Time-consistent dynamic power purchase portfolio optimization method and system

Abstract

This invention discloses a time-consistent dynamic power purchase combination optimization method and system, including: acquiring load demand, spot and option prices, and accumulated power purchase costs; constructing a power purchase optimization function with the mean and variance of the terminal accumulated power purchase cost as the objective; introducing an equilibrium strategy, treating decisions at different times as a continuous self-dynamic game, and converting it into an extended HJB equation; utilizing price homogeneity and cost translation invariance, reducing the original state to a one-dimensional equivalent model; calculating equilibrium feedback control based on the one-dimensional model to determine the optimal option and spot power purchase ratio; and numerically solving the problem using a fully implicit finite difference algorithm to output a discrete power purchase strategy. This invention solves the time inconsistency problem caused by the mean-variance objective, outputs a sustainable dynamic power purchase strategy, and achieves synergistic optimization of power purchase costs and risks.

Description

Technical Field

[0001] This invention relates to the field of power system and power market technology, and in particular to a method and system for optimizing dynamic power purchase combinations based on time consistency. Background Technology

[0002] With the ongoing reform of the power market, the power purchase methods of electricity buyers have gradually shifted from traditional static one-time procurement to dynamic joint procurement covering the entire transaction cycle. While the spot market offers advantages such as timely price discovery and flexible trading, its significant price volatility exposes power buyers to high cost uncertainty. Options and other derivatives can hedge price risk to some extent; therefore, current research typically combines the spot and derivatives markets for joint power purchase optimization. Commonly used techniques include discrete-time rolling optimization, multi-stage stochastic programming, static power purchase portfolio optimization, and proportional allocation methods based on given risk preferences. These methods generally predict future prices using historical data and then adjust the power purchase ratio at discrete points in time to reduce total power purchase costs or achieve a trade-off between cost and risk.

[0003] However, existing technologies still have at least the following drawbacks. First, most methods only focus on obtaining a seemingly optimal power purchase plan at the initial moment, without addressing the inherent time inconsistency problem caused by the mean-variance objective. Therefore, as time progresses, the optimal plan obtained at the initial moment may be actively overturned at future points, making it difficult to sustain the power purchase strategy. Second, existing methods typically ignore the increasing marginal execution cost effect brought about by the expansion of option usage, i.e., ignoring the secondary convex transaction friction, which easily leads to a borderline aggressive strategy that shifts all resources to cheaper channels. This is inconsistent with the situation of limited liquidity and execution shocks in the real electricity market. Third, if spot prices, option prices, and cumulative power purchase costs are considered simultaneously in a continuous time frame, the corresponding models are often high-dimensional and complex to solve, which is not conducive to engineering deployment and online computation. Fourth, existing technologies generally lack a unified scheme that can directly output dynamic power purchase ratios and can be embedded into the power purchase decision system.

[0004] Therefore, a new dynamic electricity purchase method is needed that can uniformly characterize the cost and risk of electricity purchase in the scenario of combined spot and option electricity purchase, solve the time inconsistency problem caused by the mean-variance objective, and realize computable and deployable dynamic electricity purchase decisions through dimensionality reduction and numerical algorithms. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a method and system for optimizing dynamic power purchase combinations based on time consistency. This method solves the problems of inconsistent strategy time, neglect of secondary convex transaction friction, and complexity of solving continuous-time multi-state models under the mean-variance power purchase objective by introducing an equilibrium strategy, a quadratic convex transaction friction term, and state dimensionality reduction and numerical solution.

