Optimization Model of Time-of-Use Electricity Pricing Considering Dynamical Time Delay of Demand-Side Response

Abstract

Traditional time-of-use (TOU) pricing models ignore the delay characteristics of user behavior; consequently, the resulting load adjustments exhibit discrete patterns, whereas actual load variations follow gradual trajectories in reality. Hence, a dynamic process is to be considered when describing user behavior and designing pricing strategy, which will, however, add to the complexity of pricing. This paper proposes a TOU pricing strategy considering user response with delay. Firstly, based on the final state of user response, the time delay of the demand response is defined. Secondly, to describe the dynamic process of load transfer, a time-varying price elasticity matrix is proposed, and its parameters are newly identified by using the weighted least squares method. Finally, based on the elasticity matrix, a mixed-integer programming model is proposed with the multi-objective of minimizing the peak–valley difference of system load and maximizing user satisfaction. An improved non-dominated sorting genetic algorithm (NSGA-II) is applied to find the Pareto front solution and obtain the optimal price of the TOU. The simulation results based on a provincial load data in China show that the proposed optimization strategy to the TOU pricing can help the system reduce peak–valley load difference and effectively smooth the load curve.

1. Introduction

With the increasingly high penetration of renewable energy and the surge in flexibility demands within power systems, conventional electricity pricing mechanisms face challenges in adapting to the dynamic supply–demand equilibrium [1,2,3,4,5]. As a demand-side management tool, TOU pricing guides users to shift consumption away from peak periods through price signals, serving as a critical means to optimize load profiles and alleviate grid stress [6,7,8]. However, existing studies predominantly focus on TOU pricing models [9,10,11,12], with limited attention to the hysteresis in user behavioral adjustments. This oversight leads to an assumption of instantaneous load transfer, which contradicts the actual dynamic and gradual evolution of demand-side responses. As the scale of demand response expands, the impacts of such time delay effects on power systems cannot be neglected. Therefore, it is imperative to develop a demand response pricing mechanism that explicitly incorporates time delay effects.

The pricing for demand-side response involves quantifying the relationship between electricity price and electricity consumption based on either the price elasticity matrix [13,14,15,16,17] or statistical theory [18,19,20,21]. Then, an optimization model is established to refine electricity pricing, as shown in Figure 1.

There have been studies on TOU pricing for demand-side response. Ref. [22] derives the time-dependent price elasticity matrix based on weights of probabilistic density function. It shows the difference in price elasticity due to different load ratios, but cannot show the change in price elasticity due to the time delay of the consumers. Ref. [23] determines the elasticity coefficient based on power consumption, which is, however, based on the precondition that the power consumption does not change with the response degree. Ref. [24] introduces fractional Brownian motion into electric power demand modeling to capture key characteristics in demand fluctuations such as self-similarity and long-range or short-range dependence. Ref. [25] defines consumers’ electricity transfer behavior in a graphical form and establishes a TOU pricing elasticity model based on graph attention networks, but it relies on extensive historical data and suffers from low computational efficiency. Ref. [26] establishes a user response model based on the UCB algorithm but fails to incorporate time delay durations as input features; thus, it is unable to characterize gradual load variation characteristics. Ref. [27] considers the time delay of the demand-side response to quantify the power price from price change to the end of the response and define the sensitivity indicators based on the price elasticity. However, the price elasticity matrix applies static parameters, which are not dynamically corrected with the time delay effect.

The difficulties to study the TOU price considering the time delay of the demand-side response lie in:

(1)

The load adjustment process is driven by the electricity price, and changes in temperature and moisture that are irrelevant to the cost. The load variation due to these factors overlaps with that caused by the time delay effect, and changes the time delay of the demand-side response.

(2)

The existing electricity price elasticity matrix considers the time delay, but is fixed; thus, it cannot show the dynamic time delay of the demand-side response.

(3)

The time delay of user response causes dynamic variation in peak–valley load differences and users’ satisfaction, and may be changed by adjusting the time-of-use price, which is a multi-objective optimization problem, and has not been defined.

