mathrm{ope}}+f_{\mathrm{VSC}.\mathrm{ope}}+f_{es.\mathrm{ope}}$$

(15)

The transformer operating cost is composed of transformer energy losses, including transformer no-load loss (iron loss) $los{s}_{Fe}$ and load loss (copper loss) $los{s}_{Cu}$, expressed as:

$$f_{\mathrm{tr}.\mathrm{ope}}=c_{\cos \mathrm{t}}(los{s}_{\mathrm{Fe}}+los{s}_{\mathrm{Cu}})$$

(16)

$$los{s}_{\mathrm{Fe}}=\sum _{t=1}^T\sum _{i=1}^n{P}_{\mathrm{Fe}.i}\mathsf{\Delta }t=\sum _{t=1}^T\sum _{i=1}^n(a{S}_{\mathrm{N}.i}^{\mathrm{tr}}+b)\mathsf{\Delta }t$$

(17)

$$los{s}_{\mathrm{Cu}}=\sum _{t=1}^T\sum _{i=1}^n{P}_{\mathrm{Cu}.i}{({\beta }_{i.t}^{\mathrm{tr}})}^2\mathsf{\Delta }t=\sum _{t=1}^T\sum _{i=1}^n(c{S}_{\mathrm{N}.i}^{\mathrm{tr}}+d){({\beta }_{i.t}^{\mathrm{tr}})}^2\mathsf{\Delta }t$$

(18)

$${\beta }_{i.t}^{\mathrm{tr}}={\displaystyle \frac{P_{i.t}^{\mathrm{tr}}}{S_{\mathrm{N}.i}^{\mathrm{tr}}\cos \phi }}$$

(19)

In (16)–(19), $c_{\cos \mathrm{t}}$ is the electricity price; $T$ is the total number of time intervals; $\mathrm{\Delta }t$ is the duration of each time interval; $P_{\mathrm{Fe}.i}$ is the no-load loss of the i-th transformer; $P_{Cu.i}$ is the rated load loss of the i-th transformer; ${\beta }_{i.t}^{\mathrm{tr}}$ is the load rate of the i-th transformer at time $t$. Fitting analysis of loss parameters for different capacity 10 kV transformers in practical engineering shows that the rated load loss and no-load loss of a transformer are typically linearly related to its capacity. $a$, $b$, $c$, $d$ are linear fitting parameters. $P_{i.t}^{\mathrm{tr}}$ is the active power flowing through the i-th transformer at time $t$, defined as positive from the high-voltage side to the low-voltage side. $\cos \phi$ is the power factor.

The converter operating cost depends on the converter power transmission loss, expressed as:

$$f_{\mathrm{VSC}.\mathrm{ope}}=c_{\cos \mathrm{t}}\sum _{t=1}^T\sum _{i=1}^n{P}_{i.t}^{\mathrm{VSC}}(1-{\eta }_i^{\mathrm{VSC}})\mathsf{\Delta }t$$

(20)

In (20), $P_{i.t}^{\mathrm{VSC}}$ is the active power on the AC side of the converter in the i-th area at time $t$, defined as positive from the AC side to the DC side; ${\eta }_i^{\mathrm{VSC}}$ is the converter efficiency.

The energy storage unit operating cost is composed of the DC/DC interface transmission loss, expressed as:

$$f_{es.\mathrm{ope}}=c_{\cos \mathrm{t}}\sum _{t=1}^T\sum _{i=1}^n(P_{i.t}^{\mathrm{ch}}+P_{i.t}^{\mathrm{dis}})(1-{\eta }_i^{es})\mathsf{\Delta }t$$

(21)

In (21), $P_{i.t}^{\mathrm{ch}}$ is the charging power of the energy storage unit in the i-th area at time $t$; $P_{i.t}^{\mathrm{dis}}$ is the discharging power of the energy storage unit in the i-th area at time $t$; ${\eta }_i^{es}$ is the efficiency of the energy storage unit DC/DC interface.

