After the distributed photovoltaic power source is connected to the distribution network, in order to observe the changes in system network losses, a simple model as shown in Figure 1 is established by referring to the connection form of the distribution network topology.

In Figure 1, d 1 represents the distance between the power station and the distributed power source, d 2 represents the distance between distributed power sources and load points, I 、 I ′ shows the output current of photovoltaic power source and power station, I ″ means the load point input current.

The relationship between the three current parameters marked in the distribution network model can be expressed as: I ″ = I + I ′ (1)

Based on the parameters given in Formula 1, two types of key network losses generated during the operation of distributed photovoltaic access distribution networks can be derived (Dengfeng et al., 2024) A 1 = 3 R d 1 I ′ 2 A 2 = 3 R d 2 − d I ″ 2

In the formula, A 1 、 A 2 stands for the network losses exhibited from the power station to the distributed power source section and from the distributed power source to the load section, respectively, R means unit resistance.

By combining the two types of losses, the comprehensive losses generated by the operation of distributed photovoltaic access to the distribution network can be calculated (Wan et al., 2022). A ^ = A 1 + A 2

In the formula, A ^ indicates comprehensive loss

When constructing the time of use electricity price optimization model, which accounts for the losses caused by distributed photovoltaic integration into the distribution network, the first optimization objective function is defined as follows: f 1 = min A ^ = min A 1 + A 2 (2)

In the equation, f 1 is the mathematical model of TOU price optimization with the goal of minimizing network loss, min means the minimum value function.

Subsequently, starting from the three aspects of load variance, equivalent load peak valley difference, and user satisfaction, the optimization objective functions of time of use electricity price are given respectively: f 2 = min ⁡ σ 2 = min 1 T ∑ t = 1 T β t − β ¯ t (3) f 3 = min max ⁡ Q − min ⁡ Q (4) f 4 = max b 1 − ∑ t = 1 T β t − β t ′ β t + 1 − b 1 − ∑ t = 1 T H t − H t ′ H t (5)

In the given, f 2 is the mathematical model of TOU price optimization aiming at load variance minimization, f 3 、 f 4 respectively represent the mathematical models for optimizing time of use electricity prices with the objectives of minimizing the peak valley difference of equivalent loads and maximizing user satisfaction, max A is the maximum value function, σ represents the variance of load, T is the period of time, t indicates the time point, β 、 β ′ 、 β ¯ respectively represent the optimized electricity consumption, pre optimized electricity consumption, and average optimized electricity consumption, Q represents equivalent load, H ′ 、 H respectively stand for the user’s electricity costs before and after optimization, b represents the weight.

Combine the four objective functions given in Formulas 2–5 to generate a multi-objective optimization model for time of use electricity pricing.

To ensure the smooth implementation of the time of use electricity price optimization method obtained by the multi-objective optimization model, considering the requirements of electricity pricing optimization (He et al., 2022), time of use electricity price optimization constraints are proposed for the three aspects of power supply company revenue, electricity consumption, and user satisfaction to assist in the subsequent optimization model solving. Changes in power supply companies can directly impact residents' electricity costs and consumption, in order to avoid affecting people’s normal daily activities. In the process of optimizing time of use electricity prices, it is necessary to ensure stable revenue (Li et al., 2022). B 1 ≤ ∑ t = 1 T C t β t ∑ t = 1 T C t ′ β t ′ ≤ B 2

In the formula, B 1 、 B 2 respectively represent the minimum and maximum set values of the revenue ratio after the implementation of the time-of-use electricity pricing strategy, while C ′ 、 C respectively stand for the electricity prices before and after optimization.

At the same time, in order to ensure the stability of power consumption and user satisfaction in the distribution system, time of use electricity price optimization constraints are set for both, as shown in Formulas 6, 7 (Li et al., 2022). G 1 ≤ ∑ t = 1 T β t ∑ t = 1 T β t ′ ≤ G 2 (6) X 1 ≤ ∑ t = 1 T C t ∑ t = 1 T C t ′ ≤ X 2 (7)

In this formula, G 1 , G 2 represents the interval of the total electricity consumption change for users before and after the optimization of time-of-use electricity pricing, and X 1 , X 2 represents the interval of the ratio of total electricity prices before and after optimization.

After determining the constraints of time of use electricity price optimization in the distribution network, an improved imperialist competition algorithm combined with Tent chaotic reverse learning algorithm (Zhang et al., 2022) is used to solve the multi-objective optimization model and generate the optimization setting results of time of use electricity price. The imperialist competitive algorithm (ICA), proposed by Atashpaz and Lucas in 2007, is an intelligent optimization algorithm based on the mechanism of imperialistic competition (Atashpaz-Gargari and Lucas, 2007). Compared to other optimization algorithms, ICA demonstrates superiority in terms of computational efficiency and optimization performance. In the actual solving process, multiple parameters such as maximum iteration times, number of empires, and number of colonies are set based on the daily load data and line operation parameters of the distribution network. Each time of use electricity price optimization scheme is treated as an initialization population individual (Bai et al., 2021), and the chaotic sequence values corresponding to each individual in the initial population are obtained through Tent mapping operation (Bai et al., 2021). γ g + 1 = 2 γ g 0 ≤ γ g ≤ 0.5 2 1 − γ g 0.5 ≤ γ g ≤ 1

In the formula, g represents the country code, and γ represents the chaotic sequence value.