To address the aforementioned issues, this paper proposes a TOU electricity pricing model that considers the dynamic time delay characteristics of user demand-side response. First, based on load data, the delay time of user demand response is quantified. Second, a price elasticity matrix with time-varying characteristics is established, and its parameters are identified through the weighted least squares method. Finally, based on the price elasticity matrix, a multi-objective mixed-integer programming model is formulated with the goals of minimizing the system’s peak–valley difference and maximizing user satisfaction. An improved Non-Dominated Sorting Genetic Algorithm (NSGA-II) is adopted to obtain TOU electricity pricing results. Case studies verify the feasibility and effectiveness of the proposed method.

The structure of this paper is as follows. Section 1 serves as the introduction, elaborating on the research background, key problems, and innovations of this study. In Section 2, a time-varying price elasticity matrix model considering time delay is proposed. Load data are preprocessed via a sliding window method, seasonal-trend decomposition eliminates natural factor impacts, and weighted least squares method identifies matrix parameters to quantify the time delay effect of user demand response. Section 3 proposes a multi-objective mixed-integer programming model based on the elasticity matrix to minimize system peak–valley load difference and maximize user satisfaction. An improved NSGA-II algorithm is applied to solve the optimal time-of-use pricing strategy by finding the Pareto front solutions. Section 4 gives the simulation results based on provincial load data of China, quantifies time delay impacts on load curve, and validates feasibility and effectiveness of the proposed method by comparing it with the existing methods without time delay. Section 5 concludes the paper with a comprehensive summary and outlines future research directions.

2. Time-Varying Price Elasticity Matrix and Its Parameter Estimation

2.1. Price Elasticity Matrix

Price elasticity reflects the impact of price changes on demand, which can be categorized into own-price elasticity coefficient and cross-elasticity coefficient [28]. The own-price elasticity coefficient ε_ii refers to the influence of electricity price changes in period i on electricity demand

$${\epsilon }_{ii}={\displaystyle \frac{\Delta q_i/q_i}{\Delta P_i/P_i}}$$

(1)

where q_i and P_i are the electricity consumption and electricity price in time period i, respectively, and Δ denotes the increment.

In addition to the electricity price in the current period, price changes in other periods can also influence electricity demand. The cross-price elasticity coefficient ε_ij refers to the degree of electricity consumption change in time period i caused by electricity price changes in time period j.

$${\epsilon }_{ij}={\displaystyle \frac{\Delta q_i/q_i}{\Delta P_j/P_j}}$$

(2)

Different load price elasticities exhibit variations across user types. Industrial users, characterized by higher electricity cost ratios in total expenditures and adjustable production processes, typically demonstrate stronger price responsiveness. In contrast, residential loads are constrained by rigid basic living demands, showing lower overall elasticity. However, technological advancements like smart home systems are progressively unlocking the adjustment potential of discrete electrical appliances.

By dividing one day into n time periods, an electricity price elasticity matrix is given in (3) based on the own-price and cross-price elasticity coefficients. The relationship between price changes and load changes is given in (4).

$$\mathit{E}=\left[\begin{array}{cccc}{\epsilon }_{11}& {\epsilon }_{12}& \cdots & {\epsilon }_{1n}\\ {\epsilon }_{21}& {\epsilon }_{22}& \cdots & {\epsilon }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\epsilon }_{n1}& {\epsilon }_{n2}& \cdots & {\epsilon }_{nn}\end{array}\right]$$

(3)

$$\left[\begin{array}{c}\Delta q_1/q_1\\ \Delta q_2/q_2\\ ⋮\\ \Delta q_n/q_n\end{array}\right]=\mathit{E}\left[\begin{array}{c}\Delta P_1/P_1\\ \Delta P_2/P_2\\ ⋮\\ \Delta P_n/P_n\end{array}\right]$$

(4)

In the aforementioned electricity price elasticity model, the load shifts immediately after price adjustments. In practice, however, load shifting induced by electricity price tends to evolve gradually. The use of a static elasticity matrix leads to unrealistic results. To accurately describe this dynamic process, it is critical to quantify the time delay in demand response and refine the price elasticity model.

2.2. Time Delay Determined by User Response Stability

After receiving electricity price adjustment signals, users exhibit delayed adjustments in actual electricity consumption behaviors due to factors such as information perception delays, solidified consumption habits, and insufficient incentives. Meanwhile, loads are also affected by factors such as temperature and humidity. The load changes induced by these factors are unrelated to electricity prices, which increases the computational complexity. Taking the time unit as a day as an example, the calculation steps for time delay are proposed:

(1) Load data collection and preprocessing. A sliding window method is employed to detect outliers. If the load in a certain time period exceeds ±3 standard deviations from the historical average for the same period, replace it with interpolated values from adjacent data points.