The load aggregation resource scheduling cost depends on the load curtailment amount managed by the aggregator, expressed as:

$$f_{agg}=c_{agg}\sum _{t=1}^T\sum _{i=1}^n{P}_{i.t}^{cut}$$

(22)

In (22), $c_{agg}$ is the compensation price per unit of curtailed power paid by the grid to the aggregator; $P_{i.t}^{cut}$ is the load curtailment amount in the i-th area at time $t$.

From a system reliability perspective, the objective is to minimize the average peak-to-valley difference of the transformer loads across all areas within the interconnected system, expressed as:

$$f_2={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{P}_i^{diff}}}{n}}$$

(23)

$$P_i^{diff}=P_i^{\max }-P_i^{\min }$$

(24)

In (23)–(24), $P_i^{diff}$ is the peak-to-valley difference of the transformer load in the i-th area; $P_i^{\max }$ is the maximum transformer load in the i-th area; $P_i^{\min }$ is the minimum transformer load in the i-th area.

4.2. Constraints

4.2.1. Equality Constraints

The active power balance constraint on the AC side of the distribution transformer area is given by:

$$P_{i.t}^{\mathrm{tr}}=P_{i.t}^{\mathrm{VSC}}+P_{i.t}^{\mathrm{AC}}$$

(25)

In (25), $P_{i.t}^{\mathrm{AC}}$ denotes the AC load in the i-th distribution transformer area at time $t$.

The active power balance constraint on the DC side of the distribution transformer area is expressed as:

$$P_{i.t}^{\mathrm{VSC}}{\eta }_i^{\mathrm{VSC}}=P_{i.t}^{\mathrm{DC}}+P_{i.t}^{\mathrm{ch}}-P_{i.t}^{\mathrm{dis}}+P_{i.t}^{\mathrm{lin}}$$

(26)

$$P_{i.t}^{\mathrm{DC}}=P_{i.t}^{\mathrm{ev}}-P_{i.t}^{cut}$$

(27)

In (26)–(27), $P_{i.t}^{\mathrm{DC}}$ is the DC load; $P_{i.t}^{\mathrm{lin}}$ is the power transmitted through the tie-line; $P_{i.t}^{\mathrm{ev}}$ is the EV charging load in the i-th distribution transformer area at time $t$.

The energy balance constraint for energy storage is:

$${\eta }_i^{es}\sum _{t=1}^{24}{P}_{i.t}^{\mathrm{ch}}\mathsf{\Delta }t-{\displaystyle \frac{1}{{\eta }_i^{es}}}\sum _{t=1}^{24}{P}_{i.t}^{\mathrm{dis}}\mathsf{\Delta }t=0$$

(28)

4.2.2. Inequality Constraints

The transformer power constraint is:

$$-{\beta }_{\max }^{\mathrm{tr}}\le {\beta }_{i.t}^{\mathrm{tr}}\le {\beta }_{\max }^{\mathrm{tr}}$$

(29)

In (29), ${\beta }_{\max }^{\mathrm{tr}}$ is the upper limit of the transformer load rate.

The converter power constraint is:

$$-P_{i.\max }^{\mathrm{VSC}}\le P_{i.t}^{\mathrm{VSC}}\le P_{i.\max }^{\mathrm{VSC}}$$

(30)

The power constraints for the DC/DC interface of the energy storage unit are:

$$0\le P_{i.t}^{\mathrm{ch}}\le U_i^{es}(t)P_{i.\max }^{es}$$

(31)

$$0\le P_{i.t}^{\mathrm{dis}}\le (1-U_i^{es}(t))P_{i.\max }^{es}$$

(32)

In (31)–(32), $U_i^{es}(t)$ is the operational status variable of the energy storage unit, taking a value of 1 during charging and 0 during discharging.

