Research on the economic feasibility of urban power grid load transfer considering heavy rainfall and waterlogging risk

1 Introduction

Climate change has intensified in recent years, and extreme rainstorm disasters have become increasingly frequent (Department of Disaster Relief and Material Support and Ministry of Emergency Management, 2023). The Zhengzhou “7.20” extreme rainstorm in 2021 triggered severe urban waterlogging. According to the investigation and assessment by the State Council disaster investigation team, the disaster caused direct economic losses of CNY 120.06 billion in Henan Province, including CNY 40.9 billion in Zhengzhou (Li et al., 2026). During the disaster, large-scale power outages occurred in many areas of Zhengzhou. Nearly 30% of power-supply facilities across the province were significantly affected; 3.74 million customers lost power, and more than 800 distribution lines were damaged (Li et al., 2026). Enterprise production stoppages and infrastructure paralysis caused by damage to power infrastructure generated incalculable indirect economic losses for the city. As the terminal link of electric-energy transmission (IEEE Guide for Electric Power, 2001), the distribution network is characterized by strong cascading propagation of outages and pronounced social impacts (Ji et al., 2025). Post-disaster microgrid-coordinated load transfer is therefore not only a core means of ensuring electricity supply, but also a key approach for reducing economic losses from disasters and building a distribution network with high disaster resilience and high economic efficiency (Yu et al., 2022).

The economic losses caused by rainstorm-waterlogging outages are multi-level and compound in nature (Zheng et al., 2015). In terms of direct losses, outages may cause shutdowns of industrial enterprises and interruptions to commercial operations. Different customers have significantly different sensitivities to outages, and their corresponding economic losses also differ markedly. In terms of indirect losses, repair costs caused by damage to distribution terminal equipment, assessment costs borne by power departments because of decreased supply-reliability indices, and switch-damage costs caused by loop-closing current impacts arising from improper transfer operations can all further increase comprehensive disaster losses. Existing research has clear deficiencies in the quantification of outage losses. Most studies simplify outage losses into a single energy-not-supplied indicator and fail to distinguish differentiated loss coefficients for different customer classes. At the same time, the influence of transfer-path selection on economic cost is generally overlooked. Differences among transfer schemes in additional network losses and loop-closing-current risk directly cause divergence in operating costs and equipment-risk costs, a dimension that is almost absent from existing evaluation systems (Li et al., 2011).

Accurate disaster-risk quantification is the foundation for improving the disaster-resistant reliability of distribution networks. Current outage-risk assessment for distribution networks often uses Monte Carlo methods for simulation and calculation (Zhang et al., 2014), or assumes constant equipment failure rates to simplify system modeling and computation (Billinton and Li, 1994). Rainstorm waterlogging exhibits clear spatiotemporal evolution. Most existing studies remain at the level of probability-driven node-failure assumptions and lack an in-depth analysis of the physical damage mechanism of the disaster. Consequently, they cannot accurately reflect the spatial impact of waterlogging on the power grid, and subsequent load-transfer strategies lack accurate scenario support (Chen and He, 2015).

At present, flexible resources such as microgrids, together with tie-switch operations, are mainly used for distribution-network load transfer and restoration under extreme disasters (Su et al., 2015). Existing studies often construct robust or stochastic optimization models to propose single-stage restoration strategies (Tang et al., 2020), or investigate dynamic reconfiguration schemes that account for changes in load status (Chen et al., 2015). Most current strategies implicitly assume that microgrids can provide continuous and stable support. However, under short-duration heavy-rainfall conditions, photovoltaic output is severely constrained, and microgrids rely almost entirely on internal energy storage for firm support. Existing studies generally lack staged countermeasures for the key time-varying feature that “microgrid energy storage gradually depletes from a fully charged state” during a disaster. If this dynamic process is ignored, the supporting power source may be exhausted in the later stage of load transfer, triggering secondary outages.

After candidate transfer paths are generated, a reasonable indicator system must also be established for optimal selection. Existing studies mostly generate transfer schemes based on topology and tie matrices, and evaluations are often limited to conventional indicators such as energy-not-supplied losses and system network losses (Zhang et al., 2022), or to comprehensive evaluation systems including voltage deviation and the number of operations (Su et al., 2026; Ma et al., 2020). However, existing evaluation systems are almost entirely limited to “steady-state operating benefits” after transfer and collectively overlook the “transient safety boundary” in practical grid operation. In actual dispatching, changing the network topology for load transfer inevitably involves a short loop-closing operating state of the tie switch. Without considering safety indicators such as high loop-closing current impacts generated at the instant of loop-closing operation, the selected path may face serious risks in practice, including protection maloperation or even switch damage.

To address these issues, this paper proposes a distribution network-microgrid coordinated two-stage load-transfer strategy considering rainstorm-waterlogging risk. First, the Chicago design storm and SCS-CN surface-runoff model are adopted to simulate short-duration heavy rainfall and finely quantify node inundation depth and outage risk. Second, considering the dynamic evolution of the power-supply capability of photovoltaic-storage microgrids, a two-stage transfer strategy is proposed: the first stage is jointly supported by microgrid energy storage and the main grid, and the second stage is independently supplied by the main grid after the energy storage is depleted. Finally, a set of transfer paths is generated using depth-first search (DFS), and a multidimensional evaluation system including loop-closing current, node voltage, and network loss is constructed. The entropy-weight TOPSIS method is used to comprehensively select the preferred scheme. Under the premise of ensuring grid operational safety, this paper aims to build a comprehensive economic-loss model covering user-side outage losses, power-department penalty costs, and additional network-loss costs. Through multi-scenario comparative quantification, the synergistic minimization of user-side losses, grid operating costs, and equipment-risk costs under waterlogging disasters is achieved, providing a decision-making reference for economic operation and maintenance of distribution networks under extreme disasters.