(2) To eliminate the influence of the natural growth in electricity consumption, seasonal and trend decomposition using locally weighted scatterplot smoothing (STL) method is applied to decompose the historical load data series Y into three components: season, trend, and residual [29].

$$Y=R+T+S$$

(5)

where R, T, and S are the residual term, trend term, and seasonal term obtained after decomposition, respectively.

Processing the load data after TOU electricity price changes to eliminate seasonal and trend components obtains load changes q, related only to electricity prices:

$$q=Y-\left(T+S\right)$$

(6)

(3) Using the average daily electricity consumption q_N from x days before the price adjustment as the baseline, normalize the post-adjustment electricity consumption $q_{t,i}^{*}$ for each period:

$$q_{t,i}^{\ast }=q_{t,i}/q_{\mathrm{N}}$$

(7)

where q_t,i is the electricity consumption of time period i on day t after price adjustment.

(4) To mitigate single-day load anomalies induced by abrupt weather changes, a d-consecutive-day average load change is calculated to eliminate sporadic interference in conclusions. Let M_t,i denote the mean electricity load during time period i from day t to t + d.

$$M_{t,i}=(q_{t,i}^{\ast }+q_{t+1,i}^{\ast }+\dots +q_{t+d-1,i}^{\ast })/d$$

(8)

(5) Calculate the total electricity consumption change ΔM_t and standard deviation σ_t on day t:

$$\Delta M_t={\displaystyle \sum _{i=1}^n\left(M_{t+1,i}-M_{t,i}\right)}$$

(9)

$${\sigma }_t=\sqrt{\sum _{t=1}^a{(\Delta M_t-\Delta {\widehat{M}}_t)}^2/a}$$

(10)

where $\Delta {\widehat{M}}_t$ is the average of ΔM_t over the first t calculations.

(6) During the initial stage after electricity price adjustment, users respond at a faster rate, and the rate of change slows over time. Therefore, when Formulas (11) and (12) are satisfied, begin judging the subsequent days for stable state. Formula (13) can be used to determine whether a stable state has been reached. When Formula (13) is not satisfied, the day counter t is increased by 1, and the procedure returns to step 4. At this point, the recalculated average electricity quantity M_t,i is updated to reflect the new time window from day t to t + d rather than reusing the previous window.

$${\sigma }_t>{\sigma }_{t-1}>{\sigma }_{t-2}$$

(11)

$${\sigma }_t>{\sigma }_{t+1}>{\sigma }_{t+2}$$

(12)

$$\left|{\sigma }_t-{\sigma }_{t-1}\right|<\delta$$

(13)

If Formula (13) is satisfied for k consecutive days, stop the calculation and record the current t. This state is the response stable state. δ is the calculation precision.

2.3. Time-Varying Price Elasticity Matrix

In traditional elasticity matrix models, price elasticity coefficients are typically set as fixed values. However, in practice, users exhibit the delayed recognition of price signals and gradual behavioral adjustments, leading to dynamic load shifts under the effects of price elasticity. To address this, each element of the price elasticity matrix is quantified as a function of time t, fitted using the weighted least squares method, thereby proposing a price elasticity matrix that accounts for the time delay in user responses.

Considering the aforementioned decay behavior, each element ε_ij(t) in the price elasticity matrix E(t) is modeled as an exponential function of the time variable t:

$${\epsilon }_{ij}\left(t\right)=\left\{\begin{array}{ll}{a}_{ij}{e}^{-b_{ij}t}+c_{ij}\hfill & i\le j\hfill \\ {\epsilon }_{ji}\left(t\right)\hfill & i>j\hfill \end{array}\right.$$

(14)

where a_ij, b_ij, and c_ij are the coefficients to be determined corresponding to the elasticity coefficient ε_ij(t). a_ij is the elasticity decay amplitude coefficient, reflecting the intensity of the initial price elasticity influenced by time delay; b_ij is the decay rate coefficient, representing the decay speed of elasticity caused by user response time delay; and c_ij denotes the price elasticity after the user response stabilizes. The coefficients a_ij, b_ij, and c_ij are determined by fitting historical load data using the weighted least squares method. The specific steps are as follows:

