The TOU electricity pricing mechanism influences the shifting of electricity load. By implementing user load transfer, it is possible to make significant improvements to load characteristics and optimize the user load curve. The paper employs the classical load transfer model to effectively characterize the user load as a piecewise linear function, which is segmented into the dead zone, linear zone, and saturated zone (LIN, 2015). In the region of low demand, the disparity in electricity prices is minimal, thereby failing to elicit a significant response from users. Conversely, in the linear region, the user’s reaction is expected to exhibit a positive correlation with the extent of the electricity price differential. However, in the saturated region, users are anticipated to cease responding to further increases in the price differential due to the restricted capacity for load transfer, thereby reaching the limit of user responsiveness.

According to the aforementioned theory, the day is segmented into peak, normal, and valley periods. According to the division of periods, the load transfer models can be categorized into peak period to normal period, peak period to valley period, and normal period to valley period, as illustrated in Figure 1.

The function expression for the load transfer model is as follows: γ i = 0 , 0 ≤ Δ p r i ≤ α i Δ p r i γ i max / β i − α i , α i < Δ p r i ≤ β i γ i max , Δ p r i > β i (1)

In the given equation, i = 1 , 2 , 3 represent the peak period to the normal period, the peak period to the valley period, and the normal period to the valley period. Additionally, γ i represents the load transfer rate, Δ P r i represents the electricity price difference, γ i max represents the maximum value of the load transfer rate, α i and β i respectively represents the upper and lower limits of the price difference in the linear region of the load transfer model.

The description of the load transfer model between each period is now complete, as represented by Eq. 1. Following the price optimization, it is necessary to consider the transfer of the load. Consequently, by the user load transfer characteristic model presented above, Eq. 2 illustrates the load value for each period under the optimized TOU electricity price strategy (Domingo et al., 2011). p load _ i = p load _ i 0 − γ 1 p ¯ load _ f 0 − γ 2 p ¯ load _ f 0 , i ∈ T 1 p load _ i 0 − γ 3 p ¯ load _ p 0 + γ 1 p ¯ load _ f 0 , i ∈ T 2 p load _ i 0 + γ 2 p ¯ load _ f 0 + γ 3 p ¯ load _ p 0 , i ∈ T 3 (2)

In the equation: p load _ i 0 and p load _ i are the load of period i under the TOU electricity price before and after optimization; p ¯ load _ f 0 and p ¯ load _ p 0 are the average load of the peak period and the normal period before the TOU electricity price optimization; T1, T2 and T3 represent the collection of peak, normal and valley periods, respectively.

In the given equation, p load _ i 0 and p load _ i represent the load during the period i under the TOU electricity price before and after optimization, p ¯ load _ f 0 and p ¯ load _ p 0 represent the average load during the peak and normal periods before the TOU electricity price optimization. T1, T2, and T3 denote the sets of peak, normal, and valley periods, respectively.

To achieve the investment benefits of the power grid and reduce users’ electricity costs, this section aims to minimize peak load and peak-valley differences in load by considering load transfer characteristics in the user response model Eqs 1, 2. The model also incorporates user-side and grid-side income as constraints and constructs Eqs 1–14 for the optimization of TOU electricity pricing.

① Objective function:

In the given equation, P load represents the load vector under the optimized TOU electricity price, where the symbol ≜ is “equivalent to” indicates that the right-hand symbol is equivalent to the left-hand equation. Both w1 and w2 serve as weight coefficients. The first component of the objective function denotes the load peak value under the optimized price, while the second component represents the load peak-valley difference under the optimized price. By modifying the weight coefficient, a balance is achieved between the two objectives.

② Constraint conditions:

Where E 0 and E A are the power company’s revenue from electricity sales before and after the implementation of TOU electricity price optimization. λ i 0 and λ i are TOU electricity prices under period i before and after optimization. h i is the marginal cost of power generation of the system in the period i . Additionally, the study considers the concatenated set of all periods, i.e., I = T 1 ∪ T 2 ∪ T 3 .

