Coordinated Dispatch Strategy of Flexible Resources in Distribution Networks for Temporary Loads

Abstract

Partial agricultural production loads exhibit significant temporality. The concentrated access of temporary loads can easily trigger operational challenges in distribution networks, such as heavy overload, terminal voltage violations, and increased network losses. To address these issues, this paper proposes a coordinated dispatch strategy for multiple flexible resources to cope with temporary loads. First, combining the operational characteristics of motor-pumped well loads, a refined model for motor-pumped well loads is constructed to fully exploit their regulation potential as flexible loads. Second, considering the supporting role of mobile energy storage systems (MESS) for heavy overload distribution networks, a spatiotemporal dispatch model for MESS is established. Then, aiming to minimize the total system operating cost, an economic dispatch model coordinating multiple flexible resources, including MESS, distributed generators (DG), and flexible loads, is developed. The original non-convex problem is transformed into a mixed-integer second-order cone programming problem using Second-Order Cone Relaxation (SOCR) method for efficient solution. Finally, the effectiveness of the proposed strategy is verified on an improved IEEE 33-bus system.

1. Introduction

Driven by the global trend towards clean energy and the comprehensive implementation of the rural revitalization strategy, the large-scale integration of new energy technologies, such as distributed photovoltaics and wind turbines, has accelerated the transformation of rural energy infrastructure into a cleaner and low-carbon paradigm [1]. As the terminal component of the power system, rural distribution networks bear the critical responsibility of supplying power for agricultural production, rural livelihoods, and township enterprises [2]. Among the various rural loads, agricultural production loads are particularly seasonal and task-oriented. Taking motor-pumped well irrigation as a representative example, these loads are highly concentrated within short windows during critical crop water demand periods. Moreover, agricultural water usage is characterized by strong temporal rigidity; operations such as wheat jointing or rice transplanting typically require continuous, full-day pumping during specific consecutive days of high temperature and low rainfall. Missing the optimal irrigation window due to power supply constraints can directly lead to reduced crop yields. Therefore, the massive access of motor-pumped well loads during peak farming seasons poses a direct challenge to the secure and economic operation of rural distribution networks.

The surge of temporary loads, particularly for pumping irrigation, creates a structural dilemma for the planning and operation of rural distribution networks. On the one hand, undersized capacity configurations fail to meet peak demands during irrigation seasons, leading to severe line overloads, excessive network losses, and sustained low voltages at terminal nodes [3,4], which jeopardize both grid safety and power quality. On the other hand, oversizing capacity to accommodate these short-term peaks results in equipment remaining in a light-load or idleness state for most of the year, drastically undermining the economic efficiency of the system. To address this contradiction, mobile energy storage systems (MESSs) emerge as a viable solution. Unlike stationary storage, MESSs offer unique flexibility across both temporal and spatial dimensions, capable of being deployed dynamically to provide temporary power support exactly where and when it is needed. This spatiotemporal transfer capability makes MESSs particularly suitable for rural distribution networks with seasonally concentrated and geographically uneven temporary loads. In this paper, this distinctive feature is leveraged to propose a coordinated dispatch strategy, offering a fresh perspective on optimizing rural distribution networks burdened by massive seasonal temporary loads [5].

Temporary loads in rural distribution networks exhibit inherent flexibility. Consequently, the optimal dispatch of such flexible loads has attracted significant attention from the academic community, leading to the proposal of various modeling methods tailored to different load types. Reference [6] put forward a price-based demand response dispatch model for residential loads. Reference [7] built a collaborative optimization model that integrates the production process and steam heat storage dispatch for high-energy-consuming loads in the textile industry, in which the time-shiftable characteristics of industrial loads are described through the task constraints of continuous production lines and the equivalent energy storage mechanism. Reference [8] proposed a flexible load modeling method based on the energy storage state of battery swap stations and hydrogen refueling stations for electricity–hydrogen integrated transportation systems, reflecting their multi-time-scale regulation ability within a day and across seasons. Reference [9] established a source–load coordinated optimization model considering the output limit and adjustment rate constraints of demand response loads for the peak shaving scenario of waste incineration power plants, which improves the consumption capacity of renewable energy and reduces wind and photovoltaic curtailment through load-side demand response. The above studies focus on the flexible modeling of typical loads in urban distribution networks or specific industrial scenarios, laying a solid theoretical foundation for the modeling of agricultural flexible loads. However, existing flexible load studies mainly provide load-side modeling ideas and do not directly answer how agricultural irrigation tasks should be represented under the operating characteristics of rural distribution networks.

In terms of the modeling and optimal operation of rural distribution systems, Reference [10] established a model for agricultural microgrids by combining pumped storage equipment with irrigation facilities. Reference [11] proposed a refined operation model based on AC power flow, aiming to address the typical characteristics of rural distribution networks such as low voltage levels, high resistance–reactance ratios, and large network loss. Reference [12] described the operation characteristics of rural power grids with radial constraints and reconfigured rural distribution networks by combining the branch-and-bound method and the primal–dual-interior-point method. Reference [13] constructed a linearized dispatch model for rural distribution networks with the aim of optimizing voltage deviation and network loss. Although the above literature considers various flexible resources such as photovoltaics, wind turbines, energy storage, and loads when modeling rural distribution network loads, most of them adopt time-series output curve models and lack in-depth analyses of typical loads and flexible regulation characteristics of rural power grids. Therefore, existing studies on rural distribution networks have clarified network-level operating characteristics, but they still provide limited support for jointly characterizing seasonal agricultural peak loads and their schedulable flexibility.

The optimal operation problem of distribution networks with energy storage equipment has also become a research hotspot. Reference [14] optimized the location and capacity of energy storage based on the optimal power flow model with SOCR, which effectively mitigated voltage violation and reduced system network loss under high-penetration photovoltaic access. Reference [15] used a genetic algorithm to optimize the energy storage configuration in order to minimize system network loss while meeting voltage security constraints. Reference [16] applied the antelope optimization algorithm to solve the problem of energy storage location and capacity, which effectively reduced network loss and improved voltage stability under complex nonlinear constraints. Reference [17] realized the dual goals of reducing system network loss and improving voltage quality through the coordinated allocation and optimal dispatch of photovoltaics and energy storage. Reference [18] proposed a two-stage optimal configuration strategy for energy storage in three-phase unbalanced distribution networks, which effectively alleviated the problems of overload, low voltage, and high network loss caused by large-scale electric vehicle access. Reference [19] determined the key buses sensitive to load change based on the loss sensitivity factor method to optimize the energy storage configuration, which significantly reduced system network loss. Reference [20] put forward an energy storage configuration method for flexibly interconnected distribution networks considering flexible constraints; this method fully considered the flexibility of energy storage and introduced flexible constraints to handle the intermittent fluctuations of distributed sources, thus reducing the net load fluctuation rate.