To simplify the expression, let vectors ΔG and ΔH denote the load changes and electricity price changes across time periods, respectively. The set of fitting points for the price elasticity matrix is defined as follows:

$$J=\left\{\left(\Delta {\mathit{G}}_1,\Delta {\mathit{H}}_1\right),\left(\Delta {\mathit{G}}_2,\Delta {\mathit{H}}_2\right),\cdots ,\left(\Delta {\mathit{G}}_t,\Delta {\mathit{H}}_t\right),\cdots \right\}$$

(15)

where ΔG_t is the electricity price changes across time periods on day t after TOU price adjustment, ΔH_t is the corresponding load changes for ΔG_t. For the N days following the electricity price adjustments, Equation (16) holds. Expanding it gives Equation (17).

$$\mathit{E}\left(t\right)\Delta {\mathit{G}}_t=\Delta {\mathit{H}}_t$$

(16)

$${\displaystyle \sum _{j=1}^n\left(a_{ij}{e}^{-b_{ij}t}+c_{ij}\right)g_{t,j}}=h_{t,j},\forall i=1,\dots ,n$$

(17)

where g_t,j and h_t,j are electricity price changes and load changes in time period j on day t after TOU price adjustment, respectively.

Based on (16) and (17), the least-squares-based parameter fitting model for the price elasticity matrix is given by the following

$$\min {{\displaystyle \sum _{\mathrm{t}=1}^N\left({\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n\left(a_{ij}{e}^{-b_{ij}t}+c_{ij}\right)g_{t,j}}}-{\displaystyle \sum _{j=1}^n{h}_{t,j}}\right)}}^2$$

(18)

When dealing with large-scale data, model error terms are prone to exhibit heteroscedastic characteristics, which may lead to significant bias in parameter estimates. To address this issue, the weighted least squares (WLS) method is introduced by assigning differential weights to correct the impact of heteroscedasticity on the error structure, thereby enhancing the accuracy of parameter estimation results [30]. The parameter fitting model for the price elasticity matrix based on the weighted least squares method is given as follows:

$$\min {\displaystyle \sum _{t=1}^N{\left(1-\frac{{\displaystyle \sum _{j=1}^n{h}_{t,j}}}{{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n\left(a_{ij}{e}^{-b_{ij}t}+c_{ij}\right)g_{t,j}}}}\right)}^2}$$

(19)

Solve Equation (19) to obtain the coefficients and reconstruct the price elasticity matrix E(t).

$$\left[\begin{array}{c}{\displaystyle \frac{\Delta q_1(t)}{q_1}}\\ {\displaystyle \frac{\Delta q_2(t)}{q_2}}\\ ⋮\\ {\displaystyle \frac{\Delta q_n(t)}{q_n}}\end{array}\right]=\left[\begin{array}{cccc}{\epsilon }_{11}(t)& {\epsilon }_{12}(t)& \dots & {\epsilon }_{1n}(t)\\ {\epsilon }_{21}(t)& {\epsilon }_{22}(t)& \dots & {\epsilon }_{2n}(t)\\ ⋮& ⋮& \ddots & ⋮\\ {\epsilon }_{n1}(t)& {\epsilon }_{n2}(t)& \dots & {\epsilon }_{nn}(t)\end{array}\right]\left[\begin{array}{c}{\displaystyle \frac{\Delta P_1}{P_1}}\\ {\displaystyle \frac{\Delta P_2}{P_2}}\\ ⋮\\ {\displaystyle \frac{\Delta P_n}{P_n}}\end{array}\right]$$

(20)

3. Multi-Objective Electricity Pricing Model Considering Dynamic Time Delay Characteristics of Demand-Side Response

Demand response incentivizes users to adjust their electricity load through price signals to achieve peak shaving and valley filling. However, user satisfaction decreases due to changes in electricity consumption habits, creating a conflict between these two objectives. The time delay in user responses further causes dynamic variations in peak–valley differences and satisfaction over time. In the initial stages of price adjustment, user responsiveness is limited, resulting in weak peak shaving and valley filling capabilities. As responses stabilize, increased load adjustments lead to reduced satisfaction. Therefore, based on the time-varying price elasticity matrix, an electricity pricing model is constructed with the objectives of minimizing the system’s peak–valley difference and maximizing user satisfaction.