The state of charge constraints for the energy storage battery are:

$$E_{i.\min }\le E_i(t)\le E_{i.\max }$$

(33)

$$E_i(t)={\displaystyle \frac{1}{E_{i\max }^{\mathrm{bat}}}}(E(0)+{\eta }_i^{\mathrm{bat}}\sum _{t=1}^T{P}_{i.t}^{\mathrm{ch}}\mathsf{\Delta }t-{\displaystyle \frac{1}{{\eta }_i^{\mathrm{bat}}}}\sum _{t=1}^T{P}_{i.t}^{\mathrm{dis}}\mathsf{\Delta }t)$$

(34)

In (33)–(34), $E_i(t)$ is the SOC of the energy storage battery in the i-th distribution transformer area at time $t$; $E_{i.\max }$ and $E_{i.\min }$ are the upper and lower SOC limits, respectively; $E(0)$ is the initial energy of the battery.

The load aggregator dispatch constraint is:

$$0\le P_{i,t}^{cut}\le P_{\mathrm{ev}\_\mathrm{disch}}$$

(35)

4.3. Solution Methods for Multi-Objective Optimization

The optimization configuration model constructed in this paper comprises two conflicting objectives: minimizing the system’s comprehensive cost and minimizing the average peak-valley difference of transformer loads across distribution transformer areas. A single solution cannot simultaneously satisfy both objectives optimally. Therefore, a multi-objective optimization method is employed to solve for the Pareto front, which represents the set of all non-dominated solutions—that is, solutions where improvement in any one objective necessarily leads to deterioration in at least one other objective.

To identify the most engineering-practical solution from the Pareto front, this paper focuses on locating the key knee point. This knee point exhibits a distinctive feature in the trade-off relationship between the objective functions: within its neighborhood, a slight sacrifice in one objective yields significant improvements in the other(s), and vice versa. This characteristic indicates that the knee point typically represents the most resource-efficient compromise solution.

This paper adopts a curvature-based method to identify the knee point. The core idea is to find the point on the Pareto front with the maximum local curvature, reflecting the region where the trade-off relationship changes most drastically. Specifically, the curvature variation of line segments formed by adjacent solutions on the Pareto front is computed numerically. The curvature $\kappa$ at each point on the Pareto front is calculated as follows:

$$\kappa ={\displaystyle \frac{|f_1^{″}\cdot f_2^{\prime }-f_2^{″}\cdot f_1^{\prime }|}{{\left({(f_1^{\prime })}^2+{(f_2^{\prime })}^2\right)}^{3/2}}}$$

(36)

where $f_1^{\prime }$, $f_2^{\prime }$ and $f_1^{″}$, $f_2^{″}$ are the first and second derivatives of the objective functions along the Pareto front, respectively. The solution corresponding to the maximum curvature is selected as the knee point, representing the most balanced trade-off between economy and reliability.

Compared to methods that subjectively assign weights or priorities, this approach more objectively reveals the inherent optimal trade-off relationship between objectives, providing decision-makers with the most resource-efficient optimization configuration scheme. Subsequent case studies apply this method to determine specific configurations and validate its effectiveness.

4.4. Solution Process

Figure 2 illustrates the overall solution flowchart of the proposed optimal configuration methodology for flexible interconnected distribution transformer areas based on load aggregation.

The process is summarized in the following key steps:

(a)

System Construction and Parameter Input: Establish the flexible interconnection architecture and input all necessary parameters, including load curves, EV travel data, and equipment costs.

(b)

Load Aggregator Model: Quantify the dispatchable potential of EV charging loads using the model established in Section 3.

(c)

Multi-objective Optimization Model Formulation: Construct the optimization model with the dual objectives of minimizing comprehensive cost and minimizing the average load peak-valley difference, subject to the system constraints outlined in Section 4.2.

(d)

Pareto Front Solving: Employ a multi-objective optimization algorithm to solve the model and obtain the set of non-dominated solutions (the Pareto front).

(e)

Knee Point Identification: Apply the curvature-based method to identify the knee point on the Pareto front, representing the most balanced trade-off between economy and reliability.

(f)

Output Optimal Configuration: Determine and output the final optimal configuration scheme.

5. Case Study

5.1. Case Scenario and Parameter Settings

The optimization object is a distribution system composed of three distribution transformer areas. The topology and load composition of the interconnected system after flexible interconnection are shown in Figure 1. The typical daily AC load power curves for each area are illustrated in Figure 3. The load curves exhibit similar trends with distinct morning and evening peaks, which are characteristic of residential and commercial electricity consumption patterns.