The above studies fully reflect the important role of energy storage in reducing distribution network loss and improving voltage quality. However, fixed energy storage is limited by its installation location, resulting in relatively limited spatial dispatch flexibility. When the load change trends vary greatly among regions, it is difficult to realize flexible energy transfer in space. Thus, while stationary energy storage can improve local balancing capability, its limited spatial transferability restricts its effectiveness in responding to temporally concentrated yet geographically uneven agricultural temporary loads.

Compared with fixed energy storage, MESSs can transfer electrical energy across space, and they also have important application value in the economic dispatch and voltage quality improvement of distribution networks. Reference [21] established a refined operation model of a large-scale MESS considering battery degradation cost, and they realized multiple objectives such as peak shaving and valley filling and delaying power grid infrastructure upgrading by optimizing the charge–discharge strategy, access location, and driving path of the MESS. Reference [22] proposed a hierarchical control strategy for active distribution networks considering the coordination of MESSs and reactive power optimization, and they introduced an improved quantum particle swarm optimization algorithm to rapidly solve multi-dimensional nonlinear problems. Reference [23] constructed a day-ahead coordinated voltage control framework for MESSs in active distribution networks with a high proportion of photovoltaics. By establishing a spatiotemporal decision model, Pareto optimization of voltage regulation cost and photovoltaic curtailment penalty was realized, which revealed its flexible value in stabilizing photovoltaic output fluctuations and delaying voltage violation. Reference [24] constructed a data-driven scenario generation method based on the denoising diffusion probabilistic model, proposed a day-ahead reactive power and voltage coordinated optimal dispatch framework, described the voltage operation risk under the uncertainty of renewable energy output through stochastic chance-constrained optimization, and dispatched MESSs under the coupling of the power grid and transportation network to achieve the dual objectives of improving voltage quality and reducing operation cost. Reference [25] addressed the uncertainty of high-proportion renewable energy output and established a two-stage robust coordinated optimization model of a dynamic reconfiguration strategy for distribution networks and the economic dispatch of MESSs, which promoted renewable energy consumption while ensuring the safe and economic operation of the power grid. Reference [26] further proposed an economic dispatch strategy for MESSs based on the equivalent reconfiguration method, which effectively reduced system network loss and operation cost by simplifying the coupling relationship between the network reconfiguration and energy storage dispatch. These studies verify that MESSs can effectively extend the flexibility of energy storage from the temporal dimension to the spatiotemporal dimension. However, most of them still focus on the optimization of MESSs themselves or their coordination with a limited number of control variables.

More recently, multi-resource coordinated dispatch studies have gradually expanded from single-resource optimization to source–grid–load–storage integrated scheduling. Reference [27] proposed a two-layer coordinated control model for active distribution networks with electric vehicle integration, jointly considering gas turbines, flexible loads, energy storage, and EV clusters. Reference [28] developed a coordinated optimization scheme for active distribution networks considering combined heat and power generation as well as orderly EV charging and discharging. Reference [29] investigated the optimal dispatch of a flexible distribution network equipped with MESSs and soft open points (SOPs), where the net dispatch benefit and voltage deviation were jointly optimized. Reference [30] further studied the routing and scheduling of MESSs in active distribution networks based on probabilistic voltage sensitivity analysis and Hall’s theorem. Reference [31] established a source–network–load–storage collaborated two-stage risk-dispatch model with conditional value-at-risk (CVaR) to quantify high-risk operational losses under renewable uncertainty. Reference [32] proposed a two-stage robust optimization model that simultaneously addresses source-side photovoltaic uncertainty and load-side demand response. Reference [33] developed a multi-objective day-ahead optimization scheduling framework for active distribution networks with distributed wind and photovoltaic integration, explicitly considering source–load interaction and demand response. In addition, recent review studies have pointed out that representative ADN scheduling methods should be compared not only by resource categories, but also from the perspectives of objective functions, key constraints, uncertainty treatment, and solution techniques [34,35]. Taken together, the above literature shows a clear development path from flexible load modeling to rural distribution network operation, stationary energy storage support, MESS-based spatial dispatch, and finally source–grid–load–storage integrated coordination. Following this logic,

Table 1. Comparison of representative recent multi-resource coordinated dispatch studies and this paper.

Ref.	Main Coordinated Resources/Scenario	Main Objective Function	Key Constraints	Uncertainty Treatment	Solution Technique
[27]	Gas turbines, flexible loads, ESSs, EV clusters in ADN	Minimize ADN operation cost and network-layer control cost	Power balance, EV charging/discharging, ESS operation, gas turbine output, network-layer constraints	Not explicitly modeled	Two-layer coordinated control, MTTA + CPLEX
[28]	CHP, EV charging/discharging, ADN resources	Improve operation economy and coordinated utilization of CHP and EVs	CHP operation, EV ordered charging/discharging, energy balance, network operation constraints	Implicitly considered, but no explicit risk/robust framework	Coordinated optimization model
[29]	MESSs + SOPs in flexible distribution network	Maximize net dispatch benefit and minimize total voltage deviation	ESS operation, time continuity, power balance, SOP capacity, DN operating constraints	Not explicitly modeled	NSGA-III
[30]	MESS routing and access-node scheduling in ADN	Improve MESS utilization and voltage support performance	Routing, access-node selection, voltage sensitivity, DN operation constraints	Probabilistic voltage sensitivity analysis	Probabilistic analysis + Hall’s theorem
[31]	Source–network–load–storage collaboration with ESS, SOP, dynamic reconfiguration	Minimize operation cost and high-risk loss	Day-ahead/intra-day coordination, dynamic reconfiguration, ESS/SOP constraints, network operation constraints	Explicitly modeled by CVaR and scenarios	Two-stage stochastic/risk dispatch
[32]	PV, DGs, demand response, reactive support in ADN	Improve economy and robustness under source–load uncertainty	Current, voltage, line current, DG/PV limits, DR constraints, uncertainty sets	Source-side PV uncertainty + load-side DR uncertainty	Two-stage robust optimization, column-and-constraint generation
[33]	Wind/PV, ESS, demand response, ADN scheduling	Coordinate economy, reliability, and safety	Renewable output models, DR constraints, ADN operation constraints	Wind/PV probabilistic modeling	MOEA/D-based multi-objective optimization
This paper	DGs + MESSs + motor-pumped well flexible loads in rural distribution network	Minimize total operating cost while reducing losses and mitigating voltage violations	Irrigation duration, conservation of shifted energy, adjustable operating windows, MESS transport–power coupling, SOC, SOCR-based power flow constraints	Task-driven temporal concentration is explicitly modeled; probabilistic uncertainty is not the main focus	SOCR-based coordinated optimization

summarizes representative recent multi-resource coordinated dispatch studies and compares them with the proposed method from the perspectives of objective functions, key constraints, uncertainty treatment, and solution techniques.