3.1. Objective Functions of the Optimization Model

Peak–Valley Difference Indicator: TOU pricing guides peak shaving and valley filling through price signals. The peak–valley difference I_f is defined as the difference between the maximum load and minimum load within a day [31]:

$$I_{\mathrm{f}}={\displaystyle \sum _{i=1}^N(\max L_t(i)-\min L_t(i))}/N$$

(21)

where L_t(i) is the load on the day t after implementing TOU pricing.

User Satisfaction Indicator on Electricity Consumption Patterns: Electricity price changes compel users to alter their consumption habits, leading to reduced satisfaction. The user satisfaction indicator on electricity consumption patterns I_m is defined as the ratio of load changes in each period to the original load

$$I_{\mathrm{m}}={\displaystyle \sum _{t=1}^N\left(1-\frac{{\displaystyle \sum _{i=1}^n}\left|L_t(i)-L_0(i)\right|}{{\displaystyle \sum _{i=1}^n}{L}_0(i)}\right)}$$

(22)

where L₀(i) is the load before implementing TOU pricing.

User Satisfaction Indicator on Electricity Cost Expenditure: When electricity price adjustments increase users’ electricity expenses, their satisfaction with cost expenditure decreases. The user satisfaction indicator on electricity cost expenditure I_a is defined as the change rate of electricity purchase costs before and after implementing TOU pricing

$$I_{\mathrm{a}}={\displaystyle \sum _{t=1}^N\left(1-\frac{{\displaystyle \sum _{i=1}^n{P}_i{L}_t(i)-U_0}}{U_0}\right)}$$

(23)

$$U_0=P_{\mathrm{f}0}{L}_{\mathrm{f}0}+P_{\mathrm{p}0}{L}_{\mathrm{p}0}+P_{\mathrm{g}0}{L}_{\mathrm{g}0}$$

(24)

where U₀ is user expenditure before electricity price adjustment. L_f0, L_p0, L_g0 are loads during peak, flat, and valley periods before implementing TOU pricing. P_f0, P_p0, P_g0 are electricity prices during peak, flat, and valley periods before implementing TOU pricing.

3.2. Constraints of the Optimization Model

Power company’s benefit: After adjusting the TOU pricing, the power company’s electricity sales revenue U_T must be greater than the difference between the original revenue U₀ and the cost savings C′ [32]:

$$U_{\mathrm{T}}⩾U_0-C^{\prime }$$

(25)

$$U_{\mathrm{T}}={\displaystyle \sum _{t=1}^N\left({\displaystyle \sum _{i=1}^n{P}_i{L}_t(i)}\right)}/N$$

(26)

User benefit: If users’ electricity purchase costs increase after adjusting the TOU pricing, their willingness to respond will decline. Users will actively participate in demand response only if they benefit from the new pricing policy [33]:

$$C_{\mathrm{T}}={\displaystyle \frac{N{U}_{\mathrm{T}}}{{\displaystyle \sum _{t=1}^N{\displaystyle \sum _{i=1}^n{L}_t(i)}}}}⩽C_0={\displaystyle \frac{U_0}{L_0}}$$

(27)

where C₀ and C_T are the average electricity purchase costs before and after implementing TOU pricing, respectively.

Generation cost constraints: The peak-period electricity price does not exceed the generator’s generation cost P_max, and the valley-period electricity price is not lower than the system’s marginal cost P_min [34]:

$$P_{\mathrm{f}}⩽P_{\max }$$

(28)

$$P_{\mathrm{g}}⩾P_{\min }$$

(29)

4. Solution of Multi-Objective Pricing Model Based on Improved NSGA-II

4.1. Improved NSGA-II Algorithm Design and Multi-Objective Optimization Solution

The TOU pricing model must consider user satisfaction while optimizing the peak–valley difference. During the optimization process, variables are mutually constrained, and the objective functions are complex, making it difficult for traditional mathematical programming to balance competing objectives.