2 Flexible-resource planning framework under rainstorm waterlogging disasters

Rainstorm waterlogging disasters pose severe challenges to power-grid stability. Considering the dynamic fluctuation characteristics of microgrid power sources under disaster scenarios, and taking the outage risk caused by waterlogging damage to node-level distribution rooms as the key basis, this paper proposes a two-stage reliability-enhancement framework for load transfer that accounts for distribution network-microgrid coordination, as shown in Figure 1. In implementation, before the disaster, the complete rainstorm disaster process is simulated by modeling short-duration heavy rainfall and establishing a dynamic sewer-flow model and a surface-runoff evolution model. After the disaster occurs, the distribution network remains connected to the main grid. Its remote-control devices are controlled by the emergency center and can flexibly change the distribution-network topology.

• First stage: microgrid support is coordinated with distribution-network tie-switch operation to complete load transfer. First, based on the collected rainstorm-waterlogging disaster data, an inundation-degree indicator is used to conduct a detailed quantitative assessment of disaster severity and damage to the distribution rooms at distribution-network nodes. The fault scenario is then further determined, and the set of load-loss nodes is identified. Considering that photovoltaic output is very small during rainstorm disasters, this study assumes that the PV system has no output at this time, while the energy-storage units in the photovoltaic-storage microgrids remain fully charged.

During this process, feasible transfer paths are identified with the aid of a transfer-path search algorithm. The optimal transfer path for this stage is selected from the candidate paths according to preset evaluation indicators. Tie switches are then operated to restore power supply to faulted loads and complete the spatiotemporal allocation of electric-energy resources. The decision variable is the switching state of the tie lines.

• 2. Second stage: load transfer is completed in linkage with switch operation. When the energy stored inside the microgrid is depleted, the distribution network has only one power-supply source. According to the evaluation indicators, the transfer path obtained for this stage is selected, and tie switches are used to supply the faulted loads while maintaining the stability of the distribution network. The decision variable is the switching state of the lines.

3 Quantification of inundation-induced outage risk under urban rainstorm waterlogging disasters

3.1 Simulation of short-duration heavy rainfall

Rainfall occupies a central position in the urban water cycle. It is not only an important link in maintaining the urban water cycle, but also the core factor inducing potential urban waterlogging disasters. Rainfall data may come from measured rainfall records or be generated by rainfall models (Chen et al., 2015). Although measured data are highly reliable, they are difficult to collect, and their recording intervals are relatively long; therefore, they cannot satisfy the refined simulation and forecasting requirements for short-duration urban heavy rainfall. Under these circumstances, this study adopts a rainfall model for analysis and further carries out assessment of urban waterlogging disasters.

Considering that urban rainfall is mostly characterized by short-duration heavy rainfall, this study selects the rainstorm-intensity formula, as shown in Equation 1, to simulate the impact of rainfall on the city.where P_r is rainfall intensity (mm/min); T is the rainfall return period (a dimensionless constant, expressed in years); t_rain is rainfall duration (min); and A, b, C, and n are formula constants whose values differ among cities.

P r = A * 1 + C ⁡ lg ⁡ T 167 * t r a i n + b n (1)

3.2 Sewer-flow model

By comparing the water level in a sewer manhole with the ground elevation, it is possible to determine whether drainage or water replenishment is required. Specifically, when the hydraulic head H at the manhole is lower than the ground elevation z, i.e., H < z, drainage can be calculated. In this case, the hydraulic head at the manhole is naturally lower than the surface-water level, and the drainage and water-replenishment processes under different hydraulic-head conditions are calculated using Equations 2–4.where h is the water depth on the ground surface; c_w is the manhole-weir flow coefficient (taken as 1); w_w is the weir crest width; and g is gravitational acceleration (m/s²). Conversely, if the water head at the manhole rises and is greater than the ground surface elevation z but is still lower than the surface-water level, i.e., H > z, drainage can be calculated using the orifice equation:where c_o is the manhole-orifice flow coefficient (taken as one in this paper), and A_m is the manhole area (mm²). Finally, if the hydraulic head at the manhole exceeds the surface-water level, i.e., H > h + z, water replenishment can be calculated using the orifice equation:

q n , d = c w * w w * h * 2 g h (2)

q n , d = c o * A m * 2 g h + z − H (3)

q n , sur = c o * A m * 2 g H − h − z (4)

The shape of the opening affects the flow exchange between the sewer system and the ground surface. Although this study does not explicitly consider the influence of manhole-opening shape because of limited data, the inlet-shape effect can be incorporated by using flow coefficients, which directly influence the magnitude of flow exchange at the inlet.

3.3 Surface-runoff calculation

Runoff is a core influencing factor in urban-waterlogging disaster simulation and a key indicator for evaluating the effectiveness of sponge-city construction. The SCS-CN model developed by the U.S. Department of Agriculture can be used to calculate surface runoff. Its simple structure and convenient computation are the main reasons for its wide application. By quantifying corresponding parameters for different soil types and land-use types, the model completes surface-runoff calculation and can also be used to analyze the action mechanism of underlying-surface characteristics on surface runoff (Chen et al., 2015). where Q and p are runoff and rainfall (mm), respectively; I_a is the initial abstraction (mm), mainly including infiltration and evaporation. Since this paper only considers flooding caused by relatively short-duration heavy rainfall, evaporation can be ignored. lambda is the initial-abstraction coefficient, usually set to 0.2; S is the regional potential maximum retention (mm); and CN is the model parameter, a dimensionless constant ranging from 1 to 100.

Q = p − I a 2 p − I a + S p > I a 0 p < I a (5)

I a = λ * S (6)

S = 25400 C N − 254 (7)

The key infiltration parameter CN in the model is mainly affected by three factors: antecedent soil moisture, hydrologic soil condition, and land-use type. Antecedent soil moisture includes dry, normal, and wet soil-moisture conditions. Based on measured data, hydrologic soil groups can be divided into four categories. The infiltration capacity decreases from group A to group D. The specific classification criteria are shown in Table 1. Because different land-use types have different runoff infiltration capacities, their corresponding numerical values also differ.

TABLE 1. SCS-CN hydrologic soil group classification criteria (Zhang et al., 2022).