[0006] To achieve the above objectives, the present invention provides the following technical solution: The present invention provides a time-consistent dynamic electricity purchase combination optimization method, which includes the following steps: Set up the electricity purchase scenario: obtain the load demand of the electricity purchaser, the spot price process, and the option price process; Construct the process of price and cumulative electricity purchase cost, and establish a stochastic process of price and dynamic cost; Constructing a mean-variance power purchase optimization objective: Based on the aforementioned spot price process, option price process, and cumulative power purchase cost process, construct a power purchase optimization objective function with the mean and variance of the terminal cumulative power purchase cost as the objective; Introducing a time-consistent equilibrium strategy (hereinafter referred to as the equilibrium strategy): The decision-making process at different times is regarded as a dynamic game between the same electricity purchaser and its continuous self in continuous time. The equilibrium strategy is defined as follows: For any given time, a small deviation is made from the original strategy within any sufficiently short local time interval. This local deviation will not improve the objective function in the first order sense. Based on the equilibrium strategy, the electricity purchase optimization objective function is transformed into the extended HJB equation. Establish a one-dimensional equivalent model after state dimensionality reduction: Utilize the common driving relationship between spot price and option price and the translation invariance of cumulative electricity purchase cost to reduce the original state system to a one-dimensional equivalent model; Calculate the explicit expression for the equilibrium feedback control: Based on the aforementioned one-dimensional equivalent model, calculate the explicit expression for the equilibrium feedback control, and determine the optimal option-based electricity purchase ratio and the optimal spot electricity purchase ratio; and Discrete solutions and discrete control are output through a fully implicit finite difference algorithm: The one-dimensional equivalent model is numerically solved using a fully implicit finite difference algorithm, and the power purchase strategy at discrete time nodes is output.

[0007] Furthermore, a quadratic convex transaction friction term related to the amount of electricity purchased via options is introduced in the process of accumulating electricity purchase costs, in order to characterize the increasing marginal execution cost effect when the scale of option purchases expands.

[0008] Furthermore, the objective function for optimizing electricity purchase is: in, Indicates at time Status is The conditional expectation of the cumulative electricity purchase cost at the terminal reflects the average level of the electricity purchase cost; This represents the corresponding conditional variance, reflecting the volatility risk of electricity purchase costs; Risk aversion coefficient; This indicates the power purchase control strategy, specifically the proportion of load demand covered by options; Indicates the terminal time under the power purchase control strategy. The cumulative cost of electricity purchases; Indicates the current decision-making moment; Indicates time State variables; Furthermore, the step of reducing the dimensionality to a one-dimensional equivalent model further includes: By leveraging the shared driving force between spot prices and option prices, we define invariant parameters: ; ; in, This represents the invariant constructed from the common driving relationship between spot price and option price; Indicates spot price negative Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. Indicates the current time; Indicates time The option price; The ratio of the volatility of the price in the option channel to the volatility of the spot price; And establish a functional relationship between option prices and spot prices; After reducing the original problem to an equivalent one-dimensional model, two functions to be solved are introduced: an auxiliary function. and equilibrium cost function Constructing a one-dimensional system: Instead, the unknown function is obtained by solving a system of one-dimensional equations and numerical algorithms. Therefore, the one-dimensional system can be written as: ; ; in, ; ; in, Represents the state variable regarding spot prices. One-dimensional differential operators; This represents an auxiliary function used to characterize the parameters. Given the conditions, from time 1 At the terminal time The expected cumulative electricity purchase cost conditions; Indicates the current time; Indicates the spot price; Indicates in the parameter One-dimensional instantaneous electricity purchase cost rate under given conditions; Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; Indicates terminal time; Indicates the spot price for end-users; This represents the equilibrium cost function, used to characterize the parameters. Given conditions, the cost of time-consistent equilibrium in the mean-variance sense; Indicates the risk aversion coefficient; express Regarding spot prices The first-order partial differential; Indicates about time and spot prices A sufficiently smooth function; express Regarding time The partial derivatives; The drift parameter represents the price movement in the spot market. express Regarding spot prices The first-order partial derivative; express Regarding spot prices The second-order partial derivative; A parameter representing the volatility of spot prices; Indicates the power purchase control strategy; Indicates time User load requirements; This represents the constant value form of the invariant parameter constructed from the common driving relationship between spot price and option price; Indicates spot price of Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. The value represents the intensity of friction in a quadratic convex transaction, used to characterize the market characteristic of increasing marginal execution costs as the size of option purchases increases; Furthermore, the balanced feedback control is calculated using the following explicit formula: in, ; in, Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; This means projecting real numbers onto an interval. Projection operator on; Indicates the intensity of friction in a quadratic convex transaction; Indicates time User load requirements; Indicates the real number Projected onto interval Projection operator on; Represent any real number; Furthermore, the spot price process and the option price process are governed by the same standard Brownian motion. Driven by, respectively, geometric Brownian motion: ; ; in, The drift parameter represents the price movement in the spot market. A parameter representing the volatility of spot prices; Indicates time The spot price; Represents standard Brownian motion; Indicates the initial spot price; Indicates time The option price; The drift parameter represents the price movement of an option. A volatility parameter representing the price movement of an option; Indicates the initial option price; Furthermore, the fully implicit finite difference algorithm includes the following steps: Introducing a logarithmic price transformation into a one-dimensional equivalence model and time inverse transformation ; in, This represents the logarithmic price variable, i.e., the spot price. The natural logarithm; Indicates the time variable after inversion; Indicates the end time of the trading cycle; Indicates the current time; The unbounded interval is truncated into a finite logarithmic price interval, and homogeneous Neumann boundary conditions are set. Uniform grids are used for discretization in both time and space directions; At each time level, first solve the tridiagonal linear equation system corresponding to the auxiliary function equation, then calculate the risk correction term, and finally solve the tridiagonal linear equation system corresponding to the equilibrium cost function equation; and The Thomas algorithm is used to solve the tridiagonal linear equations at each time level.