By inverse mapping the chaotic sequence using Formula 8, the initial position of individual countries can be derived as follows (Bai et al., 2021): V g = η + γ g η ′ − η (8)

In the equation, V represents the initial position of the countryand η ′ , η represents the search range for the optimal solution of the multi-objective optimization model. After the Tent mapping is completed, the top ranked countries within the national population are selected to serve as empires, while the remaining countries are divided into different countries as colonies, ultimately forming multiple empire groups (Wang et al., 2023). The number of colonies under each empire can be expressed as (Wang et al., 2023): K j = 1.3 max 1 ≤ a ≤ n μ a − μ j max 1 ≤ a ≤ n μ a > 0 0.7 max 1 ≤ a ≤ n μ a − μ j max 1 ≤ a ≤ n μ a ≤ 0 M j = K j ∑ a = 1 n μ a N

In the formula, j represents the empire number, n is the number of countries, μ 、 K means the empire’s power and relative power respectively, M represents the number of colonies under the empire and N represents the total number of colonies to be divided. In the above scenario, referring to the crossover and mutation operation modes of genetic algorithm, the analysis of colonial assimilation and movement towards the empire, imperial reform, and colonial revolution is completed. The specific operation modes are shown in Figures 2, 3.

Through assimilation and reform, the colonies within the empire are replaced to increase group diversity. On this basis, multiple empire groups are controlled to compete with each other. If the competition fails, the empire will perish. If the competition succeeds, the remaining colonies will be merged to enhance the empire’s power value until the algorithm termination condition is met. The empire with the highest current power value will be found, and its corresponding time of use electricity price optimization scheme will be the optimal solution of the model.

After considering the losses of distributed photovoltaic access to the distribution network, a new time of use electricity price optimization method was designed. In order to test the practical application effect of this method, referring to the distribution network structure shown in Figure 4, a power experimental system consisting of 33 nodes was built using an experimental institution.

At nodes 17 and 28, distributed photovoltaic power sources were respectively connected, and the final experimental scene is shown in Figure 5.

All nodes in the distribution network, except for node 0, are treated as residential load nodes. The typical daily load data of residents, as shown in Table 1, were obtained from the power supply bureau and injected into the distribution network. A simulation of the normal operation mode of distributed photovoltaic integration into the distribution network was then performed.

From the daily load data provided in Table 1, it can be seen that there are significant differences in residential electricity consumption at different times. In order to facilitate the optimization of time of use electricity prices, the entire operation period of the distribution network is divided into three periods. Among them, the peak period is from 07:00 to 11:00 and from 17:00 to 21:00, the valley period is from 24:00 to 03:00 and from 03:00 to 06:00 the next day, and the normal period is from 12:00–16:00 and from 22:00 to 23:00.

Apply the newly designed time of use electricity price optimization method, time of use electricity price optimization method based on multiple cost structure, and time of use electricity price optimization method based on time-varying price elasticity matrix to experimental scenarios, generate new time of use electricity price formulation strategies, and implement the new strategies to test the application performance of various methods.

When analyzing the losses of photovoltaic access to the distribution network, the access capacity of two distributed photovoltaic power sources is set to be fixed and unchanged, while the power factors are taken as 1, 0.9, 0.8, 0.7, and 0.6 respectively. The changes in network losses under different power factors are observed, and the statistical results shown in Figure 6 are obtained.

From Figure 6, it can be seen that when the power factor value is 0.6, the maximum loss of distributed photovoltaic access to the distribution network reaches 263 kW. When the power factor is 0.8, the active power loss of the power grid is only 210 kW, and the operating loss of the distribution network is the lowest at this time. According to the analysis results, adjust the topology structure of the distribution network in the experimental scenario, and based on this, carry out subsequent optimization of time of use electricity prices.

After constructing a multi-objective time of use electricity price optimization model, the optimal solution of the model can be obtained by improving the imperialist competition algorithm. In the process of determining the optimal time of use electricity pricing plan, the parameter settings related to improving the imperialist competition algorithm are shown in Table 2.

Complete the basic parameter layout according to Table 2, and based on this, solve the multi-objective time of use electricity price optimization model, and finally obtain the optimal solution variation curve as shown in Figure 7.

As shown in Figure 7, after 30 repetitions of the improved imperialist competition algorithm, the optimal solution obtained meets the requirement of stable convergence. After meeting the termination criteria for iteration, an optimized time of use electricity pricing plan is obtained, which is 0.5 yuan (kW⋅h)⁻¹ during normal periods, 0.245 yuan (kW⋅h)⁻¹ during valley periods, and 0.81 yuan (kW⋅h)⁻¹ during peak periods.

From the above tests, it can be seen that the new research on time of use electricity price optimization method is feasible. Subsequently, two additional methods were applied to optimize electricity pricing, and the optimization strategies proposed by the three methods were implemented separately, resulting in the daily load curve of the distribution network shown in Figure 8.

From Figure 8, it can be seen that after implementing the time of use electricity price optimization method that takes into account the losses of distributed photovoltaic access to the distribution network, the peak to valley difference of the power load curve is only 350 MW. Compared with the peak to valley difference of 574 MW before optimization, and the peak to valley differences of 502 MW and 534 MW presented by the other two methods after optimization, there has been a significant reduction. This proves that the new research optimization method has superior peak shaving and valley filling effects.