As summarized in Table 1, existing studies have made important progress in flexible load scheduling, active distribution network operation, MESS dispatch, and source–grid–load–storage coordination. Nevertheless, three gaps still remain for the specific problem addressed in this paper. First, most flexible load models are developed for residential, industrial, or integrated transportation-energy scenarios, while the task-oriented characteristics of agricultural motor-pumped well loads, including continuous irrigation requirements, rigid operation windows, and seasonal concentrated access, are still insufficiently modeled. Second, existing rural distribution network studies generally focus on conventional time-series load descriptions and do not fully reveal the formation mechanism and schedulable potential of seasonal temporary agricultural peak loads. Third, current multi-resource coordinated scheduling studies are mainly oriented toward general active distribution networks or MESS-centric optimization, and a unified economic dispatch framework that simultaneously coordinates DGs, motor-pumped well flexible loads, and MESSs in rural distribution networks is still lacking.

To fill the above research gap, this paper proposes an optimal dispatch strategy for rural distribution networks that coordinates multiple flexible resources, specifically targeting the scenario of massive grid access by agricultural motor-pumped wells during peak farming seasons. The main contributions of this work are threefold.

(1)

Considering the power consumption characteristics and irrigation task constraints of agricultural motor-pumped wells, a flexible load model for rural motor-pumped wells is established to fully exploit their flexible regulation capabilities.

(2)

In the context of a coupled power grid and transportation network, a spatiotemporal dispatch model for MESSs is established, considering the flexibility of energy transfer across temporal and spatial dimensions.

(3)

An economic dispatch model for distribution networks coordinating multiple flexible resources, including MESSs, flexible loads, and DGs, is established. In order to minimize the total system operating cost, the SOCR method is adopted to linearize the power flow constraints. This enables the coordinated dispatch of multiple flexible resources, including MESSs, motor-pumped well flexible loads, and DGs, to achieve safe and economic operation of the distribution network, reduce system network losses, and mitigate voltage violations.

The organization of this paper is as follows: Section 2 models the flexible loads in rural power grids represented by motor-pumped wells. Section 3 establishes the dispatch model for Mobile Energy Storage Systems (MESS). Section 4 formulates the economic dispatch model for rural distribution networks coordinating multiple flexible resources. Section 5 illustrates the flowchart for the solution methodology. Section 6 analyzes and validates the proposed strategy through specific case studies. Finally, Section 7 concludes the paper.

2. Modeling of Typical Flexible Loads in Rural Distribution Networks

Motor-pumped wells are widely distributed and numerous in rural areas, and the loading rate of distribution transformers remains persistently high during peak irrigation seasons. This section conducts modeling analysis for agricultural motor-pumped wells, and performs flexibility modeling based on their time-shiftable load characteristics.

2.1. Modeling of Agricultural Motor-Pumped Wells

Agricultural motor-pumped wells are devices that utilize electric motors to drive centrifugal pumps, extracting water from underground to the surface for irrigation purposes. The electricity consumption during irrigation operations depends on the rated power consumption of the motor-pumped wells and the irrigation time. The specific calculation formulas are shown in Equations (1)–(3):

$$P_{\mathrm{mo}}=Q_{\mathrm{m}}{H}_{\mathrm{L}}{\rho }_{\mathrm{w}}g/{\eta }_{\mathrm{m}}$$

(1)

$$t_{\mathrm{ir}}=\left(W_{\mathrm{r}}-W_{\mathrm{d}}\right)/\left(Q_{\mathrm{m}}{\eta }_{\mathrm{w}}\right)$$

(2)

$$E_{\mathrm{f}}=P_{\mathrm{mo}}{t}_{\mathrm{ir}}$$

(3)

Among them, $E_{\mathrm{f}}$ denotes the electricity consumption for a single irrigation per hectare of arable land; $P_{\mathrm{mo}}$ represents the rated power consumption of the motor-pumped well during pumping and irrigation; $t_{\mathrm{ir}}$ is the duration of a single irrigation by the motor-pumped well; $Q_{\mathrm{m}}$ indicates the flow rate of the motor-pumped well’s water pump, with the unit of m³/s; $H_{\mathrm{L}}$ refers to the equivalent head height. Considering the influence of friction resistance, the equivalent head height $H_{\mathrm{L}}$ is approximately 1.1 to 1.2 times the actual head; ${\rho }_{\mathrm{w}}$ denotes the density of the liquid, with the unit of kg/m³; g represents the acceleration due to gravity, with the unit of m/s²; ${\eta }_{\mathrm{m}}$ and ${\eta }_{\mathrm{w}}$ characterize the efficiency of the water pump and the utilization rate of irrigation water, respectively; $W_{\mathrm{r}}$ and $W_{\mathrm{d}}$ denote the daily water requirement per hectare of crops and the daily effective precipitation, respectively.

2.2. Flexibility Modeling of Motor-Pumped Wells

As a typical flexible load in rural distribution networks, motor-pumped wells possess time-shiftable characteristics. The irrigation periods of motor-pumped wells can be flexibly adjusted within the irrigation time windows allowed by crop growth, and intra-day load shifting does not affect agricultural production. Under the incentive of flexible load regulation subsidy mechanisms, motor-pumped well loads can achieve electricity consumption period shifting, while the total electricity consumption of motor-pumped well loads remains unchanged. The constraints shown in Equation (4) are constructed as the flexible load constraints for motor-pumped wells, which are used to specify the total electricity consumption of motor-pumped wells across all time-shiftable periods and to determine the operating status and power consumption of motor-pumped wells in individual time periods.

$$E_{s,T}^{\mathrm{load}}={\displaystyle \sum }_{m=1}^T{\delta }_{s,m}{P}_{s,m}^{\mathrm{mo}}{t}_{s,m}^{\mathrm{ir}}$$

(4)

Among them, $E_{s,T}^{\mathrm{load}}$ denotes the total electricity consumption of motor-pumped well s across all time-shiftable periods T; ${\delta }_{s,m}$ represents a binary variable indicating whether motor-pumped well s is operating in time period m, where 1 indicates that motor-pumped well s is working; $P_{s,m}^{\mathrm{mo}}$ denotes the power consumption of motor-pumped well s in time period m; $t_{s,m}^{\mathrm{ir}}$ represents the operating duration of motor-pumped well s in time period m.