[0009] Furthermore, the auxiliary function equation and the equilibrium cost function equation are discretized at the interior points using backward time difference and spatial central difference, respectively forming the following tridiagonal linear equation system: The discrete format corresponding to the auxiliary function equation is: ; Organized into ; in, Describing auxiliary functions Discrete value at the nth time level and the i-th spatial grid node; Indicates the time grid step size; Represents the convection term coefficient; Represents the diffusion term coefficient; Indicates the spatial grid step size; This represents the discrete value of the source term at the (n+1)th time level and the ith spatial grid node; Represents the discrete diffusion coefficient; Represents the discrete convection coefficient; Indicates the spatial grid node index; Indicates the number of spatial grid divisions; The discrete format corresponding to the equilibrium cost function equation is: ; Represents the equilibrium cost function Discrete value at the nth time level and the i-th spatial grid node; This represents the discrete value of the risk correction term at the (n+1)th time level and the ith spatial grid node; The present invention provides a time-consistent dynamic power purchase combination optimization system for the electricity market, used to implement the above method, including: The data acquisition module is used to acquire the load demand, spot price process, option price process, and cumulative electricity purchase cost process of the electricity purchaser. The model building module, connected to the data acquisition module, is used to construct an electricity purchase optimization objective function based on the data acquired by the data acquisition module, with the mean and variance of the cumulative electricity purchase cost of the terminal as the objective. The equilibrium strategy solution module, connected to the model construction module, is used to introduce an equilibrium strategy. It treats the decision-making process at different times as a dynamic game between the same electricity purchaser and its continuous self over continuous time. The equilibrium strategy is defined as follows: for any given moment, a small deviation is made from the original strategy within any sufficiently short local time interval, such a local deviation does not improve the objective function in the first-order sense. Based on the equilibrium strategy, the electricity purchase optimization objective function is transformed into an extended HJB equation. The dimensionality reduction module, connected to the equilibrium strategy solution module, is used to reduce the original state system to a one-dimensional equivalent model by utilizing the common driving relationship between spot prices and option prices and the translation invariance of cumulative electricity purchase costs. The feedback control calculation module, connected to the dimensionality reduction processing module, is used to calculate the equilibrium feedback control based on the one-dimensional equivalence model, and determine the optimal option-based power purchase ratio and the optimal spot power purchase ratio; and The numerical solution module, connected to the feedback control calculation module, is used to numerically solve the one-dimensional equivalent model using a fully implicit finite difference algorithm and output the electricity purchase strategy at discrete time nodes.

[0010] Furthermore, the numerical solution module further includes: Transformation unit, used to perform logarithmic price transformation and time inversion transformation on a one-dimensional equivalent model; Discretization unit, used to discretize the transformed model into a uniform grid in both time and space; The equation system construction unit is used to construct a tridiagonal linear equation system corresponding to the auxiliary function equation and the equilibrium cost function equation at each time level; and The solution unit is used to solve the tridiagonal linear equations sequentially using the Thomas algorithm to obtain discrete equilibrium feedback control solutions.