It is assumed that the departure time during the morning commuting period, grid-connection time during the daytime period, departure time during the evening commuting period, and grid-connection time during the evening period for electric vehicles (EVs) all follow normal distributions. The EV travel parameters, listed in

Table 1. EV Travel Parameters.

Parameter	Distribution
Travel distance (km)	$\ln ~N(5.6,{0.8}^2)$
Travel threshold	$N(0.4,{0.1}^2)$
Departure time (morning)	$N(8,{0.8}^2)$
Charging time (morning)	$N(9,{0.8}^2)$
Departure time (evening)	$N(18,{0.8}^2)$
Charging time (evening)	$N(19,{0.8}^2)$
Initial travel SOC	$N(0.5,{0.1}^2)$

, are based on typical user behavior patterns and charging characteristics, supported by empirical data [15].

Each distribution transformer area is assumed to contain 50 EVs. The DC load power curves for each area, primarily composed of EV charging loads, are shown in Figure 4. These curves are generated based on the Monte Carlo simulation of EV charging behaviors, considering user travel patterns and charging characteristics. The observed peaks in the DC load during evening hours align with typical EV charging behaviors after evening commutes.

In this case, the equivalent annual value conversion coefficient is set to 0.1, the electricity price is 0.636 CNY/kWh, and the power factor of the distribution transformer areas is 0.95. Based on [16] and practical engineering considerations, the parameters of distribution equipment are configured as shown in

Table 2. Parameters of Distribution Equipment.

Equipment Type	Parameter	Value
Transformer	Linear fitting parameter $a$	0.0012
	Linear fitting parameter $b$	0.077
	Linear fitting parameter $c$	0.0091
	Linear fitting parameter $d$	0.61
	Maximum load factor (%)	80
Converter	Unit capacity investment cost (CNY/kW)	2000
	Efficiency (%)	0.98
Energy storage unit	Unit capacity investment cost of DC/DC interface (CNY/kW)	500
	Unit capacity investment cost of battery (CNY/kWh)	500
	Efficiency of DC/DC interface (%)	99
	Upper limit of battery SOC (%)	90
	Lower limit of battery SOC (%)	10

.

To comparatively analyze the capabilities of the AC/DC hybrid supply flexible interconnected distribution system and the load aggregation strategy in reducing system economic costs and distribution losses, and to clearly demonstrate the advances of this research compared to previous studies, four scenarios are established as follows. Among them, Scenario 2 and Scenario 3 are configured to correspond to the methods presented in [16,17] respectively, while Scenario 4 represents the novel approach proposed in this paper:

• Scenario 1: Three areas operate independently, each achieving AC/DC hybrid supply via area converters, without load aggregation for EV charging loads.
• Scenario 2 (Corresponding to the method in [17]): Three areas operate independently, each achieving AC/DC hybrid supply via area converters, with load aggregation for EV charging loads.
• Scenario 3 (Corresponding to the method in [16]): Three areas operate in full interconnection, without load aggregation for EV charging loads.
• Scenario 4 (Proposed method): Three areas operate in full interconnection, with load aggregation for EV charging loads.

5.2. Comparative Analysis of Distribution Transformer Area Operation

To evaluate the impact of flexible interconnection and load aggregation strategies on distribution transformer area operation, this section compares the load rate variations and equipment losses of distribution transformers across areas under Scenarios 1, 2, and 3. Figure 5 illustrates the load rates of distribution transformers in each area under Scenarios 1–3. The analysis reveals that compared to independent area operation, both load aggregation and flexible interconnection significantly improve transformer operating conditions.

In Scenario 1, load curves in each area remain dominated by local load characteristics and exhibit pronounced peak-valley phenomena. Transitioning to Scenario 2, centralized scheduling of EV charging loads by the load aggregator significantly flattens evening charging peaks. This effect is clearly demonstrated in Area 3, where the high evening load observed in Scenario 1 is effectively mitigated with a notable reduction in peak load rate. The successful shift of partial charging loads from peak to off-peak periods achieves localized peak shaving and valley filling. Despite this improvement, Scenario 2 shows limited effectiveness in reducing AC-load-dominated peaks due to the absence of inter-area power support, highlighting the inherent optimization constraints when relying solely on load aggregation.