3.4 Quantification of inundation risk

This study constructs a rainstorm-waterlogging model to simulate the urban waterlogging disaster process. Inundation depth is selected as the core analysis indicator. Combined with the results of rainfall and surface-runoff simulation, the “equal-volume method” is adopted; that is, the rainfall volume is assumed to be equal to the total water volume in the inundated area, so that the inundation volume and related quantities are calculated using Equations 8–10 and inundation depth at distribution-network nodes can be estimated (Tseng et al., 2025).

Assuming that the city is a passive inundation area, the distribution-network region is divided into cells. Any cell whose terrain elevation is lower than the inundation elevation is included in the inundated region. The specific procedure for inundation-risk quantification is as follows: where V_flood is the flood volume in the study area (m³); s is the study-area area (m²); V_flow is the drainage volume in the study area (m³); v is the drainage rate (mm/h); and V_remain is the final inundation volume of the area (m³).

• The total rainfall is obtained from the short-duration rainstorm formula, and surface runoff is calculated using the SCS-CN model, as given in Equations 5–7.
• When calculating waterlogging volume, the drainage capacities of the urban underlying surface and pipeline network are first considered. The drainage volume of both facility types during the rainfall period is calculated, and the corresponding waterlogging volume is then obtained after subtracting both drainage volumes.
• Considering the terrain conditions of each distribution-network node, this paper combines terrain coefficients to obtain the rainstorm inundation depth of each load node.
• The natural-breaks method is adopted to classify inundated nodes by risk level, and the distribution of inundation risk in the distribution network under rainstorm waterlogging conditions is obtained.

V flood = Q * s (8)

V flow = s * v * t (9)

V remain = V flood − V flow (10)

4 Quantitative model of economic losses caused by outages

The economic losses caused by outages triggered by rainstorm waterlogging involve three levels: user-side outage losses, power-department penalty costs, and additional network-loss costs. To quantitatively compare the economic benefits of different transfer strategies, this section constructs a comprehensive economic-loss model to provide a computational basis for subsequent multi-scenario comparisons.

4.1 User-side outage losses

Different customer types have significantly different sensitivities to outages. Based on the zoning structure of the industrial, commercial, and residential areas in the distribution network, this paper uses the differentiated unit-outage-loss coefficient method for modeling, as shown in (Equation 11).where lambda_i is the unit outage-loss coefficient of the area to which node i belongs (CNY/kWh); P_i is the load lost at node i (kW), using the actual load value at the disaster occurrence time (10:00 a.m.) in Appendix A; Delta T_i is the actual outage duration at node i (h); and Omega_fault is the set of faulted nodes.

C u s e r = ∑ i ∈ Ω f a u l t λ i · P i · Δ T i (11)

Based on typical values reported in domestic studies on power-supply reliability assessment and outage loss estimation, and considering the load characteristics of the industrial, commercial, and residential areas in the distribution-network case (Li et al., 2011), the unit outage loss coefficients for the three user categories are set as shown in Table 2. Industrial-area loads are mainly composed of production equipment, power loads, and continuous-process loads, and are characterized by relatively high electricity demand and strong requirements for production continuity. Power outages may cause production-line shutdowns, equipment stoppages, raw-material waste, scrap of work-in-process products, and production schedule delays. In some continuous production processes, sudden power interruptions may even lead to equipment damage or high restoration costs. Therefore, the outage losses of industrial users are mainly reflected in production interruption losses and equipment-related risks. Accordingly, the unit outage loss coefficient for industrial users is set to 20 CNY/kWh in this paper.

TABLE 2. Unit outage-cost coefficients for three categories of users.

Commercial-area loads mainly include those of shopping malls, office buildings, restaurants, retail stores, financial services, and other public service facilities. These loads are characterized by concentrated business hours, high requirements for service continuity, and strong time sensitivity. Power outages may directly lead to transaction interruptions, customer loss, reduced service capacity, and decreased operating revenue. In some commercial premises, additional losses may also arise from the shutdown of lighting, communication, cashier systems, cold-chain equipment, and security systems. Because commercial users generally produce relatively high economic output per unit of electricity, and outages have a more direct impact on real-time business activities, the unit outage loss coefficient for commercial users is set to 30 CNY/kWh, which is higher than those of industrial and residential users.

Residential-area loads mainly consist of lighting, air conditioning, heating, communication, household appliances, and community public electricity consumption. The impacts of outages on residential users are mainly reflected in reduced living convenience and interruptions to basic electricity services. Outages may also cause elevator shutdowns, failure of access-control and security systems, food spoilage due to refrigeration interruption, and other problems, thereby resulting in certain living costs and public safety risks. However, compared with industrial production interruptions and commercial business disruptions, the direct economic losses caused by residential outages are generally lower. Therefore, the unit outage loss coefficient for residential users is set to 10 CNY/kWh in this paper. In summary, the three user categories differ significantly in load characteristics, electricity-use purposes, outage sensitivity, and forms of economic loss. The use of differentiated unit outage loss coefficients can more accurately reflect the economic consequences of outages for different load types under rainstorm-induced waterlogging disasters.

Referring to typical values in domestic power-reliability assessment literature, the unit outage-loss coefficients for the three customer categories are shown in Table 2.

In the two-stage transfer strategy proposed in this paper, the restoration time of each node changes dynamically with the energy-storage state. Therefore, the outage duration is not a fixed constant but is coupled with the two-stage transfer strategy. During the first stage, when microgrid energy storage provides support, some nodes restore supply immediately after the transfer operation is completed. During the second stage, after the storage is depleted, improper path switching may trigger secondary outages, further causing cumulative user-side losses.

4.2 Power-department penalty costs

According to the relevant provisions of the Measures for Power Supply Supervision, the power department must bear assessment and penalty responsibilities for outage incidents. The penalty cost consists of an energy-penalty term and a customer-penalty term, as calculated in (Equation 12).where alpha is the penalty coefficient for unit energy not supplied, set to 1.5 CNY/kWh; beta is the penalty coefficient for each customer outage, set to 50 CNY/customer; and N_customer is the total number of customers affected by the outage. The parameters of the power utility penalty cost model are summarized in Table 3.