[0011] The beneficial effects of this invention are as follows: This invention provides a time-consistent dynamic power purchase combination optimization method and system, comprising: acquiring the load demand, spot price process, option price process, and cumulative power purchase cost process of the power purchase entity; constructing a power purchase optimization objective function with the mean and variance of the terminal cumulative power purchase cost as the objective; introducing an equilibrium strategy, treating the decision-making process at different times as a dynamic game between continuous self-entities, and converting it into an extended HJB equation; utilizing the common driving relationship between spot and option prices and the translation invariance of cumulative power purchase cost, reducing the original state system to a one-dimensional equivalent model; calculating equilibrium feedback control based on the one-dimensional model to determine the optimal option power purchase ratio and the optimal spot power purchase ratio; and numerically solving the one-dimensional equivalent model using a fully implicit finite difference algorithm to output the power purchase strategy at discrete time nodes. This invention solves the time inconsistency problem caused by the mean-variance objective, outputs a sustainable dynamic power purchase strategy, and achieves synergistic optimization of power purchase cost and risk. Compared with the prior art, this invention has the following specific beneficial effects: This method is the first to integrate mean-variance risk characterization, time consistency constraints, and quadratic convex transaction friction constraints into a continuous-time power purchase optimization framework, simultaneously addressing power purchase costs, risk control, and strategy sustainability in the same model.

[0012] This invention addresses the time inconsistency problem in existing electricity purchase strategies, which often exhibit initial optimality but subsequent failure under a mean-variance objective. By introducing the concept of time-consistent equilibrium, a dynamic electricity purchase strategy solution framework that can be continuously executed throughout the entire transaction cycle is constructed.

[0013] This method introduces a quadratic convex transaction friction term related to the amount of electricity purchased via options during the process of accumulating electricity purchase costs. This term is used to characterize the market characteristic of increasing marginal execution costs when the scale of option purchases expands. This effectively suppresses boundary-oriented aggressive strategies that are detached from actual market conditions and improves the real-world applicability of the model results.

[0014] This method utilizes the structural characteristics of cumulative cost variables and the shared driving relationship between spot prices and option prices to transform the original multi-state power purchase optimization problem into a one-dimensional solution model. This significantly reduces the complexity of the model solution and improves the computability and engineering feasibility of the method.

[0015] This method provides a directly calculable equilibrium feedback control formula, enabling electricity purchasers to directly determine the option purchase ratio and the spot purchase ratio based on the current spot price, user load demand function, and related parameters, thereby enhancing the real-time response capability of the strategy.

[0016] This method further constructs a fully implicit finite difference solution algorithm applicable to the above one-dimensional equivalent model, explicitly giving the auxiliary function equation, risk correction term, equilibrium cost function equation and its corresponding tridiagonal linear equation system, and outputting discrete auxiliary function solution, discrete equilibrium cost function solution and discrete control results, thus forming a complete and reproducible numerical implementation path.

[0017] The purpose of this invention is to provide a time-consistent dynamic power purchase combination optimization method for power market purchasers, in order to solve the technical problems existing in the current power purchase optimization method, such as the inconsistency of strategy time under the mean-variance objective, the overly aggressive control due to ignoring the secondary convex transaction friction, and the complexity and difficulty in engineering implementation of the continuous time multi-state model. Thus, under the premise of meeting the user load demand, it outputs a dynamic power purchase strategy that can be continuously executed throughout the entire transaction cycle, and achieves the synergistic optimization of the terminal's cumulative power purchase cost and power purchase risk.

[0018] The above and other objects, advantages, and features of the present invention will be more fully set forth and demonstrated through the following detailed description of specific embodiments in conjunction with the accompanying drawings. Those skilled in the art, upon referring to the following detailed description and the accompanying drawings, will be able to better understand and realize the above advantages of the present invention. Other objects, features, and advantages of the present invention will become clearer after being described in detail in the detailed description section in conjunction with the accompanying drawings. Attached Figure Description

[0019] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following drawings are provided for illustration.