Scenario 3 introduces flexible interconnection, which breaks the traditional islanded operation mode by enabling flexible surplus power transfer between areas. This configuration yields smoother load curves and significantly reduced peak-valley differences compared to Scenario 1. It is noteworthy that without load aggregation, the local evening peaks induced by EV charging—particularly evident in Area 3—are alleviated through power interchange, though to a lesser extent than achieved under Scenario 2’s direct charging load control. This comparative outcome underscores the unique advantages of load aggregation in fine-grained flexible load management while demonstrating its functional complementarity with flexible interconnection technologies.

Table 3. Annual Energy Losses of Distribution Equipment Under Scenarios 1–3.

	Scenario 1	Scenario 2	Scenario 3
Transformerlosses(MWh)	40.04	39.21	35.39
Converter losses (MWh)	17.50	17.51	10.26
Storage unit DC/DC interface losses (MWh)	2.40	2.91	1.80

quantifies the annual energy losses of distribution equipment under Scenarios 1–3, where interconnected operation in Scenario 3 reduces total transformer losses by 11.6%, converter losses by 41.4%, and energy storage interface losses by 25.0% compared to Scenario 1. This reduction stems from two key mechanisms. Flexible interconnection channels overcome islanded constraints by enabling peak-load areas to draw support from valley-load areas, thereby reducing transformer underloading and overloading. Simultaneously, cross-area power optimization minimizes frequent adjustments of converters and storage units, enhancing overall equipment efficiency. Conversely, while load aggregation in Scenario 2 reduces transformer losses, the increased charging/discharging frequency of local storage raises interface losses by 21.3%, exposing the inherent limitations of single-technology applications.

5.3. Comparative Analysis of Distribution System Optimization Configuration Results

To comprehensively evaluate the synergistic benefits of flexible interconnection and load aggregation technologies,

Table 4. Optimal Configuration Results of Distribution Equipment Capacities Under Different Scenarios.

		Scenario 1	Scenario 2	Scenario 3	Scenario 4
Converter capacity (kW)	Area 1	44.22	56.30	65.71	65.71
	Area 2	54.59	72.85	55.09	46.78
	Area 3	50.15	66.63	52.19	46.93
Energy storage battery capacity (kWh)	Area 1	109.66	150.33	0	0
	Area 2	136.68	179.48	240.29	35.06
	Area 3	139.05	169.01	53.87	38.90

compares the optimal configuration capacities of converters and energy storage batteries under different scenarios. In scenarios with independent distribution transformer area operation, each area requires large-capacity local energy storage to mitigate load fluctuations. After introducing flexible interconnection, cross-area power mutual support significantly reduces dependence on local storage: In Scenario 3 (without load aggregation), the energy storage battery capacity in Area 1 drops to 0 kWh through interconnected scheduling. Further integrating load aggregation in Scenario 4 achieves efficient consolidation of global energy storage resources, with battery capacities in all areas reduced to below 40 kWh—a reduction exceeding 80% compared to independent operation scenarios. Converter configurations also exhibit coordinated optimization: In Scenario 4, the converter capacities in Areas 2 and 3 decrease significantly compared to Scenario 2, indicating that flexible interconnection channels enhance the utilization efficiency of power transmission equipment.

5.4. Comparative Analysis of Comprehensive Costs and Load Peak-Valley Differences in Distribution System

To comprehensively evaluate the impact of different operational strategies on distribution transformer areas, Figure 6 presents the Pareto fronts obtained through multi-objective optimization for the four scenarios. Using a curvature-based knee point identification method, the optimization scheme with the highest resource allocation efficiency is selected from the Pareto front. Key performance indicators for each scenario are then extracted based on this scheme for comparative analysis, with results summarized in

Table 5. Comprehensive Costs and Average Load Peak-valley Differences of the Distribution System under Different Scenarios.