The operating duration of motor-pumped wells across all time-shiftable periods equals the sum of actual irrigation time in each time period. The constraints shown in Equation (5) are constructed as the total operating duration constraints for motor-pumped wells, which are used to specify the total operating time of motor-pumped wells across all time-shiftable periods and to determine the operating status of motor-pumped wells in individual time periods.

$$t_{\mathrm{total},s}^T={\displaystyle \sum }_{m=1}^T{\delta }_{s,m}{t}_{s,m}^{\mathrm{ir}}$$

(5)

Among them, $t_{\mathrm{total},s}^T$ denotes the total operating time of motor-pumped well s across all time-shiftable periods T.

The total electricity consumption of motor-pumped wells across all time-shiftable periods equals the sum of electricity consumption in individual time periods. The constraints shown in Equation (6) are constructed as the energy conservation constraints for motor-pumped wells, which are used to specify the total electricity consumption of motor-pumped wells across all time-shiftable periods and to determine the electricity consumption of motor-pumped wells in individual time periods.

$$E_{s,T}^{\mathrm{load}}-{\displaystyle \sum }_{m=m_{xia}}^{m_{\mathrm{shang}}}{E}_{s,m}^{\mathrm{load}}=0$$

(6)

Among them, $E_{s,m}^{\mathrm{load}}$ denotes the electricity consumption of motor-pumped well s in time period m; $m_{xia}$ represents the lower bound of the time-shiftable periods; $m_{shang}$ denotes the upper bound of the time-shiftable periods.

To further exploit the regulation capability of flexible loads, variable-frequency drives and multi-gear power regulation modules can be integrated into the motor-pumped well control systems. This module operates in coordination with the frequency converter controller, allowing operators to flexibly select among preset gear levels such as “high flow-medium flow-low flow” based on the grid load conditions, thereby achieving refined power control through adjusting the water pump flow rate. The constraints shown in Equation (7) are constructed as the flexible load power constraints, which are used to specify the maximum and minimum power consumption values of motor-pumped wells and to determine the power consumption of motor-pumped wells in individual time periods.

$$P_s^{\min }\le P_{s,m}^{\mathrm{mo}}\le P_s^{\max }$$

(7)

Among them, $P_s^{\max }$ and $P_s^{\min }$ denote the maximum and minimum power consumption of motor-pumped well s, respectively.

3. Dispatch Model of MESS

Under the background of coupled power and transportation networks, considering the temporal and spatial energy transfer characteristics of MESS, the spatiotemporal dispatch model of MESS is precisely modeled by accounting for multiple constraints including spatiotemporal scheduling constraints, charging/discharging constraints, and state-of-charge constraints.

In the coupled power-transportation network system, the charging/discharging power and location of MESS jointly determine its dispatch state. A node traversal distance adjacency matrix is established, and the shortest path between node j and node k is obtained through Dijkstra’s shortest path algorithm [34]. Considering the congestion conditions of the transportation network, the dynamic speed of MESS is acquired, and the travel time of MESS between two nodes is calculated based on node traversal distance and speed [35]. The architecture of the coupled power-transportation network is shown in Figure 1.

$$D_{i,j,t}=D_{i,j,0}\left(1+{\displaystyle \frac{1}{v_{k,t}^{\mathrm{ME}}}}\right)$$

(8)

$$v_{k,t}^{\mathrm{ME}}=v_{k,0}{\mathrm{e}}^{-1.7c}$$

(9)

$$T_{k,i,j,t}^{\mathrm{ME}}={\displaystyle \frac{D_{i,j,t}}{v_{k,t}^{\mathrm{ME}}}}$$

(10)

Among them, $D_{i,j,0}$ represents the static shortest travel distance between node i and node j in the power network; $D_{i,j,t}$ represents the actual shortest travel distance of MESS between node i and node j at time t considering congestion conditions; $v_{k,t}^{\mathrm{ME}}$ represents the actual speed of MESS k at time t; $v_{k,0}$ represents the ideal speed of MESS k without considering congestion; c represents the congestion coefficient of the transportation network, which can be estimated based on road traffic conditions; $T_{k,i,j,t}^{\mathrm{ME}}$ represents the actual shortest travel time of MESS k from node i to node j starting at time t. Equation (8) represents the actual shortest travel distance of MESS, Equation (9) represents the dynamic real-time speed of MESS, and Equation (10) represents the actual shortest travel time of MESS.

When the power grid issues dispatch instructions to MESS, the MESS needs to select the optimal path based on the congestion conditions of the transportation network, and transfer from node i to node j for charging/discharging operations with the minimum travel time. MESS can be connected to at most one node at any given time, and is not in a connected state while traveling between nodes i and j; it can only charge or discharge when connected to a node. Accordingly, the constraints shown in Equation (11) through (13) are formulated as the spatiotemporal scheduling constraints for MESS.

$${\alpha }_{k,i,t}^{\mathrm{ME}}+{\alpha }_{k,j,t+\Delta t}^{\mathrm{ME}}\le 1,\hspace{1em}\forall \Delta t<T_{k,i,j,t}^{\mathrm{ME}}+T_0^{\mathrm{ME}}$$

(11)

$${\displaystyle \sum }_{i\in N}{\alpha }_{k,i,t}^{\mathrm{ME}}\le 1$$

(12)

$$U_{k,t}^{\mathrm{Mch}}+U_{k,t}^{\mathrm{Mdch}}\le {\displaystyle \sum }_{i\in N}{\alpha }_{k,i,t}^{\mathrm{ME}}{\alpha }_{k,i,t+1}^{\mathrm{ME}}$$

(13)

where ${\alpha }_{k,i,t}^{\mathrm{ME}}$ is a binary variable representing the connection status between MESS unit k and node i at time t. Similarly, ${\alpha }_{k,j,t+\Delta t}^{\mathrm{ME}}$ represents the connection status between MESS k and node j at time $\left(t+\Delta t\right)$. $T_0^{\mathrm{ME}}$ denotes the installation time of MESS k. Furthermore, $U_{k,t}^{\mathrm{Mch}}$ and $U_{k,t}^{\mathrm{Mdch}}$ represent the charging and discharging states of MESS k at time t, respectively, both of which are adjustable binary variables (a value of 1 indicates an active charging/discharging state, and 0 indicates inactive). Equation (11) indicates that when the operation time of the MESS is less than the sum of the travel time between two nodes and the installation time of the MESS, the MESS remains disconnected from any node. Equation (12) ensures that one MESS unit can be connected to at most one node at any given time. Equation (13) represents the coupling relationship between the charging/discharging state and the location of the MESS, permitting charging or discharging operations only when the MESS is physically connected to either node i or node j.