[0020] Figure 1 Here is a flowchart of the time-consistent dynamic power purchase combination optimization method; Figure 2 This is a schematic diagram of a time-consistent dynamic power purchase combination optimization system. Figure 3 The flowchart for solving the problem using the fully implicit finite difference algorithm is shown below. Figure 4 A three-dimensional surface plot of the equalization control under the reference parameters; Figure 5 This is a schematic diagram simulating the spot price path and the corresponding equilibrium control trajectory. Detailed Implementation

[0021] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention. Example 1

[0022] like Figure 1As shown in the figure, the time-consistent dynamic power purchase combination optimization method provided in this embodiment is characterized by including the following steps: Set up the electricity purchase scenario: obtain the load demand of the electricity purchaser, the spot price process, and the option price process; Construct the process of price and cumulative electricity purchase cost, and establish a stochastic process of price and dynamic cost; Constructing a mean-variance power purchase optimization objective: Based on the aforementioned spot price process, option price process, and cumulative power purchase cost process, construct a power purchase optimization objective function with the mean and variance of the terminal cumulative power purchase cost as the objective; An equilibrium strategy is introduced: the decision-making process at different times is regarded as a dynamic game between the same electricity purchaser and its continuous self over continuous time. The equilibrium strategy is defined as: for any given time, a small deviation is made from the original strategy within any sufficiently short local time interval. This local deviation will not improve the objective function in the first order sense. Based on the equilibrium strategy, the electricity purchase optimization objective function is transformed into the extended HJB equation. Establish a one-dimensional equivalent model after state dimensionality reduction: Utilize the common driving relationship between spot price and option price and the translation invariance of cumulative electricity purchase cost to reduce the original state system to a one-dimensional equivalent model; Calculate the explicit expression for the equilibrium feedback control: Based on the aforementioned one-dimensional equivalent model, calculate the explicit equilibrium feedback control to determine the optimal option-based power purchase ratio and the optimal spot power purchase ratio; and Discrete solutions and discrete control are output through a fully implicit finite difference algorithm: The one-dimensional equivalent model is numerically solved using a fully implicit finite difference algorithm, and the power purchase strategy at discrete time nodes is output.

[0023] In this embodiment, a quadratic convex transaction friction term related to the amount of electricity purchased via options is introduced in the process of accumulating electricity purchase costs, in order to characterize the increasing marginal execution cost effect when the scale of option purchases expands.

[0024] The objective function for optimizing electricity purchase in this embodiment is: in, Indicates at time Status is The conditional expectation of the cumulative electricity purchase cost at the terminal reflects the average level of the electricity purchase cost; This represents the corresponding conditional variance, reflecting the volatility risk of electricity purchase costs; Risk aversion coefficient; This indicates the power purchase control strategy, specifically the proportion of load demand covered by options; Indicates the terminal time under the power purchase control strategy. The cumulative cost of electricity purchases; Indicates the current decision-making moment; Indicates time State variables; The step of reducing the dimension to a one-dimensional equivalent model described in this embodiment further includes: By leveraging the shared driving force between spot prices and option prices, we define invariant parameters: ; ; in, This represents the invariant constructed from the common driving relationship between spot price and option price; Indicates spot price negative Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. Indicates the current time; express; The ratio of the volatility of the price in the option channel to the volatility of the spot price; And establish a functional relationship between option prices and spot prices; Introducing two functions to be solved: an auxiliary function and equilibrium cost function Constructing a one-dimensional system: Instead, the unknown function is obtained by solving a system of one-dimensional equations and numerical algorithms. Therefore, the one-dimensional system can be written as: ; ; in, ; ; in, Represents the state variable regarding spot prices. One-dimensional differential operators; This represents an auxiliary function used to characterize the parameters. Given the conditions, from time 1 At the terminal time The expected cumulative electricity purchase cost conditions; Indicates the current time; Indicates the spot price; Indicates in the parameter One-dimensional instantaneous electricity purchase cost rate under given conditions; Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; Indicates terminal time; Indicates the spot price for end-users; This represents the equilibrium cost function, used to characterize the parameters. Given conditions, the cost of time-consistent equilibrium in the mean-variance sense; Indicates the risk aversion coefficient; express Regarding spot prices The first-order partial differential; Indicates about time and spot prices A sufficiently smooth function; express Regarding time The partial derivatives; The drift parameter represents the price movement in the spot market. express Regarding spot prices The first-order partial derivative; express Regarding spot prices The second-order partial derivative; A parameter representing the volatility of spot prices; Indicates the power purchase control strategy; Indicates time User load requirements; This represents the constant value form of the invariant parameter constructed from the common driving relationship between spot price and option price; Indicates spot price of Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. The value represents the intensity of friction in a quadratic convex transaction, used to characterize the market characteristic of increasing marginal execution costs as the size of option purchases increases; The balanced feedback control described in this embodiment is calculated using the following formula: in, ; in, Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; This means projecting real numbers onto an interval. Projection operator on; Indicates the intensity of friction in a quadratic convex transaction; Indicates time User load requirements; Indicates the real number Projected onto interval Projection operator on; Represent any real number; In this embodiment, the spot price process and the option price process are governed by the same standard Brownian motion. Driven by, respectively, geometric Brownian motion: ; ; in, The drift parameter represents the price movement in the spot market. A parameter representing the volatility of spot prices; Indicates time The spot price; Represents standard Brownian motion; Indicates the initial spot price; Indicates time The option price; The drift parameter represents the price movement of an option. A volatility parameter representing the price movement of an option; Indicates the initial option price; The fully implicit finite difference algorithm described in this embodiment includes the following steps: Introducing a logarithmic price transformation into a one-dimensional equivalence model and time inverse transformation ; in, This represents the logarithmic price variable, i.e., the spot price. The natural logarithm; Indicates the time variable after inversion; Indicates the end time of the trading cycle; Indicates the current time; The unbounded interval is truncated into a finite logarithmic price interval, and homogeneous Neumann boundary conditions are set. Uniform grids are used for discretization in both time and space directions; At each time level, first solve the tridiagonal linear equation system corresponding to the auxiliary function equation, then calculate the risk correction term, and finally solve the tridiagonal linear equation system corresponding to the equilibrium cost function equation; and The Thomas algorithm is used to solve the tridiagonal linear equations at each time level.