 2) Customer-side benefit constraint: To prevent an increase in the cost of electricity for the customer resulting from changes in electricity prices, it is necessary to establish that following the TOU electricity price optimization, the average cost of electricity to the customer does not exceed the pre-optimization cost, as represented by Eq. 8:

The optimized electricity prices for the peak hour, the usual hour, and the valley hour are denoted by λ f , λ p and λ g respectively.

 4) The constraint on the difference between peak and valley prices is essential to maintain the rationality of the TOU electricity pricing strategy. It is typically limited to a specific range, as represented by Eq. 10.

Where τ max and τ min represent the upper and lower limits of the peak-to-valley electricity price scaling factor.

 5) Generation cost constraint: To prevent financial losses for the grid company due to the introduction of the new TOU electricity pricing strategy, it is necessary to establish that the optimized TOU electricity price exceeds the marginal cost of electricity generation, as represented by Eq. 11:

Where P load 0 represents the load vector be located in the context of TOU electricity pricing before any optimizations? Equations 1–14 depict essential models for price optimization, which are capable of reducing price peaks and filling troughs.

Furthermore, the exclusion of grid investment costs in Eq. 5 limits the optimization space for TOU electricity prices to only the grid company’s power sales revenue. This limitation hinders the achievement of enhanced load characteristics, reduced grid company investment costs, and lower electricity consumption costs for customers. Therefore, to address the aforementioned issues, this section enhances Eq. 5 and formulates a model for sharing the benefits of grid investment. The objective is to minimize the grid company’s investment expenses by devising a viable TOU electricity pricing strategy. The goal is to minimize the investment expenditure of the power grid company by designing a feasible time-of-use electricity price strategy, as shown in Eqs. 15–16. Additionally, a portion of the reduced investment costs is allocated to benefit the users, thereby broadening the scope for optimizing the TOU electricity pricing and achieving a mutually beneficial outcome for both the grid company and the users. The specific enhancements are outlined below: E A + 1 − ∂ Δ H ≥ E 0 (15) H 0 − Δ H = H (16)

Where ∂ represent the concession coefficient, denoting the proportion of the investment cost saved by the power grid company that translates to profit for the users; Δ H represents the investment cost saved by the power grid company after the optimization of TOU electricity price; H represents the investment cost saved by the power grid company after the optimization of TOU electricity price; H0 represents the investment cost of the power grid company before the optimization of TOU electricity price.

H in Eq. 16 can be determined using Eqs 17–24 from the grid investment planning optimization model, which is widely employed in industry. This model takes into account both conventional units and each new energy field station, as outlined in (LIN, 2015).

Objective function: min ⁡ H = α α + 1 Y α + 1 Y − 1 ∑ l ∈ N l c l C l L l + ∑ y = 1 Y ∑ t = 1 T ∑ g = 1 N g a g P f , g , t Δ t + ∑ y = 1 Y ∑ t = 1 T ∑ w = 1 N w χ w P w , t * Δ t (17)

Where H represents the total cost of the grid, the first term denotes the annual equipment investment of the grid, the second term signifies the operation cost of thermal power, hydropower, and other traditional units, and the third term indicates the penalty cost of new energy sources such as abandoned wind and light. α represents the discount rate, c l stands for the investment cost of the line per unit of capacity per unit of length, L l represents the length of the lth branch, a g represents the offer of the g t h traditional unit, χ w represents the penalty cost of the unit of energy abandonment, C l represents the l t h branch’s capacity, P f , g , t represents the power of the g t h conventional unit in the time period t , P w , t * represents the amount of energy discarded by the w t h new energy station in the time period t , N l represents the set of routes to be optimized, N g and N w represent the number of conventional units and new energy stations, respectively, and Y represents the number of years of planning; T represents the total number of time periods, while Δ t represents the interval between time periods.

① Constraint conditions:

Where t = 1 , 2 … , T ; N l o a d represents the set of nodes accessing the load; P w , t represents the actual power of the w t h new energy station in time period t; P n , load , t represents the load of the nth node in time period t.