The Non-dominated Sorting Genetic Algorithm (NSGA-II), as a heuristic optimization algorithm, can effectively identify high-quality solutions in the objective space that cannot be further improved. However, the random initialization strategy for the initial population may lead to insufficient diversity and local optima. To address this, a logistic chaotic mapping is adopted to generate higher-quality initial solutions:

$$x_{n+1}=r{x}_n(1-x_n)$$

(30)

where x_n+1 and x_n are the values of the next and current iterations, respectively, and r is the control parameter of the chaotic mapping.

Each solution in the Pareto front set is optimal, and the final scheme must be selected based on practical requirements. To eliminate the influence of subjective factors on the final decision, the entropy weight method [35] and technique for order of preference by similarity to ideal solution (TOPSIS) method [36] are employed.

For m Pareto front solutions obtained after optimizing n objective functions, a judgment matrix Z is constructed and standardized:

$$\left\{\begin{array}{l}{z^{\prime }}_{ij}={\displaystyle \frac{\max \left\{z_{1j},\cdots ,z_{mj}\right\}-z_{ij}}{\max \left\{z_{1j},\cdots ,z_{mj}\right\}-\min \left\{z_{1j},\cdots ,z_{mj}\right\}}}\hfill \\ \mathit{Z}={\left(z_{ij}\right)}_{m\times n}\hfill \end{array}\right.$$

(31)

where z_ij is the value of the j-th objective function corresponding to the i-th Pareto front solution. $z_{ij}^{’}$ is the normalized objective function value.

Entropy reflects the differences among solutions for the same objective function, eliminating the need to determine the variation degree of objective function values based on personal experience. The entropy A_j and weight value ω_j of the j-th objective function are given by:

$$\left\{\begin{array}{l}{A}_j=-{\displaystyle \frac{1}{\ln m}}{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{z_{ij}}{{\displaystyle \sum _{i=1}^m}{z}_{ij}}}\ln {\displaystyle \frac{z_{ij}}{{\displaystyle \sum _{i=1}^m}{z}_{ij}}}\hfill \\ {\omega }_j={\displaystyle \frac{1-A_j}{n-{\displaystyle \sum _{j=1}^n}{A}_j}}\hfill \end{array}\right.$$

(32)

The weighted standardized judgment matrix elements v_ij are calculated using the weights:

$$v_{ij}={\omega }_j{z^{\prime }}_{ij}$$

(33)

The positive ideal solution $v_j^{+}$ and negative ideal solution $v_j^{-}$ for the j-th objective are determined as follows:

$$\left\{\begin{array}{c}{v}_j^{+}=\max \left\{v_{1j},v_{2j},\cdots ,v_{nj}\right\}\\ v_j^{-}=\min \left\{v_{1j},v_{2j},\cdots ,v_{nj}\right\}\end{array}\right.$$

(34)

The distances $D_i^{+}$ and $D_i^{-}$ between the i-th Pareto front solution and the positive/negative ideal solutions are calculated. The overall evaluation value S_i is given in (36):

$$\left\{\begin{array}{c}{D}_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^n}{\left(v_{ij}-v_j^{+}\right)}^2}\\ D_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^n}{\left(v_{ij}-v_j^{-}\right)}^2}\end{array}\right.$$

(35)

$$S_i={\displaystyle \frac{D_i^{-}}{D_i^{+}+D_i^{-}}}$$

(36)

According to the magnitude of the overall evaluation value, the optimal Pareto front solutions are obtained.

4.2. Model Development and Solution

The model establishment and solving process are shown in Figure 2, which specifically includes the following:

Pricing model:

(1)

Collect historical load data from a provincial power grid in China. Apply sliding window preprocessing to the historical load data to filter out outliers. Use the STL decomposition method to eliminate seasonal fluctuations from the data.

(2)

Iteratively determine the user response time delay through analysis. Select load data that exceed the duration of the determined time delay for subsequent analysis.

(3)

Fit the time-varying price elasticity matrix using Equation (19) based on the selected load data.

(4)

Construct a multi-objective electricity pricing model with the optimization targets of user satisfaction and peak–valley load difference.

(5)

Use typical daily load data from the region as inputs and optimize the load curve through the established pricing

Optimization problem solving:

(1)

Parameter initialization and population generation: Set population size, crossover probability, mutation probability, and maximum evolution generations. Generate the initial population using logistic chaotic mapping.

(2)

Objective evaluation and population sorting: Calculate objective function values for all individuals. Perform non-dominated sorting based on these values.