C p e n a l t y = α ∑ i ∈ Ω f a u l t P i · Δ T i + β · N c u s t o m e r (12)

TABLE 3. Parameters of the power utility penalty cost model.

4.3 Additional network-loss cost

Different load-transfer path schemes change the network topology and thereby redistribute line power flows. Monetizing the additional network-loss rate makes it possible to quantify the influence of transfer-path selection on operating economy, as shown in (Equation 13).where P_l is the system network loss before transfer (kW); Delta T is the duration of transfer (h); and c_elec is the on-grid electricity price, set to 0.7 CNY/kWh.

C g r i d = N 2 · P l · Δ T · c e l e c (13)

4.4 Comprehensive economic loss

Combining the above components, the comprehensive economic loss is obtained as shown in (Equation 14).

C t o t a l = C u s e r + C p e n a l t y + C g r i d (14)

To ensure the additivity of all cost components in the comprehensive economic loss model, different types of losses are monetized and uniformly converted into economic costs. Specifically, the user-side outage loss is obtained by multiplying the unit outage loss coefficient, lost load, and outage duration, with the dimension of ((CNY/kWh) × kW × h = CNY). The power utility penalty cost consists of an energy-not-supplied penalty term and an affected-customer penalty term. The dimension of the energy-not-supplied penalty term is ((CNY/kWh) × kW × h = CNY), while that of the affected-customer penalty term is ((CNY/customer) × customer = CNY). The additional network-loss cost is calculated by multiplying the additional network-loss rate, the pre-transfer system network loss, the load-transfer duration, and the on-grid electricity price. Since the additional network-loss rate is dimensionless, its dimension is (kW × h × (CNY/kWh) = CNY). Therefore, all cost components in the model have the same monetary dimension and can be summed.

Meanwhile, the three types of costs correspond to different sources of economic impacts during load loss and load-transfer operations under rainstorm-induced waterlogging disasters. The user-side outage loss reflects the direct economic impact of outages on industrial, commercial, and residential users. The power utility penalty cost represents the equivalent cost borne by the power utility due to reduced supply reliability, customer complaint handling, and related assessment responsibilities. The additional network-loss cost reflects the impact of different load-transfer paths on the operating economy of the distribution network after network power flows are redistributed. These three costs characterize disaster-related losses from the user side, the power utility side, and the system operation side, respectively. They are economically independent and are all expressed in monetary terms. Therefore, combining them into a comprehensive economic loss indicator enables a more complete evaluation of the economic performance of different load-transfer strategies under disaster scenarios and provides a unified quantitative basis for subsequent multi-scenario comparative analysis.

5 Load-transfer path search based on active-power flow direction and depth-first search

Designing the optimal load-transfer path requires preprocessing the distribution-network topology and performing network-topology analysis. This paper uses a directed graph G = (V, E) to represent the distribution-network topology, where V = {n_1, n_2, …, n_N} is the vertex set and E = {l_1, l_2, …, l_M} is the edge set; N and M denote the numbers of vertices and edges, respectively. An adjacency matrix is used to describe the relationship between V and E in the directed graph G. For a directed graph, the adjacency matrix is an N x M matrix A_NM, and its elements are calculated as follows the elements of the adjacency matrix are defined in Equation 15:

a i j = 1 if an edge exists from node i to node j 0 otherwise (15)

This paper uses sparse-matrix technology to optimize the storage of the adjacency matrix. In the specific research scenario, the function of the adjacency matrix is limited to describing adjacency relationships among components within the system, and the related numerical calculations are relatively simple. Therefore, processing the matrix with sparse technology meets practical requirements and is reasonable and effective.

When a station device fails or is overloaded, fault-location technology is first used to identify the specific fault location and determine the central main transformer to which it belongs. The load-transfer problem can then be transformed into a directed-graph path-search problem (Chen et al., 2015; Su et al., 2026), namely, the search for a path from the power-source point to the set of downstream load nodes of the faulted line. In this paper, active-power flow direction is taken as the primary basis, and the depth-first search algorithm is used to obtain the transfer path with the best economy under comprehensive conditions. The detailed steps are as follows:

• Remove the faulted line section, close the primary or secondary tie switch of the main transformer in the adjacency matrix, and keep other tie switches in the network open. This effectively avoids redundant topology-search operations and further shortens the time required to read network-topology information.
• Starting from the main transformer to which the fault location belongs, take it as the initial search node. Based on the adjacency matrix, search for and mark the next-level nodes connected to the current visited node.
• Determine whether there are unmarked nodes among the next-level nodes of the current node that are connected to it. If such nodes exist, continue to visit the next-level node and mark it. Otherwise, return to the previous-level node and continue to determine whether there are connected and unmarked nodes among the previous-level nodes. If such nodes exist, continue the above operation.
• Based on the recursive algorithm, further repeat step (3) to search for similar nodes found in the previous iteration until the target node set required for the transfer path is reached.
• Add the required constraints, including transfer-path length and the number of tie lines traversed by the transfer path, and screen the searched transfer paths to obtain the path with the optimal comprehensive evaluation.
• Obtain the final candidate schemes.

The specific steps of the transfer-path search algorithm are shown in Figure 2.

6 Comprehensive analysis and evaluation of load-transfer schemes based on entropy-weight TOPSIS

6.1 Evaluation indicators for transfer paths

During load-transfer operations, nearby resources are dispatched preferentially to restore loads, and distribution-network and user-side resources are optimized to formulate a reasonable transfer strategy. Meanwhile, important loads associated with distribution-network nodes are excluded from the transfer adjustment range to ensure the safety and reliability of supply to important users and reduce overall economic losses. Under the above premise, this paper proposes load-transfer evaluation indicators based on the Power System Safe Operation Manual and uses them as optimization objectives for load transfer. The transfer paths obtained by search are comprehensively evaluated from reliability and economy perspectives to screen the optimal transfer path under each load scenario: where S is the rated capacity of the main transformer; S_L and S_NL denote the apparent power of the main transformer before and after the fault, respectively. When N_1 = 0, the change in main-transformer load rate is optimal. where P_l and P_rl represent the network-loss values in the main-transformer area before and after transfer, respectively. When N_2 = 0, the additional network-loss rate is optimal.