Due to the presence of bilinear terms in Equation (13), a linearization method is employed to transform Equation (13) into the linear constraints [36]. By introducing auxiliary variables, the bilinear constraints are transformed into linear constraints, as shown in Equation (14).

$$\left\{\begin{array}{l}{\alpha }_{k,i,t}^{\mathrm{MCS}}\le {\alpha }_{k,i,t+1}^{\mathrm{ME}}\hfill \\ {\alpha }_{k,i,t}^{\mathrm{MCS}}\le {\alpha }_{k,i,t}^{\mathrm{ME}}\hfill \\ {\alpha }_{k,i,t}^{\mathrm{MCS}}\ge {\alpha }_{k,i,t}^{\mathrm{ME}}+{\alpha }_{k,i,t+1}^{\mathrm{ME}}-1\hfill \\ U_{k,t}^{\mathrm{Mch}}+U_{k,t}^{\mathrm{Mdch}}\le {\displaystyle {\displaystyle \sum }_{i\in N}}{\alpha }_{k,i,t}^{\mathrm{MCS}}\hfill \end{array}\right.$$

(14)

where ${\alpha }_{k,i,t}^{\mathrm{MCS}}$ denotes the connection status between MESS k and node i during the time interval from t to t + 1, which is a binary variable; a value of 1 indicates that MESS k is connected to node i during the corresponding time interval, and a value of 0 indicates a non-connection state.

The charging and discharging constraints of the MESS are given by Equations (15)–(18).

$$0\le P_{k,t}^{\mathrm{Mch}}\le U_{k,t}^{\mathrm{Mch}}{P}_{k,\max }^{\mathrm{Mch}}$$

(15)

$$0\le P_{k,t}^{\mathrm{Mdch}}\le U_{k,t}^{\mathrm{Mdch}}{P}_{k,\max }^{\mathrm{Mdch}}$$

(16)

$$0\le Q_{k,t}^{\mathrm{Mch}}\le U_{k,t}^{\mathrm{Mch}}{Q}_{k,\max }^{\mathrm{Mch}}$$

(17)

$$0\le Q_{k,t}^{\mathrm{Mdch}}\le U_{k,t}^{\mathrm{Mdch}}{Q}_{k,\max }^{\mathrm{Mdch}}$$

(18)

where $P_{k,t}^{\mathrm{Mch}}$ and $Q_{k,t}^{\mathrm{Mch}}$ denote the active and reactive power output of MESS k during charging at time t; $P_{k,\max }^{\mathrm{Mch}}$ and $Q_{k,\max }^{\mathrm{Mch}}$ denote the maximum allowable active and reactive power output of MESS k during charging; $P_{k,t}^{\mathrm{Mdch}}$ and $Q_{k,t}^{\mathrm{Mdch}}$ denote the active and reactive power output of MESS k during discharging at time t; $P_{k,\max }^{\mathrm{Mdch}}$ and $Q_{k,\max }^{\mathrm{Mdch}}$ denote the maximum allowable active and reactive power output of MESS k during discharging. Equations (15) and (16) define the operating ranges for the active charging and discharging power of the MESS, while Equations (17) and (18) impose the upper and lower bound constraints on the reactive power during charging and discharging operations.

The state-of-charge (SOC) constraints for the MESS are given by Equations (19) and (20).

$$E_{k,t+\Delta t}^{\mathrm{ME}}=E_{k,t}^{\mathrm{ME}}+P_{k,t}^{\mathrm{Mch}}{\eta }_k^{\mathrm{Mch}}\Delta t-{\displaystyle \frac{P_{k,t}^{\mathrm{Mdch}}}{{\eta }_k^{\mathrm{Mdch}}}}\Delta t$$

(19)

$$E_{k,\min }^{\mathrm{ME}}\le E_{k,t}^{\mathrm{ME}}\le E_{k,\max }^{\mathrm{ME}}$$

(20)

where $E_{k,t}^{\mathrm{ME}}$ denotes the state of charge of MESS k at time t; ${\eta }_k^{\mathrm{Mch}}$ and ${\eta }_k^{\mathrm{Mdch}}$ denote the energy conversion efficiency of MESS k during the charging and discharging process, respectively; $E_{k,\max }^{\mathrm{ME}}$ and $E_{k,\min }^{\mathrm{ME}}$ denote the maximum and minimum capacity of MESS k, respectively. Equation (19) represents the state-of-charge evolution constraint of the MESS, while Equation (20) represents the upper and lower bound constraints on the capacity of the MESS.

4. Optimal Dispatch Model for Rural Distribution Networks Coordinating Multiple Flexible Resources

With the objective of minimizing total system operating costs, MESS are optimally dispatched under multiple constraints including power flow constraints and DG output constraints, thereby achieving safe and economic operation of distribution networks and alleviating voltage violations.

4.1. Objective Function

The objective function of the model is shown in Equation (21).

$$\min f=f_L+f_V$$

(21)

Among them, f denotes the total operating cost of the system; f_L represents the network loss cost of the system; f_V denotes the electricity loss cost caused by voltage violations.

(1)

Network loss cost of the system

$$f_L=c_L{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{ij\in E}{I}_{ij,t}^2}{r}_{ij}\Delta t}$$

(22)

Among them, $c_L$ denotes the cost per unit of network loss in the system; T represents the total dispatch period; E denotes the set of all branches in the system; $I_{ij,t}$ indicates the current value of branch ij at time t; $r_{ij}$ represents the resistance value of branch ij; Δt denotes the time step length.

(2)

Electricity loss cost caused by voltage violations

$$f_V=c_V{\displaystyle \sum _{t=1}^T\Delta U}$$

(23)

$$\Delta U_{i,t}=\left\{\begin{array}{l}{V}_{i,t}^2-V_{i,\mathrm{nom}\max }^2,V_{i,t}\ge V_{i,\mathrm{nom}\max }\\ 0,V_{i,\mathrm{nom}\min }\le V_{i,t}\le V_{i,\mathrm{nom}\max }\\ V_{i,\mathrm{nom}\min }^2-V_{i,t}^2,V_{i,t}\le V_{i,\mathrm{nom}\min }\end{array}\right.$$

(24)

Among them, $c_V$ denotes the electricity loss cost per unit of voltage violation in the system; $\Delta U_{i,t}$ represents the degree of voltage violation; $V_{i,t}$ denotes the voltage value at node i at time t; $V_{i,\mathrm{nom}\max }$ and $V_{i,\mathrm{nom}\min }$ denote the maximum and minimum values of the normal operating voltage range at node i that avoids power losses. The decision variables of the model are the connection status between MESS and nodes ${\alpha }_{k,i,t}^{\mathrm{ME}}$, the charging and discharging active power of MESS $P_{k,t}^{\mathrm{Mch}}$ and $P_{k,t}^{\mathrm{Mdch}}$, the reactive power $Q_{k,t}^{\mathrm{Mch}}$ and $Q_{k,t}^{\mathrm{Mdch}}$ the operating status of motor-pumped well s ${\delta }_{s,m}$, and the power consumption $P_{s,m}^{\mathrm{mo}}$.