(3)

Parent selection: Adopt binary tournament selection strategy, combining non-dominated ranking and crowding distance metrics to select high-quality parents.

(4)

Genetic operations: Execute simulated binary crossover to generate offspring and apply mutation operators to diversify the population.

(5)

Population merging and selection: Merge parent and offspring populations. Re-perform non-dominated sorting and crowding distance calculation. Select the new generation based on Pareto dominance and diversity metrics.

(6)

Termination check: If current iterations are less than maximum generations, return to Step (2); otherwise, output optimization results.

(7)

Select the optimal solution: Extract the non-dominated solution set from the final population. Determine the optimal TOU pricing scheme using entropy weight method combined with TOPSIS.

The method proposed in this paper only analyzes single-type loads. In scenarios with mixed multi-type loads, the response time delay characteristics of different load types vary, leading to the inability of the time-varying price elasticity matrix to accurately describe the adjustment processes of all loads. Consequently, after electricity price optimization, some users exhibit lower satisfaction metrics, and the peak-shaving and valley-filling effects are diminished. To address this, the proposed method can be applied to different load datasets to obtain distinct time delay durations for each load type. Loads can then be clustered based on their time delay differences, and time-varying price elasticity matrices can be separately calculated for each cluster to enhance algorithm accuracy. Under this modification, the proposed method remains applicable, albeit with a slight increase in computational time.

5. Case Analysis

The load data of a province in China in one month is applied to validate the proposed optimization strategy to the TOU pricing [37]. The prices in the peak hour, the flat hour, and the valley hour are 0.8, 0.5, and 0.3 CNY/kW·h, respectively.

5.1. Load Response to Time-of-Use Price

The original load data in one day is shown in Figure 3. Then, with the TOU price, the load curve changes due to user response. With the TOU price, the loads on the 7th, 23rd, and 30th days are selected to describe the time delay. Due to the hysteresis of user behavior, the load curve changes slightly at the initial stage. In day 7, the peak load decreases by 2.5% compared with the original load, and the valley load increases by 2.8%. Afterwards, the peak load reduces, and the valley load increases gradually. In day 23, the response approaches stable, with little difference from the load curve in day 30 where the peak load decreases by 4.3%, and the valley load increases by 7.0%. Hence, the load data across 23 days are applied to fit their time delay response to the time-of-use price.

5.2. Fitting of the Time-Varying Price Elasticity Matrix

Based on load data with the TOU price, the relation of the load and the TOU price is fit according to (19) to obtain the time-varying price elasticity coefficients, as given in

Table 1. Time-dependent coefficients for model electricity price elasticity.

	Peak Period	Flat Period	Valley Period
Peak period	0.130 × e−0.104t − 0.206	−0.027 × e−0.0721t + 0.0509	−0.012 × e−0.063t + 0.036
Flat period	−0.027 × e−0.072t + 0.051	0.123 × e−0.116t − 0.195	−0.024 × e−0.099t + 0.048
Valley period	−0.012 × e−0.063t + 0.036	−0.024 × e−0.099t + 0.048	0.099 × e−0.090t − 0.199

.

The fitting of the total load during peak, valley, and flat periods in a month is shown in Figure 4. The peak and valley periods can more accurately fit the dynamic process of load transfer caused by time delay. The fitting during the flat period is poorer than those during the peak and valley periods. This is because the load variation during the flat period is relatively smaller due to the minor fluctuation of electricity price. The daily fluctuation in load further reduces the signal-to-noise ratio, contributing to poor fitting. Consequently, load variation distribution is relatively dispersed and lacks distinct change. The overall mean absolute percentage error is 0.66%, indicating a relatively good fitting effect.

5.3. Optimization Results to Time-of-Use Price

A 24 h load dataset is analyzed, and the improved NSGA-II algorithm is adopted for solving. The NSGA-II parameter settings are as follows: population size N = 400, maximum iterations G = 200, crossover probability p_cross = 0.8, mutation probability p_m = 0.2. The initial electricity price data is shown in

Table 2. Initial electricity price parameters.

Period	Initial Price/(CNY·(kW·h)−1)	Price Range/(CNY·(kW·h)−1)
Peak period	0.8	0.8–1.2
Flat period	0.5	0.3–0.75
Valley period	0.3	0.15–0.3

.