[0025] In this embodiment, the auxiliary function equation and the equilibrium cost function equation are discretized at the interior points using backward time difference and central spatial difference, respectively forming the following tridiagonal linear equation system: The discrete format corresponding to the auxiliary function equation is: ; Organized into ; in, Describing auxiliary functions Discrete value at the nth time level and the i-th spatial grid node; Indicates the time grid step size; Represents the convection term coefficient; Represents the diffusion term coefficient; Indicates the spatial grid step size; This represents the discrete value of the source term at the (n+1)th time level and the ith spatial grid node; Represents the discrete diffusion coefficient; Represents the discrete convection coefficient; Indicates the spatial grid node index; Indicates the number of spatial grid divisions; The discrete format corresponding to the equilibrium cost function equation is: ; in, Represents the equilibrium cost function Discrete value at the nth time level and the i-th spatial grid node; This represents the discrete value of the risk correction term at the (n+1)th time level and the ith spatial grid node; like Figure 2 As shown in this embodiment, the time-consistent dynamic power purchase combination optimization system for the electricity market is characterized by comprising: The data acquisition module is used to acquire the load demand, spot price process, option price process, and cumulative electricity purchase cost process of the electricity purchaser. The model building module, connected to the data acquisition module, is used to construct an electricity purchase optimization objective function based on the data acquired by the data acquisition module, with the mean and variance of the cumulative electricity purchase cost of the terminal as the objective. The equilibrium strategy solution module, connected to the model construction module, is used to introduce an equilibrium strategy. It treats the decision-making process at different times as a dynamic game between the same electricity purchaser and its continuous self over continuous time. The equilibrium strategy is defined as follows: for any given moment, a small deviation is made from the original strategy within any sufficiently short local time interval, such a local deviation does not improve the objective function in the first-order sense. Based on the equilibrium strategy, the electricity purchase optimization objective function is transformed into an extended HJB equation. The dimensionality reduction module, connected to the equilibrium strategy solution module, is used to reduce the original state system to a one-dimensional equivalent model by utilizing the common driving relationship between spot prices and option prices and the translation invariance of cumulative electricity purchase costs. The feedback control calculation module, connected to the dimensionality reduction processing module, is used to calculate the equilibrium feedback control based on the one-dimensional equivalence model, and determine the optimal option-based power purchase ratio and the optimal spot power purchase ratio; and The numerical solution module, connected to the feedback control calculation module, is used to numerically solve the one-dimensional equivalent model using a fully implicit finite difference algorithm and output the electricity purchase strategy at discrete time nodes.