 2) Branch power constraint

Where t = 1 , 2 … , T ; l = 1 , 2 … , Nl 0 ; Nl0 represent the total number of branch circuits in the grid; Cl0 represents the capacity of branch circuit l.

 3) Branch-planned capacity constraints

Where ∈ N l ; C l max and C l min are the upper and lower limits of the plannable capacity of branch l respectively.

 4) Conventional unit constraint

Where t = 1,2…,T; g = 1,2…,Ng; P f , g , ⁡ max and P f , g , ⁡ min represent the upper and lower limit values of the power of the g t h traditional unit; r g , up and r g , down represent the upper and lower limit values of the traditional unit to climb the slope, respectively.

5)  New energy station constraints

Where t = 1,2…,T; P w , t , ⁡ max represents the maximum generating power of the w t h new energy station in t period.

In summary, the model proposed in this paper is essentially a two-layer optimization model, and in the actual grid model, firstly, with Eq. 17 as the objective function and Eqs 18–24 as the constraints, the inner layer optimization model is solved to obtain the investment cost H of the grid company after the optimization of TOU electricity price and to achieve the saving of the grid investment cost. Then, we substitute it into Eq. 16 and solve the outer optimization model with Eq. 3 as the objective function and Eqs 5–16 as the constraints. The load transfer is guided to realize the “peak shaving and valley filling” of the load curve.

In summary, the model presented in this paper is fundamentally a two-layer optimization model. In the actual grid model, the inner layer optimization model is solved first, with Eq. 17 serving as the objective function and Eqs 18–24 as the constraints, to obtain the investment cost H of the grid company after optimizing the TOU electricity price and achieving savings in grid investment costs. Subsequently, the substitution is made into Eq. 16 and the outer optimization model is solved, with Eq. 3 serving as the objective function and Eqs 5–16 as the constraints. The load transfer is directed towards achieving “peak shaving and valley filling” of the load curve.

The section proposes a time slot division method based on the GMM clustering algorithm to address issues related to objectivity and adaptability in traditional time slot division methods. This approach aims to provide technical support for the reasonable division of time-sharing tariff time slots and mitigate the influence of subjective factors on the results.

The GMM clustering relies on the probability of classification members, which falls under “soft classification.” This method does not explicitly assign members to a specific category but instead provides the probability of a member belonging to each category. The information conveyed by this approach is considerably greater than that of K-means and other “hard classification” clustering methods, resulting in improved clustering effectiveness. Additionally, GMM clustering can more effectively uncover correlations between various attributes. As a result, GMM clustering algorithms are currently extensively employed across diverse fields (DANG et al., 2015).

The GMM is derived by linearly combining multiple Gaussian distribution functions. The r-dimensional sample dataset D r = x 1 , x 2 , . . . , x r be classified into S classes, and the Gaussian mixture model, which comprises a mixture of S Gaussian distributions, is defined by an Eq. 25: P x = ∑ s = 1 S α s p x | β s (25) p x | β s = 1 ∑ s 2 π r / 2 e − x − μ s 2 2 ∑ s (26) ∑ s = 1 S α s = 1 , 0 ≤ α s ≤ 1 (27)

Where P x represent the probability density function of GMM; p x | β s represent the probability density function of the s t h Gaussian distribution; α s 、 μ s 、 ∑ s represent the weight, expectation and covariance matrix of the s t h Gaussian distribution, respectively.

The output of a GMM consists of a sequence of probability values, and the category with the highest probability is considered as the one to which the member belongs. Determining the probability value necessitates identifying the Gaussian distribution to which the member belongs, and ensuring that the Gaussian distribution accurately represents the sample data by fitting it as closely as possible, thereby ensuring that its parameters are accurate. The Expectation-Maximization algorithm (EM) (Zanetti et al., 2015) is employed for parameter estimation of the GMM. The fundamental concept of EM is to iteratively optimize the likelihood estimation of the model distribution parameters, converging on the model parameters by repeatedly iterating until the likelihood function value reaches convergence, thereby completing the parameter estimation process. Furthermore, due to the complexity of solving the likelihood estimation function for sample data in the Gaussian mixture model, it is common to use the logarithm of the likelihood function, as shown in Eq. 28: ln ∏ i = 1 r P x i | α , μ , ∑ = ∑ i = 1 r ln ∑ s = 1 S α s p x | β s (28)

The EM algorithm primarily engages in iterative parameter estimation through two steps: the E-Step (Expectation Step) and the M-Step (Maximization Step). The E-Step involves calculating the probability that each data point is generated by each Gaussian distribution using the initial values of α 、 μ 、 ∑ or the value obtained from the previous iteration. The M-Step entails solving and updating the model parameters based on the value obtained from the E-Step.