4.2. Constraints

(1)

DG output constraints

$$0\le P_{h,t}^{\mathrm{DG}}\le P_{h,t,\max }^{\mathrm{DG}}$$

(25)

$$0\le Q_{h,t}^{\mathrm{DG}}\le Q_{h,t,\max }^{\mathrm{DG}}$$

(26)

where $P_{h,t}^{\mathrm{DG}}$ and $Q_{h,t}^{\mathrm{DG}}$ denote the active and reactive power output of DG h at time t, respectively; $P_{h,t,\max }^{\mathrm{DG}}$ and $Q_{h,t,\max }^{\mathrm{DG}}$ denote the maximum active and reactive power output of DG h at time t, respectively.

(2)

Power flow constraints of the distribution networks

The SOCR method [37,38] is employed to model the power flow constraints of distribution networks. In the branch power flow model, the squared terms of voltage and the squared terms of current are replaced with new variables, and the non-convex power flow equations are convexified through convex relaxation, thereby transforming them into convex problems that can be efficiently solved. The power flow constraints of distribution networks can be expressed as follows:

$${\displaystyle \sum }_{j\in \delta \left(i\right)}{P}_{ij,t}-{\displaystyle \sum }_{j\in \gamma \left(i\right)}\left(P_{ji,t}-R_{ji}{I}_{ji,t}^{\mathrm{sqr}}\right)={\alpha }_{h,i}^{\mathrm{DG}}{P}_{h,t}^{\mathrm{DG}}+{\alpha }_{k,i,t}^{\mathrm{ME}}(P_{k,t}^{\mathrm{Mdch}}-P_{k,t}^{\mathrm{Mch}})-P_{i,t}^{\mathrm{L}}$$

(27)

$${\displaystyle \sum }_{j\in \delta \left(i\right)}{Q}_{ij,t}-{\displaystyle \sum }_{j\in \gamma \left(i\right)}\left(Q_{ji,t}-X_{ji}{I}_{ji,t}^{\mathrm{sqr}}\right)={\alpha }_{h,i}^{\mathrm{DG}}{Q}_{h,t}^{\mathrm{DG}}+{\alpha }_{k,i,t}^{\mathrm{ME}}(Q_{k,t}^{\mathrm{Mdch}}-Q_{k,t}^{\mathrm{Mch}})-Q_{i,t}^{\mathrm{L}}$$

(28)

$$V_{i,t}^{\mathrm{sqr}}-V_{j,t}^{\mathrm{sqr}}=2\left(R_{ij}{P}_{ij,t}+X_{ij}{Q}_{ij,t}\right)+\left(R_{ij}^2+X_{ij}^2\right){\displaystyle \frac{P_{ij,t}^2+Q_{ij,t}^2}{V_{i,t}^{\mathrm{sqr}}}}$$

(29)

$$I_{ij,t}^{\mathrm{sqr}}={\displaystyle \frac{P_{ij,t}^2+Q_{ij,t}^2}{V_{i,t}^{\mathrm{sqr}}}}$$

(30)

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

(31)

$$0\le I_{ij,t}^{\mathrm{sqr}}\le I_{ij,t,\max }^{\mathrm{sqr}}$$

(32)

Among them, $P_{ij,t}$ and $Q_{ij,t}$ denote the active and reactive power transmitted through branch ij at time t, respectively; $R_{ij}$ denotes the resistance value of branch ij; $X_{ij}$ indicates the reactance value of branch ij; $P_{i,t}^{\mathrm{L}}$ and $Q_{i,t}^{\mathrm{L}}$ denote the active and reactive power of the load at node i at time t, respectively; $I_{ij,t}^{\mathrm{sqr}}$ indicates the squared value of the current in branch ij at time t; ${\alpha }_{h,i}^{\mathrm{DG}}$ is a binary variable representing the connection status between distributed generator h and node i; $V_{i,t}^{\mathrm{sqr}}$ denotes the squared value of the voltage at node i at time t; $V_{i,t,\max }^{\mathrm{sqr}}$ and $V_{i,t,\min }^{\mathrm{sqr}}$ represent the squared value of the maximum and minimum allowable node voltage at node i in the system at time t, respectively; $I_{ij,t,\max }^{\mathrm{sqr}}$ indicates the maximum value of the squared current in branch ij at time t. Equations (27) and (28) represent the power balance constraints of the power grid; Equation (29) is the node voltage difference constraint; Equation (30) is the nonlinear constraint regarding voltage, current, and power; Equation (31) is the upper and lower bound constraint for node voltage; and Equation (32) is the upper and lower bound constraint for branch current.

After introducing the new variables, Equation (30) still contains squared terms of the decision variables, constituting a quadratic nonlinear constraint. If the optimization objective is a strictly monotonically increasing function of $I_{ij,t}^{\mathrm{sqr}}$ then the following convex relaxation can be applied:

$$I_{ij,t}^{\mathrm{sqr}}\ge {\displaystyle \frac{P_{ij,t}^2+Q_{ij,t}^2}{V_{i,t}^{\mathrm{sqr}}}}$$

(33)

$${(2P_{ij,t})}^2+{(2Q_{ij,t})}^2\le 4I_{ij,t}^{\mathrm{sqr}}{V}_{i,t}^{\mathrm{sqr}}$$

(34)

$${‖\begin{array}{c}2P_{ij,t}\\ 2Q_{ij,t}\\ I_{ij,t}^{\mathrm{sqr}}-V_{i,t}^{\mathrm{sqr}}\end{array}‖}_2\le I_{ij,t}^{\mathrm{sqr}}+V_{i,t}^{\mathrm{sqr}}$$

(35)

Equations (33)–(35) provide a detailed mathematical derivation of the transformation from non-convex constraints to second-order cone constraints through relaxation.

(3)

Constraints for motor-pumped wells

Refer to Equations (1)–(7).

(4)

Constraints for MESS dispatch

Refer to Equations (6)–(18).

5. Solution Procedure of the Model

The flowchart of the model solution process in this article is shown in Figure 2.

Step 1: Determine system parameters.

This method first requires determining various parameters including distribution network topology, line parameters, distributed generator capacity, MESS technical parameters, and load parameters, providing data support for economic dispatch.

Step 2: Establish flexible load model for rural distribution networks.

Construct the motor-pumped well load model, including flexible load power constraints, total operating duration constraints, energy conservation constraints, and others. Motor-pumped well loads can achieve electricity consumption period shifting, while the total electricity consumption of flexible loads remains unchanged.

Step 3: Establish MESS dispatch model.