• N₁ denotes the main-transformer load-rate variation, which represents the difference between the load rate of the main transformer after load transfer and that before load transfer. A smaller value of N₁ indicates a smaller impact of the transfer operation on the main-transformer load-rate variation and the additional network-loss rate are calculated using Equations 16, 17, respectively.

N 1 = S N L − S L S (16)

• 2. N₂ denotes the additional network-loss rate, which represents the variation in network loss caused by load transfer. It is used to evaluate the impact of different transfer paths on the operating economy of the distribution network. A smaller value of N₂ indicates lower additional network loss.

N 2 = P l − P r l P l (17)

• 3. N₃ denotes the load loss, namely, the amount of load that still cannot be restored to normal power supply after load transfer. When N₃ = 0, the load-loss index reaches its optimal value, indicating that all transferable faulted loads have been restored.
• 4. N₄ denotes the voltage deviation, which is defined as the difference between 1.0 p.u. and the minimum per-unit voltage of load nodes after load transfer. When N₄ = 0, the voltage deviation reaches its optimal value, indicating that the node voltage is closest to the rated voltage.
• 5. N5 denotes the number of switching operations, which refers to the total number of tie-switch operations required during the load-transfer process. When N5 = 0, the switching-operation index reaches its optimal value, indicating that the number of tie-switch operations is minimized.
• 6. N6 denotes the loop-closing current, namely, the loop current generated in the closed loop formed after load transfer. When N6 = 0, the loop-closing-current index reaches its optimal value, indicating that no loop-closing current impact is imposed on the distribution network.

6.2 Indicator weight calculation based on the entropy-weight method

As an indicator-weighting method based on information entropy, the entropy-weight method quantifies the degree of variation in indicator data to assess information content and ultimately determine the weights of indicators in the comprehensive evaluation system. This method uses the dispersion characteristics of the data themselves as the judgment basis and therefore has strong objectivity. Information entropy can be used to measure the dispersion degree of indicator data. If the data under an indicator are more dispersed, the corresponding entropy value is lower, the indicator contains richer information, and the indicator has a greater effect on the comprehensive evaluation (Zhu et al., 2023). The steps for determining objective weights of evaluation indicators using the entropy-weight calculation procedure, including the original matrix, normalization, information entropy, and indicator weights, is given in Equations 18–23. where Y_ij denotes the normalized result of the evaluation data. where n is the number of load-transfer schemes, and j is the jth load-transfer indicator. where W_j is the final weight of each evaluation indicator and k is the number of indicators. The weight is calculated from the self-variation characteristics of the indicator data. When applied to the evaluation of distribution-network transfer schemes, this ensures the objectivity of indicator weights.

• Determination of the original sequence matrix. Let the total number of transfer-path schemes be n, and let each candidate scheme have m evaluation indicators. Taking x_ij as the jth indicator of the ith transfer scheme, the original sequence matrix is:

X = x 11 ⋯ x 1 m ⋮ ⋮ x n 1 ⋯ x m n (18)

• 2. Data normalization. Evaluation indicators have different assessment orientations and measurement units. To ensure that the original evaluation data can be used to calculate indicator weights, the indicator data must be normalized. The normalization formulas for positive and negative indicators are as follows:

Y i j = x i j − min x i max x i − min x i (19)

Y i j = max x i − x i j max x i − min x i (20)

• 3. Calculation of information entropy.

E j = − ln n − 1 ∑ i = 1 n p i j ⁡ ln ⁡ p i j (21)

p i j = Y i j ∑ i = 1 n Y i j (22)

• 4. Determination of the weights of each indicator. After the information entropy is obtained, the objective weight of each evaluation indicator is calculated using the following equation:

W j = 1 − E j k − ∑ j = 1 k E j (23)

6.3 Scheme priority determination based on TOPSIS

• Weighted matrix calculation. After the required weights corresponding to the commonly used evaluation indicators of transfer paths are obtained, the indicators in the known normalized matrix B are weighted to obtain matrix C. The TOPSIS procedure, including the weighted matrix, positive and negative ideal solutions, Euclidean distances, and relative closeness, is given in Equations 24–29.

c i j = b i j w j (24)

• 2. Determination of positive and negative ideal solutions. The calculation formulas for the positive and negative ideal solutions are as follows:

c j + = max ⁡ c j , for benefit attributes min ⁡ c j , for cost attributes , j = 1 , 2 , … , m (25)

c j − = min ⁡ c i j , for benefit attributes max ⁡ c i j for cost attributes , j = 1 , 2 , … , m (26)

• 3. Calculation of distances to the best and worst solutions. Euclidean distance is used to calculate the distances from each transfer scheme to the positive and negative ideal solutions, as follows:

d j + = ∑ j = 1 m c i j − c j + 2 , i = 1 , 2 , . . . , n (27)

d j − = ∑ j = 1 m c i j − c j − 2 , i = 1 , 2 , . . . , n (28)

• 4. Relative closeness calculation.

f i = d i − d i − + d i + , i = 1 , 2 , . . . , n (29)

Relative closeness can be used to comprehensively evaluate and rank transfer-path schemes. By calculating the relative-closeness value, the relative distance relationship between each transfer scheme and the positive and negative ideal solutions can be understood, and the superiority of each scheme can be analyzed.

The entropy-weight TOPSIS method combines the advantages of the entropy-weight method and the TOPSIS method. It not only considers the weights of each indicator but also evaluates the superiority of each object by calculating its distance from the ideal and negative ideal solutions. The evaluation results are both objective and comprehensive, enabling a more accurate reflection of the actual situation of the evaluated objects.

7 Case study

7.1 Parameter settings

This paper builds a distribution-network case based on the modified IEEE 69-bus system shown in Figure 3. In power-flow calculation, the main grid is treated as a generator. Nodes 27 and 35 each have a photovoltaic-storage microgrid, denoted as MG1 and MG2, respectively. Considering that PV output is very small under rainstorm disasters, this paper assumes that the PV units in the microgrids do not output power during the rainstorm-waterlogging disaster. The output power limits of the microgrid energy storage at nodes 27 and 35 are 400 kW and 200 kW, respectively.