Based on the aforementioned theory, to achieve a precise division of time-sharing tariff periods, this section presents a method for time-sharing tariff period division based on the GMM clustering algorithm. The flowchart of this method is illustrated in Figure 2 and described below.

Step 1: First, give an r-dimensional sample set of typical daily load data D r = x 1 , x 2 , . . . , x r (e.g., one data point at 1 h intervals, a day has 24 time periods, r = 24); second, given the number of clusters S (e.g., if a day is divided into 3 time periods, peak, normal, and valley, then S = 3), the GMM consists of a mixture of S Gaussian distributions.

Step 2: First, the initial GMM parameters α 0 、 μ 0 、 ∑ 0 are randomly given. Second, E-Step is used to calculate the posterior probability of each data sample x i i = 1 , 2 , . . . , r in the sample set D r = x 1 , x 2 , . . . , x r generated by the sth Gaussian distribution γ s , i as in Eq. 29. Finally, on the basis of the values obtained by E-Step, the GMM parameters are updated by M-Step as in Eqs 30–32.

Step 3: Determine whether the stopping condition is satisfied, i.e., when the maximum number of iterations is reached or the log-likelihood function Eq. 28 converges go to Step 4 to complete the estimation of the GMM parameters; otherwise, go to Step 2 to repeat E-Step and M-Step.

Step 4: First, according to the a posteriori probability γ s , i that the data sample x i i = 1 , 2 , . . . , r may be generated by each Gaussian distribution, we select the cluster with the largest a posteriori probability γ s , i as the cluster to which the time period belongs, and mark the cluster σ i as Eq. 33; second, according to the cluster marking, divide the data sample D r = x 1 , x 2 , . . . , x r set into S clusters T a l l = T 1 , T 2 , . . . , T S , and the resulting S clusters T a l l are the S classes of the time period, with the time period belonging to that class included in each class, thus completing the division of the time-sharing tariff time period. γ s , i = α s p x i | β s / ∑ s = 1 S α s p x i | β s (29) α s = ∑ i = 1 r γ s , i / n , s = 1 , 2 , . . . , S (30) μ s = ∑ i = 1 r γ s , i x i / ∑ i = 1 r γ s , i , s = 1 , 2 , . . . , S (31) ∑ s = ∑ i = 1 r γ s , i x i − μ s 2 / ∑ i = 1 n γ s , i , s = 1 , 2 , . . . , S (32) σ i = argmax γ s , i , i = 1 , 2 , . . . , r ; s ∈ 1 , 2 , . . . , S (33)

The design of a time-of-day tariff encompasses two primary components: time slot division and price determination. This subsection consolidates the time-of-day tariff price optimization model, which takes into account the grid investment benefit outlined in Section 1, and the time-of-day tariff time slot division method based on the GMM clustering algorithm discussed in Section 2.1. Furthermore, it introduces a comprehensive optimization strategy for time-of-day tariff price and time slot, considering the grid investment benefit, as depicted in Figure 3. The specific steps are outlined below:

Step 1: First, give a typical daily load profile as input and set it to divide a day into S categories such as peak, normal, valley, etc. second, adopt the time-sharing tariff time period division method based on GMM clustering algorithm proposed in Section 2.1 to obtain S clusters T a l l = T 1 , T 2 , . . . , T S such as peak, normal, valley, etc. where each of the clusters contains a time period belonging to the cluster, so as to realize the division of time-sharing tariff time periods.