[0026] The numerical solution module described in this embodiment further includes: Transformation unit, used to perform logarithmic price transformation and time inversion transformation on a one-dimensional equivalent model; Discretization unit, used to discretize the transformed model into a uniform grid in both time and space; The equation system construction unit is used to construct a tridiagonal linear equation system corresponding to the auxiliary function equation and the equilibrium cost function equation at each time level; and The solution unit is used to solve the tridiagonal linear equations sequentially using the Thomas algorithm to obtain discrete equilibrium feedback control solutions. Example 2

[0027] This embodiment further illustrates the method with specific illustrations and implementation details. Specifically: The basic principle of this method is to address the time inconsistency problem faced by electricity purchasers when dynamically co-purchasing electricity through spot and option transactions. A continuous-time electricity purchase optimization method based on the mean-variance criterion is constructed. Assume the electricity purchaser operates within a single trading cycle... The user simultaneously participates in both spot and option-based electricity purchases, and the user load demand is a known deterministic continuous function. The option-based electricity purchase ratio is denoted as The spot purchase ratio is The corresponding option purchase volume is Spot purchase volume is To characterize the key information that needs to be tracked during the electricity purchase decision-making process, the initial state process is defined as follows: in, Indicates time The spot price, Indicates time The option price, Indicates the end time The cumulative cost of electricity purchase. The spot price process and the option price process are driven by the same standard Brownian motion, respectively satisfying: in, The drift parameter represents the price movement in the spot market. A parameter representing the volatility of spot prices; Indicates time The spot price; Represents standard Brownian motion; Indicates the initial spot price; Indicates time The option price; The drift parameter represents the price movement of an option. A volatility parameter representing the price movement of an option; Indicates the initial option price; The cumulative electricity purchase cost process meets ; in, It is a quadratic convex transaction friction term introduced regarding the electricity purchased through options.

[0028] Indicates the end time The cumulative cost of electricity purchases; Indicates time Electricity purchase control strategy; Indicates time User load requirements; The term represents the intensity of quadratic convex transaction friction, used to characterize the market characteristic of increasing marginal execution costs as the scale of option purchases expands. Without this term, overly aggressive boundary control can easily be achieved when the price difference is large.

[0029] Electricity purchasers not only focus on the average level of cumulative electricity purchase costs at the end of the transaction cycle, but also on the volatility risk of these costs. Therefore, they use the cumulative electricity purchase cost at the end as a benchmark. To optimize the target, a mean-variance electricity purchase objective is constructed: in, Indicates at time Status is The conditional expectation of the cumulative electricity purchase cost at the terminal reflects the average level of the electricity purchase cost; This represents the corresponding conditional variance, reflecting the volatility risk of electricity purchase costs; Risk aversion coefficient; This indicates the power purchase control strategy, specifically the proportion of load demand covered by options; Indicates the terminal time under the power purchase control strategy. The cumulative cost of electricity purchases; Indicates the current decision-making moment; Indicates time State variables; Indicates time The spot price; Indicates time The option price; Indicates the end time The cumulative cost of electricity purchases; Unlike objectives that only aim to minimize expected cost, when a variance term is introduced into the objective function, the objective function generally no longer satisfies the recursive structure required by the standard Bellman optimality principle.

[0030] For any ,have in, Indicates the current moment Current state conditions The following is a power purchase control strategy. The corresponding mean-variance objective function value for electricity purchase; Indicates a future moment Future state Under these conditions, the power purchase control strategy will continue to be implemented. The corresponding mean-variance objective function value for electricity purchase; Indicates at time Conditional expectation when the state is ; Indicates a future moment Future state Conditional expectation under given conditions; Indicates the corresponding conditional variance; Risk aversion coefficient; It is evident that the recursive relation required by standard dynamic programming generally does not hold under a mean-variance objective, because the objective function contains a non-zero additional term that cannot be absorbed by the recursive structure. This means that decision-makers at the current moment, when evaluating subsequent electricity purchasing strategies, must consider not only the local value of a particular realized future state, but also the dispersion of expected values ​​between different future states. This information cannot be recursively absorbed by a single future state, thus disrupting the recursive relationship upon which traditional dynamic programming relies. The time inconsistency here is not caused by changes in external information or sudden price fluctuations, but rather by an endogenous property of the mean-variance objective itself.