As shown in Figure 3, the distribution network is divided into three regions: an industrial area, a commercial area, and a residential area. In the industrial area, node 39 has a switching station with two incoming feeders (L38 and L72), and its feeder nodes are 38, 39, 48, 32, and 35. In the residential area, node 21 has a switching station with two incoming feeders (L20 and L70), and its feeder nodes are 20, 21, 22, and 23. Although the switching stations guarantee reliable power supply for the distribution network, they also increase the difficulty of load transfer. Therefore, multiple factors must be comprehensively considered to obtain the optimal transfer path. The load characteristics of each region are shown in Figure 4, and the partition details are shown in Table 4.

TABLE 4. Partitioning details of the distribution-network test case.

7.2 Rainstorm waterlogging simulation

The duration of this rainstorm simulation is 3 h. The 3 h rainfall-intensity trend based on the Chicago design storm is shown in Figure 5. Based on the rainstorm-waterlogging model, the process of ponding generated under the effect of rainfall within 3 h can be obtained for the set distribution-network scenario.

A terrain coefficient is defined as a real number between 0 and one to describe the terrain condition of each load point in the distribution network. The inundation depth considering terrain effects is obtained by dividing the inundation depth without considering terrain effects by the terrain coefficient. The terrain coefficients of load points 1-69 are 0.6, 0.78, 0.51, 0.67, 0.4, 0.44, 0.52, 0.92, 0.73, 0.86, 0.72, 0.7, 0.32, 0.7, 0.92, 0.58, 0.64, 0.7, 0.75, 0.63, 0.65, 0.32, 0.93, 0.54, 0.85, 0.61, 0.74, 0.57, 0.6, 0.74, 0.62, 0.61, 0.77, 0.7, 0.31, 0.79, 0.65, 0.93, 0.88, 0.54, 0.78, 0.51, 0.67, 0.7, 0.71, 0.87, 0.46, 0.73, 0.59, 0.92, 0.87, 0.47, 0.91, 0.68, 0.91, 0.54, 0.78, 0.6, 0.64, 0.4, 0.85, 0.63, 0.79, 0.65, 0.93, 0.68, 0.71, 0.85, and 0.65.

After 1 h of rainfall, surface ponding begins to occur. In most areas, the ponding depth is less than 0.1 m, while only a small portion of the area has a ponding depth of 0.1–0.3 m. After 2 h of rainfall, the ponding depth in most areas reaches 0.1–0.3 m (Zhu et al., 2023), and only a very small portion of the area reaches 0.3–0.5 m, indicating a gradual increase in the disaster risk to distribution terminals. After 3 h of rainfall, the ponding depth in most areas of the distribution-network region remains within 0.1–0.3 m, while some areas reach 0.3–0.5 m. Distribution terminals at load points are generally installed at a certain elevation to prevent water inundation. According to typical design criteria for distribution terminals, the bottom of a distribution terminal should be 0.3–0.5 m above the ground Xiao et al., 2026. When the ponding depth exceeds the bottom elevation of a distribution terminal, equipment damage is likely to occur; therefore, the affected equipment should be promptly taken out of service for inspection and repair.

After the rainstorm process simulation is completed, the inundation depths of the 69 load points in the distribution-network case are obtained. Based on the inundation depth of load points and the sum of the inundation depths at both ends of each line, the natural-breaks method is used to divide load points and lines into five categories: high risk, relatively high risk, medium risk, relatively low risk, and low risk. Load points 13, 22, and 35 and lines 12, 13, 21, and 34 are high-risk elements. The inundation depths at load points 13, 22, and 35 all exceed 0.5 m. Therefore, the distribution terminals at load points 13, 22, and 35 are considered damaged, although power supply can be restored through load transfer. The fault distribution in the distribution-network case is shown in Figure 6.

As shown in Figure 6, nodes 13, 22, and 35 and lines 12, 13, 21, and 34 fail directly because of the rainstorm disaster, while load nodes 23–27 fail indirectly because they lose connection to the main grid. As rainfall continues, the inundation depth and extent in the distribution-network area gradually increase, and waterlogging progressively worsens. Terrain factors have a significant influence: the lower the terrain of the distribution terminal, the higher the inundation risk. Therefore, flood-prevention construction for distribution terminals in medium- and high-risk areas should receive particular attention.

7.3 Path-evaluation results

7.3.1 First stage

The disaster occurs at 10:00 a.m. The load at each node is listed in Appendix A. The energy storage in the microgrids is fully charged. Based on the transfer-path search algorithm, the transfer paths for the distribution-network case are obtained. When transferring load for nodes 22–27, tie switch 73 or 74 must be closed. For transfer schemes in which tie switches 73 and 74 are not closed simultaneously, the evaluation result of paths closing tie switch 73 is significantly better than that of paths closing only tie switch 74. The transfer-path schemes are screened again, and paths that close only tie switch 74 are removed. Finally, 12 transfer-path schemes are obtained, as shown in Table 5.

TABLE 5. Path search results of the distribution network.

Basic power-flow calculation and loop-closing-current calculation are performed for each transfer path, and the node voltage magnitudes, network losses, loop-closing currents, and other values of each scheme are obtained. The evaluation-index values for each scheme are shown in Table 6.

TABLE 6. Evaluation-index values of load-transfer schemes in the first stage.

Table 6 shows that the voltage magnitudes at some nodes of schemes 4, 6, and seven are lower than 0.95 and therefore fail to satisfy voltage-stability requirements. Thus, schemes 4, 6, and seven are removed in subsequent evaluation, and the evaluation-index values of the remaining schemes are normalized, as shown in Table 7.

TABLE 7. Normalized evaluation-index values of transfer schemes in the first stage.

During load transfer, minimizing system load loss is generally the primary objective. Therefore, the load loss of all transfer schemes in Table 7 is 0, and no load loss occurs. The main-transformer load-rate change is also 0. Consequently, the main-transformer load-rate-change indicator and the load-loss indicator are omitted when evaluating the transfer schemes.