The optimization model yields the optimal TOU electricity prices considering the time delay characteristics of user response: the price in peak hours is 0.897 CNY/kW·h, the price in flat hours is 0.508 CNY/kW·h, and the price in valley hours is 0.163 CNY/kW·h. The load curves before and after TOU pricing implementation is shown in Figure 5, and the comparative data are given in

Table 3. Comparison of daily load before and after TOU pricing implementation.

Index	Before Implementation	7th Day After Implementation	30th Day After Implementation
Maximum load/(GW)	38.485	37.373	36.927
Minimum load/(GW)	30.078	32.171	32.931
Peak–valley difference/(GW)	8.407	5.203	3.395
Load fluctuation rate	8.1%	4.3%	3.1%
Satisfaction indicator on electricity cost expenditure	1	0.965	0.950
Satisfaction indicator on electricity consumption patterns	1	0.995	1.005

.

As shown in Figure 4 and Table 3, after implementing TOU pricing, compared to the original load: on Day 7, the peak load decreased from 38.485 GW to 37.373 GW, a reduction of 2.9%; the valley load increased from 30.078 GW to 32.171 GW, an increase of 7.0%, the peak–valley difference decreased from 8.407 GW to 5.203 GW, a reduction of 38.1%; and the load fluctuation rate dropped to 4.3%. After the response stabilized, on Day 30, the peak load further decreased to 36.927 GW, a reduction of 4.0%; the valley load increased to 32.931 GW, an increase of 9.5%; the peak–valley difference decreased to 3.395 GW, a reduction of 59.6%; and the load fluctuation rate dropped to 3.1%, a reduction of 5%. Electricity cost satisfaction initially decreased after the price adjustment but exceeded the satisfaction level of the original pricing policy once stability was achieved.

A comparison is conducted between the time-of-use electricity price optimization method proposed in this paper and the optimization method in Reference [38], which does not consider time delay. The load comparison diagram is shown in Figure 6. Under the TOU pricing without response delay consideration, the peak load reached 36.869 GW, the valley load was 32.188 GW, and the maximum peak-to-valley difference was 4.681 GW. Using the method proposed in this paper, the peak load slightly increased to 36.927 GW, the valley load rose to 32.931 GW, and the maximum peak-to-valley difference narrowed to 3.395 GW. This resulted in a 59.6% reduction in the peak-to-valley difference compared to the original load, a 27.5% further reduction compared to ordinary TOU pricing, a 4.0% decrease in daily peak load, a 9.5% increase in daily valley load, and a 2.3% higher increase in valley load compared to ordinary TOU pricing.

Figure 7 shows the variation patterns of daily load peak–valley characteristics within the statistical period under the two methods. During the initial phase of price adjustment, changes in peak and valley loads were minor. Over time, user responses intensified, causing peak and valley loads to converge inward until stabilization, achieving smooth load transitions. Under the time-delay-considered TOU scheme, valley load increased more rapidly, and the average peak–valley difference over the statistical period decreased from 8.407 GW to 4.591 GW.

6. Conclusions

This paper proposes a TOU pricing strategy that considers the time delay characteristics of user demand response. The time delay duration is determined based on load data. Then, a time-varying price elasticity matrix is established to describe the dynamic process of user demand response. Finally, a multi-objective TOU pricing model is constructed with the goals of minimizing peak–valley load difference and maximizing user satisfaction. Based on simulation analysis to the load data from a provincial power grid in China, the following conclusions are obtained:

(1)

The proposed time-varying price elasticity matrix model can effectively describe the dynamic response process of users and better reflect the variation of electricity demand with price changes.

(2)

The proposed TOU pricing strategy considering users’ delay response significantly reduces the peak–valley load difference. Without reducing user satisfaction, the peak load is reduced by 4.0%, and the valley load is increased by 9.5%. Compared with conventional TOU pricing methods without considering the time delay, the peak–valley load difference is further reduced by 27.5%. The load change using the proposed pricing strategy is smoother than that using the existing methods and ignoring the time delay.

The proposed time-of-use pricing strategy provides a basis for the coordination of multiple users with different delay responses to the TOU pricing, especially with the bulk integration of stochastic wind and solar powers, which will be explored in future research.