Step 2: Based on the results of the obtained time slot division T a l l , the time slots are included in the categories of peak, normal, and valley are determined to complete the construction of the time-sharing tariff optimization model Eqs 1–4 and Eqs 6–24 proposed in this paper considering the investment benefits of the power grid.

Step 3: First, solve the time-sharing tariff price optimization model by adopting algorithms such as the out-point method (cited in the literature), and obtain the time-sharing tariff price; secondly, combine with the time slot division method obtained in step 1, and finally obtain the time-sharing tariff strategy. For example, set the number of clusters S = 3, the 24 h of the day are divided into peak hours, weekdays and valleys, then according to the results of the time division, the following time-sharing tariff strategy can be constructed. A time-of-use electricity price strategy can be constructed as shown in Eq. 34: λ i = λ f , i ∈ T 1 λ p , i ∈ T 2 λ g , i ∈ T 3 (34)

Step 4: In order to assess the impact of the designed time-sharing tariff strategy on the benefits of the grid company and users. First, calculate the grid company’s benefit E company , the sum of the grid company’s revenue from electricity sales and the investment cost saved by the grid company after the concessions, as in Eq. 35, the larger the value, the better the grid company’s benefit; second, calculate the price of electricity per unit of the user to characterize the user’s benefit, as in Eq. 36, the larger the reduction in the price, the larger the benefit to the user. E company = E A + 1 − ∂ Δ Q = ∑ i ∈ I λ i p load _ i − ∑ i ∈ I c i p load _ i + 1 − ∂ Δ Q (35) E user = ∑ i ∈ I λ i p load _ i / ∑ i ∈ T p load _ i (36)

The above analysis culminates in the development of a time-sharing tariff strategy that takes into account the grid’s investment benefits. This strategy offers potential solutions for enhancing the load curve, reducing grid investment costs, and lowering electricity consumption expenses for users. Furthermore, through an analysis of the advantages for both the grid company and users under the optimized time-sharing tariff, this study offers insights that can inform the appropriate adjustment of future time-sharing tariffs. This adjustment is beneficial for mitigating the conflict between power supply and demand.

To assess the efficacy of the TOU electricity price period segmentation method proposed in this study, a set of typical daily load data from a provincial-level power system in China was selected for period segmentation, similar to that of a real power grid (DONG et al., 2023; DING et al., 2001b). The typical daily load data is depicted in Figure 4, with the number of clusters designated as S = 3, indicating the division of the 24 time periods of a day into peak periods, off-peak periods, and normal periods.

The existing TOU electricity pricing strategy in this province delineates time periods as follows: peak periods are from 11:00 to 17:00 and 20:00 to 22:00, off-peak periods are from 8:00 to 11:00, 17:00 to 20:00, and 22:00 to 24:00, and the valley period is from 0:00 to 8:00. Nevertheless, as depicted in Figure 4, the current time period segmentation method does not adequately capture the peak and off-peak characteristics of the province’s load. There are several instances of inconsistency, such as the load showing a significant increase at 10:00 with a high load value, yet being categorized as an off-peak period based on the current time segmentation. Similarly, at 21:00, the load demonstrates a distinct decrease but is designated as a peak period according to the current time segmentation. Consequently, it is imperative to modify the segmentation of time periods in this province.

To comprehensively illustrate the efficacy of the TOU electricity price period segmentation method based on the GMM clustering algorithm proposed in this paper, this section sets up four comparative analysis scenarios, labeled as M0 to M3.

M0: Method for segmenting the current time period.

M1: A method for segmenting TOU electricity price periods based on membership functions has been proposed (DING et al., 2001a).

M2: A method for segmenting TOU electricity price periods based on the K-means clustering algorithm (Nedal, 2011).

M3: proposes a method for segmenting TOU electricity price periods based on the Gaussian Mixture Model (GMM) clustering algorithm.

In M1 to M3, M1 denotes the conventional TOU electricity price period segmentation approach, characterized by its simplicity and efficiency, but also susceptible to significant influence from human factors. The M2 method is a traditional clustering approach frequently employed for segmenting TOU electricity price periods. It is recognized for its straightforward principles and straightforward implementation, although it has constraints regarding the types of samples to which it can be applied. The method proposed in this paper is referred to as M3.