All evaluation indicators are cost-type indicators. Therefore, when the entropy-weight method is used to calculate objective weights, the cost-type indicator calculation method is adopted. Based on (16), the objective weight vector under the entropy-weight method is obtained as:

W = (0.1361, 0.1732, 0.2656, 0.4251).

Based on the above objective weights, the indicators in the normalized matrix B are weighted, yielding the weighted vector:

C = (0.093, 0.3333, 0.1802, 0.4421, 0.3148, 0.2469, 0.8683, 0.5742, 0.3333).

Based on the TOPSIS method, the relative closeness values of the transfer schemes are:

F = (1, 0.6901, 0.8876, 0.5498, 0.714, 0.8016, 0, 0.3793, 0.6901).

The evaluation indicators in this paper are all minimum-type indicators and are not converted into maximum-type indicators. Therefore, a smaller relative-closeness value indicates a better scheme. According to the relative-closeness results, scheme 10 is the optimal transfer path. It requires only four tie-switch operations and does not form a loop, so no loop-closing current impact is imposed on the distribution network. Scheme 11 ranks second. Compared with scheme 10, it adds one switch operation and forms one loop; however, compared with other loop-forming schemes, scheme 11 produces the smallest loop current. Because line 21 loses supply capability due to the fault, the switching station at node 21 provides more transfer-path options for load nodes 22–27, improving supply reliability. The energy storage at node 35 can supply only 200 kW of load at that node. Without the switching station, part of the load at node 35 would have to be curtailed.

7.3.2 Second stage

It is assumed that the energy stored at nodes 27 and 35 is depleted after 2 h. At this time, the distribution network is supplied only by the main grid. The searched transfer paths are the same as those in the first stage. Basic power-flow calculation and loop-closing-current calculation are carried out again to obtain the node voltage magnitudes, network losses, loop-closing currents, and other values of each scheme. The second-stage transfer paths are evaluated, and the evaluation-index values of each scheme are shown in Table 8.

TABLE 8. Evaluation-index values of load-transfer schemes in the second stage.

In Table 8, the voltage magnitudes at some nodes of schemes 4-8, 10, and 11 are lower than 0.95, and those of schemes 4, 6, and seven even fall as low as 0.92, indicating poor voltage stability. Therefore, schemes 4-8, 10, and 11 are removed from subsequent evaluation. The evaluation-index values of the remaining second-stage schemes are normalized, as shown in Table 9.

TABLE 9. Normalized evaluation-index values of load-transfer schemes in the second stage.

Table 9 shows that schemes 2 and 12 have identical values for all evaluation indicators, but the differences among transfer schemes under each indicator are relatively large, especially in terms of loop-closing current. The entropy-weight method is then used to calculate the objective weights of the indicators, yielding the objective weight vector:

W = (0.2405, 0.2191, 0.2408, 0.2996).

Based on the above objective weights, the indicators in normalized matrix B are weighted to obtain the weighted vector:

C = (0.28, 0.7269, 0.4504, 0.5404, 0.7269).

Based on the TOPSIS method, the relative closeness values of the transfer schemes are:

F = (1, 0, 0.6188, 0.4175, 0).

According to the above relative-closeness results, schemes 2 and 12 are the optimal transfer paths. Among the five evaluated transfer schemes, schemes 2 and 12 have the smallest additional network-loss-rate and switch-operation-number indicators and are therefore the best transfer paths for this stage.

After the energy storage in the microgrids is depleted, only the main grid supplies electricity. At this time, the first-stage optimal transfer path, scheme 10, has node voltage magnitudes below 0.95 and is not suitable for load transfer. Therefore, while considering microgrid power-supply support, it is also necessary to consider changes in the state of distributed power sources within the microgrid. The transfer path should be selected comprehensively and staged transfer should be implemented for different scenarios to maximize system reliability.

7.4 Comparative analysis of economic benefits

To quantitatively verify the economic benefits of the two-stage transfer strategy proposed in this paper, three comparative scenarios are established according to the comprehensive economic-loss model developed in Section 3. Scenario A performs no transfer operation, and all faulted nodes are without power for the entire 3 h, serving as the benchmark upper bound for economic losses. Scenario B implements single-stage pure main-grid transfer and ignores the time-varying characteristics of microgrid energy storage; scheme 10 is used in both the first and second stages. However, as shown in Table 8, after the storage is depleted in the second stage, scheme 10 causes the node voltage magnitude to fall below 0.95, triggering secondary outages. The secondary-outage loss is superimposed on the original loss. Scenario C is the two-stage distribution network-microgrid coordinated transfer strategy proposed in this paper. Scheme 10 is implemented in the first stage, and the strategy switches to scheme 2 or scheme 12 in the second stage. Voltage stability is maintained throughout, and no secondary outage occurs.

Based on the load data of each faulted node in Appendix A and substituting them into the comprehensive economic-loss model, the economic-loss results of the three scenarios are shown in Table 10.

TABLE 10. Comparison of comprehensive economic losses across three scenarios.

Table 10 shows that the comprehensive economic loss of the two-stage transfer strategy proposed in this paper (Scenario C) is CNY 19,806. Compared with the no-transfer Scenario A, it saves CNY 38,307 (65.9%); compared with the single-stage strategy Scenario B, it saves CNY 15,320 (43.6%).

The economic cost of Scenario B deserves particular attention. Because the time-varying feature of energy-storage depletion is ignored, using scheme 10 in the second stage leads to node voltage-limit violations and triggers secondary outages. Its user-side outage loss is three times that of Scenario C. This result directly quantifies, from an economic perspective, the cost of “ignoring storage time variability,” and verifies the necessity and economic rationality of staged evaluation in the proposed two-stage strategy.