Under the context of power grid and transportation network coupling, considering the electricity transfer characteristics of MESS in temporal and spatial scales, the spatiotemporal dispatch model for MESS is precisely modeled by accounting for multiple constraints including spatiotemporal dispatch constraints, charging and discharging constraints, and state of charge constraints.

Step 4: Establish an operational optimization model for rural distribution networks coordinating multiple flexible resources.

Based on the established flexible load model and MESS dispatch model, construct a multi-resource coordinated operational optimization model for rural distribution networks. This model takes minimizing total system operating costs as the optimization objective, fully coordinates flexible resources including MESS, DG, and flexible loads, ensuring economic system operation and alleviating voltage violations. The model needs to consider multiple constraints including MESS dispatch constraints, flexible load constraints, and distributed generator output constraints.

Step 5: Transform mixed-integer nonlinear programming model into convex optimization model through SOCR method.

By adopting the SOCR method, the original problem is transformed into a mixed-integer second-order cone programming problem through variable substitution and cone relaxation of the squared voltage terms and squared current terms in power flow constraints.

Step 6: Solve the model by invoking the Gurobi solver to obtain the optimal economic dispatch strategy.

Invoke the Gurobi solver to solve the model and obtain the distributed generator output scheme, MESS dispatch scheme, and electricity consumption behavior of flexible loads, ensuring safe and economic operation of the power system and alleviating voltage violations.

6. Case Study Analysis

To verify the effectiveness and feasibility of the economic dispatch strategy for rural distribution networks considering the synergy between MESS and flexible loads, this study establishes a case study in the MATLAB R2022b simulation platform and solves the model by calling the Gurobi solver.

6.1. Case Study Parameters

A modified IEEE 33-bus system is adopted for case study analysis, and the system diagram is shown in Figure 3. The load curve is shown in Figure 4, where nodes 1–22 belong to Area 1 and nodes 23–33 belong to Area 2. The MESS is initially located at node 1, with a capacity of 1.0 MWh and a maximum active power of 200 kW. The charging and discharging efficiencies of the MESS are both 0.98. Photovoltaic (PV) systems are connected at nodes 30 and 31, with a rated active power of 50 kW. Wind turbines (WT) are connected at six nodes including 7, 8, 19, 20, 24, and 25, with a rated active power of 200 kW. Time-series output curves of photovoltaic and wind turbines is shown in Figure 5 Motor-pumped Wells (MWs) are connected at twelve nodes including 4, 5, 6, 9, 10, 11, 24, 25, 30, 31, 32, and 33. Each MW has a rated active power of 30 kW and an adjustable power range of 25–35 kW. The MWs require a total daily operating time of 12 h, but the specific operating periods and power levels can be flexibly selected. The power regulation range of the motor-pumped well load is 25–35 kW. The transportation network topology is identical to the power grid, and the electrical distance between adjacent nodes is 1 km [39]. The cost per unit of network loss in the system $c_L$ is set to 1 CNY/kW, the electricity loss cost per unit of voltage violation in the system $c_V$ is set to 3.12 CNY/kV². The system safe voltage range is [0.9, 1.1] p.u., and the normal voltage optimization range, which avoids significant power losses, is [0.95, 1.05] p.u.

6.2. Analysis of Results

6.2.1. System Voltage and Network Loss Analysis

The network losses at each time period in Area 1 are shown in Figure 6. The network losses are relatively low during the 1:00–4:00 time period. Due to the increase in load power, the network losses gradually rise during the 4:00–9:00 time period. Due to the decrease in the load curve, the network losses gradually decline during the 9:00–13:00 time period. The network losses show an overall upward trend during the 13:00–22:00 time period, and gradually decrease during the 22:00–24:00 time period. The overall variation trend of network losses in Area 1 is similar to the load curve. The network losses at each time period in Area 2 are shown in Figure 7. The network losses gradually increase with the rise in the load curve during the 1:00–13:00 time period, and show a general downward trend with the decrease in load after 13:00.

The voltage extrema at each node of the system are shown in Figure 8. The maximum and minimum voltages at nodes 1 to 18 decrease sequentially, with the lowest voltage at node 18 being 0.95 p.u., which is exactly at the minimum voltage limit. Similarly, the maximum and minimum voltages at nodes 19 to 33 also decrease sequentially, with the lowest voltage at node 33 being 0.95 p.u., just without violating the lower limit. The voltage extrema at each time step of the system are presented in Figure 9. During the period from 1:00 to 5:00, due to the relatively low load curve, the system minimum voltage is maintained at approximately 0.96 p.u. During the period from 6:00 to 22:00, the total system load curve is relatively high, resulting in a lower system minimum voltage, which is maintained at approximately 0.95 p.u. but without voltage violations. From 22:00 to 24:00, the total system load curve decreases, and the system minimum voltage begins to gradually increase.

6.2.2. Analysis of Flexible Resource Dispatch Results

The power and access location of the MESS are shown in Figure 10, where the horizontal axis represents the time, the left vertical axis represents the power of the MESS, and the right vertical axis represents its access location. During the 7:00–9:00 time period, as the load curve in Area 1 gradually increases, the MESS moves from node 1 to node 18 and continues to discharge to reduce system network losses and provide voltage support for the terminal nodes. During the 12:00–22:00 time period, the MESS moves from node 18 to node 33 to discharge, reducing network losses and ensuring that node 33 does not experience a lower voltage limit violation due to motor-pumped well irrigation.

The power profiles of the motor-pumped wells are shown in Figure 11 and Figure 12. Since the load curve in Area 1 is very high during the 7:00–9:00 and 16:00 time periods, the motor-pumped wells in Area 1 mainly operate for pumping and irrigation during the 1:00–6:00, 11:00–15:00, and 23:00–24:00 time periods. As the daytime load curve in Area 2 is consistently high, most motor-pumped wells in Area 2 operate for pumping and irrigation at night. However, due to the large photovoltaic output at noon, motor-pumped wells near photovoltaic nodes operate during the noon period. Additionally, due to the large wind turbine output at 19:00, the motor-pumped wells in Area 2 operate intensively at 19:00. The output of distributed generators is shown in Figure 13 and Figure 14. The distributed generators operate at full capacity, which also contributes to reducing system network losses and maintaining node voltage.

6.2.3. Comparative Analysis

To verify the effectiveness of the proposed strategy, the following four strategies are set up for comparison:

• Strategy 1: Energy storage is connected at a fixed location, and flexible loads consume electricity during fixed time periods.
• Strategy 2: Energy storage is connected at a fixed location, and the flexibility of flexible load electricity consumption periods is considered.
• Strategy 3: MESS is connected, and flexible loads consume electricity during fixed time periods.
• Strategy 4: MESS is connected, and the flexibility of flexible load electricity consumption periods is considered.