In this section, a comparative analysis method, as cited in (QIAO, 2011), is employed to determine the percentage of peak load across various time period segments. The rationality of the proposed time period segmentation method is assessed through a comprehensive analysis. The time period divisions from M0 to M3 are presented in Table 1, while the proportion of electricity usage during peak hours within these divisions is depicted in Figure 5.

The comparison of the proposed TOU electricity price period segmentation method M3 with M1 and M2, based on the results obtained from Table 1, yields the following observations: In the time period segmentation result of M1, 9:00 is not designated as a peak period. Nevertheless, as illustrated in Figure 4, the load at 9:00 demonstrates a noticeable increasing pattern and is characterized by a relatively high load value. The power generation is only 189.14 MW less than the peak value recorded at 12:00, representing approximately 86.88% of the peak load. Consequently, if the period is designated as off-peak, it has the potential to result in a new peak load at that time due to user load shifting, which may not be favorable for peak shaving. The time period segmentation result of M2 does not designate 13:00 as a peak period. Despite the current decrease in load, the load value remains relatively high, with a difference of only 46.82 MW compared to the peak load at 12:00, representing approximately 96.75% of the peak load. If the period is categorized as off-peak, it could lead to a reduction in transferable load or the creation of a new peak load at that time following user load shifting, posing challenges to effectively accomplish peak shaving and valley filling.

In summary, the study confirms the effectiveness of the time-sharing tariff period division based on the GMM clustering algorithm proposed in this paper, demonstrating its ability to achieve a rational division of time-sharing tariff periods.

To assess the efficacy of the time-sharing tariff price optimization method proposed in this study, taking into account the grid investment benefit, this section utilizes the actual load data of a province in China from 2022 and the locally implemented time-sharing tariff policy for verification. This is combined with the typical daily load curve, as depicted in Figure 6. The time-sharing tariff policy implemented in the province on an average day is presented in Table 2.

Three TOU electricity tariff optimization methods M4-M6 are set up in this section, as shown in Table 3:

The load profiles after optimization by different methods are shown in Figure 7.

The conclusions that can be drawn from Figure 7 are as follows:

1) When comparing M4, M5, and M6 with the original load curves, it is evident that M4-M6 has successfully achieved the objective of peak reduction and valley filling.

By optimizing the current TOU electricity pricing, users’ load curves have been enhanced, leading to peak load reduction and off-peak load increase, as well as a decrease in the investment cost of the power grid. In comparison to M4 and M5, the load curve of M6 exhibits a more pronounced peak reduction and valley filling, thereby enhancing the optimization of user load curves.

To provide additional evidence of the efficacy of the proposed approach, an analysis of the advantages for power grid companies and users is conducted, and the findings are presented in Table 4 and Table 5.

The following conclusions can be drawn from Table 4:

1) After the implementation of the M4 TOU electricity price optimization method, the income of the power grid company has not changed compared with the original TOU electricity price strategy, the power grid company has not reduced its income due to the change of the price strategy.

The conclusions drawn from Table 4 are as follows:

1) Following the implementation of the M4 TOU electricity price optimization method, there has been no significant change in the income of the power grid company compared to the original TOU electricity price strategy. The power grid company has not experienced a reduction in income as a result of the change in the price strategy.

As indicated in Table 5, the unit electricity consumption for M5 users proposed in this study is 0.001 yuan less than the average price for M4 users and the original pricing strategy. Similarly, the unit electricity consumption for M6 users is 0.003 yuan lower than the average price for M4 users and the original pricing strategy. This suggests that, under the TOU electricity price period segmentation method based on the GMM, customers are experiencing cost savings on their electricity bills.

The study confirms that the TOU electricity pricing strategy proposed in this paper effectively empowers users to actively participate in peak load reduction and off-peak load utilization. This not only decreases the power grid’s investment costs but also lowers electricity expenses for users, thereby achieving a mutually beneficial outcome for both the power grid company and the users.