7.5 Sensitivity analysis of key assumptions

Since photovoltaic output, the initial state of charge of energy storage, the energy storage depletion time, and rainfall intensity all affect the available support capability of the microgrid and the range of nodal load loss, a sensitivity analysis is conducted in this section to further demonstrate the robustness of the proposed two-stage load-transfer strategy. Scenario C in Section 6.4 is taken as the baseline scenario. The photovoltaic output ratio, initial state of charge of energy storage, sustainable support duration of energy storage, and rainfall intensity are varied to analyze their impacts on load-transfer path selection and comprehensive economic losses.

Specifically, the photovoltaic output ratio is set to 0%, 10%, and 20%, representing three operating conditions: extreme rainstorm, weak solar irradiance, and partial photovoltaic availability. The initial state of charge of energy storage is set to 60%, 80%, and 100% to analyze the impact of pre-disaster energy-storage reserve levels on load restoration. The sustainable support duration of energy storage is set to 1 h, 2 h, and 3 h to reflect differences in depletion time under different energy-storage capacities and support power levels. Rainfall intensity is perturbed by changing the rainfall return period or total rainfall amount, so as to analyze the influence of increased or reduced waterlogging risk on the identification of faulted nodes.

The sensitivity analysis results show that variations in these parameters do not change the basic applicability of the proposed two-stage load-transfer strategy, but they affect the switching time between the two stages, the feasibility of candidate load-transfer paths, and the level of comprehensive economic losses. When the photovoltaic output ratio or the initial state of charge of energy storage increases, the available support capability of the microgrid is enhanced, the depletion time of energy storage is delayed, and some loads can receive longer power-supply support during the first stage, thereby reducing user-side outage losses. Conversely, when the initial state of charge decreases or the energy storage depletion time shortens, the system enters the main-grid-only supply stage earlier. If the first-stage load-transfer path continues to be used, nodal voltage-limit violations or secondary outage risks may occur. Therefore, load-transfer paths should be dynamically adjusted according to changes in the energy-storage state.

In addition, as rainfall intensity increases, the nodal inundation depth and the range of high-risk nodes expand, and the number of faulted nodes and indirectly affected loads may increase, resulting in greater load-transfer pressure and higher comprehensive economic losses. In this case, relying solely on a single-stage load-transfer scheme makes it difficult to ensure operational security throughout the entire process. In contrast, the two-stage strategy considering the dynamic evolution of energy storage can adjust load-transfer paths in a timely manner according to changes in microgrid support capability, thereby exhibiting better operational adaptability. Therefore, the proposed strategy shows a certain degree of robustness with respect to key parameters such as photovoltaic output, energy-storage state, and rainfall intensity. It also indicates that, in practical applications, the two-stage switching time and load-transfer paths should be updated in a rolling manner by considering the pre-disaster state of charge of energy storage, real-time photovoltaic output forecasts, and rainfall intensity forecasts.

8 Conclusion

This study focuses on the load loss caused by the inundation and forced outage of node-level distribution rooms in urban distribution networks under rainstorm-induced waterlogging disasters. The operational feasibility and economic performance of load-transfer paths are investigated under coordinated distribution-network and microgrid operation. Considering the limited photovoltaic output during disasters and the gradual decline in the support capability of microgrid energy storage over time, a two-stage load-transfer strategy accounting for the dynamic evolution of energy storage is proposed. An integrated analytical framework is further developed, including rainstorm-induced waterlogging risk quantification, load-transfer path search, multi-criteria comprehensive evaluation, and economic loss assessment. The main conclusions are as follows:

• A method for quantifying nodal load-loss risk in distribution networks under rainstorm-induced waterlogging disasters is established. The Chicago design storm model is used to simulate short-duration heavy rainfall, and the SCS-CN surface runoff model is adopted to calculate regional runoff. In addition, terrain coefficients are introduced to account for their influence on nodal inundation depths, enabling refined assessment of inundation risk at distribution-network nodes. The case study shows that rainstorm-induced waterlogging may cause some distribution terminals to be taken out of service when inundation depths exceed their safety thresholds, which further leads to indirect load loss at adjacent nodes. This method couples the physical evolution of rainstorm-induced waterlogging with distribution-network fault scenarios, thereby providing more realistic scenario support for post-disaster load-transfer decision-making.
• The coordination between the two stages, namely, “microgrid energy storage supplying the distribution network as a distributed power source” and “main-grid-only supply after microgrid energy storage depletion,” effectively optimizes microgrid-assisted load-transfer paths under urban waterlogging disasters.
• A load-transfer path generation and selection method integrating active-power-flow direction analysis, depth-first search, and the entropy-weighted TOPSIS method is developed. Candidate load-transfer paths are generated using the depth-first search algorithm, and a comprehensive evaluation system is constructed by incorporating nodal voltage, additional network loss, the number of switching operations, and loop-closing current. This enables multidimensional comparison among different load-transfer schemes. Compared with conventional methods that mainly consider steady-state operating indicators such as network loss or voltage, the proposed method incorporates loop-closing current into the path evaluation process, which can reflect the transient safety risk during the momentary loop-closing operation of tie switches. This improves the operational security and engineering feasibility of the selected load-transfer path.
• A comprehensive economic loss model for disaster scenarios is constructed to quantitatively evaluate the economic benefits of different load-transfer strategies. The model incorporates user-side outage losses, power utility penalty costs, and additional network-loss costs, thereby providing a relatively comprehensive representation of the economic impacts of load loss and load-transfer operations under rainstorm-induced waterlogging disasters. Simulation results based on a modified IEEE 69-bus distribution network show that the proposed two-stage distribution-network–microgrid coordinated load-transfer strategy reduces comprehensive economic losses by 65.9% compared with the no-transfer baseline scenario and by 43.6% compared with the single-stage strategy that ignores the time-varying characteristics of energy storage. In addition, because the single-stage strategy neglects the energy storage depletion process, it leads to voltage-limit violations and secondary outages in the second stage, resulting in significantly higher user-side outage losses than those of the two-stage strategy. These results economically verify the effectiveness and necessity of the proposed staged load-transfer strategy.

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Footnotes

Edited by: Yu Jin, Tianjin University, China

Reviewed by: Jiayan Liu, Hunan University, China

Yiqun Zou, Central South University, China