	Scenario 1	Scenario 2	Scenario 3	Scenario 4
Fixed-asset investment cost (10,000 CNY)	5.33	6.98	5.25	3.70
Equipment operating cost (10,000 CNY)	3.81	3.79	3.02	2.61
Load aggregation resource scheduling cost (10,000 CNY)	0	0.12	0	0.12
System comprehensive cost (10,000 CNY)	9.14	10.89	8.27	6.43
Average load peak-valley difference (kW)	107.14	65.56	63.96	52.77

.

Analysis of the data in Table 5 reveals significant differences in system comprehensive cost and average load peak-valley difference across the four scenarios. Under the independent operation mode (Scenarios 1–2), while introducing load aggregation significantly reduces the average load peak-valley difference by 38.8% compared to Scenario 1, the system comprehensive cost has a 19.1% rise over Scenario 1 due to energy storage capacity expansion and the addition of load aggregation scheduling costs. The independent application of flexible interconnection (Scenario 3) demonstrates stronger economic advantages, reducing the system comprehensive cost by 9.5% and the average load peak-valley difference by 40.3% compared to Scenario 1.

In Scenario 4, through the deep integration of flexible interconnection and load aggregation technologies, the system comprehensive cost drops to its minimum value. This cost advantage stems from two key mechanisms: Firstly, flexible interconnection enables cross-area power mutual support, significantly reducing the demand for local energy storage investment, driving the fixed-asset investment cost down to 3.70 × 10⁴ CNY—the lowest among all scenarios. Secondly, the refined scheduling of EV charging loads by the load aggregator, combined with the global power optimization allocation of the interconnected system, effectively reduces operating losses in transformers, converters, and other equipment, lowering the equipment operating cost to 2.61 × 10⁴ CNY. These results align well with the imposed performance objectives of minimizing comprehensive cost and flattening load profiles, demonstrating the effectiveness of the proposed coordinated strategy. Simultaneously, Scenario 4 achieves optimal operational stability, with the average load peak-valley difference further reduced to 52.77 kW. This represents reductions of 50.8%, 19.5%, and 17.5% compared to Scenarios 1, 2, and 3, respectively. The obtained metrics not only meet but exceed the anticipated synergistic benefits, validating that the integration of flexible interconnection and load aggregation effectively balances economic and reliability objectives.

6. Conclusions

This paper addresses the problems of insufficient regulation capacity and low resource coordination efficiency in low-voltage distribution transformer areas caused by the large-scale integration of new power loads. It proposes an optimal configuration model for flexible interconnected distribution transformer areas based on load aggregation. Through theoretical modeling and case study verification, the following conclusions are drawn:

(1)

Load aggregation technology significantly reduces the load peak-valley difference within distribution transformer areas by integrating dispersed electric vehicle charging loads into dispatchable resources. Flexible interconnection technology, on the other hand, enhances equipment utilization and power supply reliability through cross-area power mutual support. The deep integration of these two technologies demonstrates significant synergistic advantages in improving both system economy and operational stability.

(2)

The constructed multi-objective optimization model aims to minimize the system comprehensive cost and minimize the average peak-valley difference of transformer loads across distribution transformer areas. By employing Pareto front analysis and a knee point identification method, the model enables quantitative evaluation of resource allocation efficiency. Case study results demonstrate that under Scenario 4, the system comprehensive cost is reduced by 29.6% compared to Scenario 1, and the average load peak-valley difference is reduced by 50.8%. This verifies the effectiveness of the model in balancing economy and reliability.

It should be noted that this study has certain limitations. The model assumes ideal communication and control conditions, relies on simplified EV behavior distributions, and does not fully consider market price fluctuations. Additionally, the case study is based on a three-area system, and the scalability to larger networks warrants further validation.

Future research will therefore focus on addressing these limitations by developing more realistic models that incorporate communication delays and uncertainties in EV behavior, investigating the interaction between the proposed configuration strategy and dynamic electricity market mechanisms, and exploring fault reconfiguration strategies and scalability for larger-scale interconnected systems. These efforts are essential to strengthen the practicality and resilience of distribution transformer areas in supporting the ongoing energy transition.