The operating costs of the four strategies are presented in

Table 2. Operating costs of the distribution network under different strategies.

Operating Cost (CNY)	Strategy 1	Strategy 2	Strategy 3	Strategy 4
Total Operating Cost	2079.08	1632.05	1451.1	1281.74
Network Loss Cost	1514.5	1454.1	1329.15	1281.74
Voltage Violation Cost	564.58	177.95	121.95	0

. A comparative analysis of the optimization results reveals that the total operating cost of the strategy incorporating MESS is lower than that of the strategy with energy storage fixed at specific locations. Furthermore, the voltage violation cost of the strategy allowing flexible loads to flexibly select operation periods is lower than that of the strategy where flexible loads operate in fixed time slots. The strategy considering the synergy between MESS and flexible loads for economic operation achieves the minimum total operating cost with no voltage violations. Compared to the strategy with energy storage fixed at specific locations and flexible loads operating in fixed time slots, this coordinated strategy reduces the total operating cost by 38.35% and the voltage violation cost by 797.34 CNY. These results fully validate the effectiveness and feasibility of coordinating multiple flexibility resources to reduce network losses and improve voltage quality in rural distribution networks.

The network losses at each time step in Area 1 under the four strategies are shown in Figure 15, and the network losses at each time step in Area 2 are shown in Figure 16. The minimum voltage at each node is presented in Figure 17, and the minimum voltage at each time step is shown in Figure 18. As can be observed from the figures, Strategy 4 achieves the lowest total network losses with no voltage violations, while the other strategies exhibit higher total network losses and experience varying degrees of voltage violations. This further confirms the effectiveness of the proposed distribution network economic dispatch strategy coordinating multiple flexible resources.

6.2.4. Sensitivity Analysis

Given the inherent uncertainty of wind turbine and photovoltaic power outputs, a sensitivity analysis was conducted focusing on prediction errors. Three scenarios were established for comparative analysis.

• Scenario 1: ±3% prediction error for DG output and ±5% prediction errors for load.
• Scenario 2: ±5% prediction error for DG output and ±5% prediction errors for load.
• Scenario 3: ±8% prediction error for DG output and ±5% prediction errors for load.

The operational costs for these three scenarios are presented in

Table 3. Sensitivity Analysis under Different Forecasting Error Scenarios.

Scenarios	Scenario 1	Scenario 2	Scenario 3
Operating Cost (CNY)	1284.57	1286.1	1288.51

. By comparing the optimization results under varying prediction errors of distributed generation (DG), it is observed that the uncertainty of DG output has a limited impact on grid operational costs. Specifically, the grid operational cost increases slightly as the prediction error of distributed generation grows.

6.2.5. Scalability Analysis

To verify the scalability and generalization capability of the proposed algorithm in larger-scale distribution networks, a modified IEEE 69-bus system is established for validation. The topology of the system is shown in Figure 19. The system is divided into two areas: Area 1 includes nodes 1–35, 51–52, and 68–69; Area 2 includes nodes 36–50 and 53–67. The initial position of the MESS is set at node 1, with a capacity of 1.0 MWh and a maximum active power of 200 kW. Photovoltaic (PV) units are connected to nodes 16 and 17, with a rated active power of 50 kW. Wind Turbines (WT) are connected to nodes 6, 7, 61, and 62, with a rated active power of 200 kW. Furthermore, to simulate the temporary agricultural load scenario, Motor-pumped Wells (MWs) are connected to 12 nodes: 18–20, 25–27, and 60–65. Each MW has a rated active power of 30 kW, with an adjustable power range of 25–35 kW. The total daily operating time for each MW is set to 12 h, which can be flexibly scheduled. The traffic network topology is assumed to be identical to the power grid topology, with a fixed electrical distance of 0.5 km between adjacent nodes. The system safety voltage range is set to [0.9, 1.1] p.u., while the normal optimization range is [0.95, 1.05] p.u.

To verify the effectiveness of the proposed strategy, the following four strategies are set up for comparison:

• Strategy 1: Energy storage is connected at a fixed location, and flexible loads consume electricity during fixed time periods.
• Strategy 2: Energy storage is connected at a fixed location, and the flexibility of flexible load electricity consumption periods is considered.
• Strategy 3: MESS is connected, and flexible loads consume electricity during fixed time periods.
• Strategy 4: MESS is connected, and the flexibility of flexible load electricity consumption periods is considered.

The operating costs of the four strategies are presented in

Table 4. Operating Costs of the 69-bus System under Different Strategies.

Operating Cost (CNY)	Strategy 1	Strategy 2	Strategy 3	Strategy 4
Total Operating Cost	1669.17	1447.13	1246.56	1133.35
Network Loss Cost	1318.69	1305.44	1191.64	1131.77
Voltage Violation Cost	290.48	141.68	54.92	1.58

. A comparison of the optimization results reveals that the total operating cost of the strategy considering MESS integration is lower than that of the fixed energy storage strategy. Furthermore, the voltage violation cost under the flexible load strategy with flexible time-slot selection is lower than that with fixed operating periods.

The strategy considering the coordinated economic operation of MESS and flexible loads yields the minimum total cost and exhibits almost no voltage violations. Compared with the strategy employing fixed energy storage and fixed-schedule flexible loads, the total cost is reduced by 32.1%, and the voltage violation cost is decreased by 288.9 CNY. These results further validate the effectiveness and feasibility of the coordinated operation of MESS and flexible loads in reducing network losses and improving voltage quality in rural distribution networks.

7. Conclusions

This paper proposes an optimal dispatch strategy for rural distribution networks that coordinates multiple flexible resources while accounting for temporary agricultural loads. Specifically, a spatiotemporal dispatch model for MESS and a load model for motor-pumped wells were established. Based on these models, an economic dispatch framework was constructed with the objective of minimizing the total system operating cost. The main conclusions are summarized as follows:

(1)

MESS effectively leverages its energy transfer capabilities across both temporal and spatial dimensions. By being deployed at the ends of feeders, MESS significantly reduces network losses and provides critical voltage support during peak load periods and concentrated irrigation intervals.

(2)

Motor-pumped wells fully exploit their inherent flexibility for regulation. By flexibly scheduling irrigation timing and adjusting power consumption during periods of high DG output or low general load, these loads further reduce network losses and effectively mitigate voltage violation issues.

(3)

The optimal dispatch strategy for rural distribution networks based on the synergy of multiple flexibility resources, including MESS, flexible loads, and DG, reduces the total operating cost of the system by 38.35% compared to the strategy with energy storage fixed at specific locations and flexible loads operating in fixed time slots. This fully validates the effectiveness of the proposed strategy.