[0031] In simple terms, time inconsistency means that a power purchaser initially formulates a seemingly optimal power purchase plan for the entire transaction period. However, as time progresses and they recalculate their purchase strategy for the remaining transaction period, they find that the initial plan is no longer the optimal choice under current conditions, and thus they will actively abandon the original plan. In other words, the optimal decision at the initial moment and the optimal decision at future moments are not evaluated by the same criteria. For power purchasers, this is not merely a theoretical problem, but a practical issue directly related to the stable implementation of the power purchase plan. If a strategy is only valid at the initial moment and requires frequent adjustments during execution, it not only increases transaction costs but may also lead to additional risk exposure due to the disconnect between planning and execution.

[0032] To address the aforementioned time inconsistency problem, this invention no longer seeks pre-optimal control that only holds true at the initial moment, as is the traditional approach. Instead, it introduces the concept of time-consistent equilibrium, viewing the decision-making process at different moments as a dynamic game between the same electricity-purchasing entity and its continuous self over continuous time. An equilibrium strategy refers to a strategy where, for any given moment, a small deviation from the original strategy is made only within a sufficiently short local time interval, and the original strategy is subsequently continued. This local deviation will not improve the objective function in the first-order sense. Specifically, for any given... and real numbers Define the perturbation strategy ; If for any All ; Then it is called For the equilibrium strategy in the electricity purchase optimization problem, the corresponding time-consistent equilibrium cost function is: ; in, Indicates at time Start, length is The perturbation strategy obtained by perturbing the original strategy within a local time interval; Indicates the local disturbance range Any feasible power purchase control adopted internally; This indicates that the candidate time-consistent equilibrium power purchase strategy is at time [time]. Control values; This represents the length of the local disturbance time interval; it is a small positive parameter. This represents the overall power purchase control strategy derived from local disturbances; Indicates time State variables; This indicates a consistent, balanced power purchase strategy based on candidate timing. Indicates at time ,state Under the condition corresponding to the equilibrium strategy The equilibrium cost function value; Indicates at time ,state Execution strategy under conditions The mean-variance objective function value for electricity purchase at that time; This definition indicates that making local modifications to the strategy at any given moment will not lead to an improvement in the objective in the first-order sense, thus the strategy has dynamic and sustainable execution.

[0033] After defining the concept of equilibrium, this invention employs the extended HJB equations to solve the aforementioned equilibrium strategy, transforming the time inconsistency problem, which originally did not satisfy the standard recursive structure, into a set of analyzable coupled partial differential equations. However, the extended HJB system in its original state still contains three state variables: spot price, option price, and cumulative electricity purchase cost, making direct solution difficult. To reduce the solution complexity, this invention further utilizes two structural features inherent in the original state system for continuous dimensionality reduction. First, the cumulative electricity purchase cost variable exhibits translation invariance; that is, adding a constant to the initial cumulative cost does not change the ranking of different controls, thus allowing the cumulative cost variable to be separated from the state system. Second, the spot price and option price are driven by the same Brownian motion, and their randomness originates from only the same random factor, thus possessing a linkage structure that can be used to eliminate redundant dimensions. Based on these two structural features, this invention first reduces the original three-dimensional state system to a two-dimensional system, and then further constructs motion invariants, rewriting the two-dimensional system as a one-dimensional system.

[0034] To obtain numerical solutions applicable to engineering implementation, this invention constructs a fully implicit finite difference algorithm on the dimensionality-reduced one-dimensional system. The fully implicit finite difference method is considered because the equations obtained after variable transformation in the one-dimensional system belong to the convection-diffusion type parabolic equations, and the equilibrium cost function equation contains a nonlinear risk correction term induced by the derivative. The fully implicit finite difference algorithm exhibits good numerical stability for this type of parabolic equation, avoiding the strict time step constraints of explicit schemes; simultaneously, it helps maintain the overall consistency of the discrete system at each time level and reduces the discrete system to a tridiagonal linear system of equations at each level, facilitating efficient solution using the Thomas algorithm. Example 3

[0035] The method provided in this embodiment is specifically constructed according to the following steps: 1. Set up electricity purchase scenarios Assume the electricity purchaser is in a trading cycle It participates in both spot and option-based electricity purchases. User load demand is a known deterministic continuous function. And for any have ;time The proportion of load demand covered by options is denoted as The proportion of demand met through spot electricity purchases is: Correspondingly, the option-based electricity purchase volume is Spot purchase volume is ; 2. Construction of Price and Cumulative Electricity Purchase Cost Process Construct the spot price process and the option price process, and make both follow the same standard Brownian motion. drive: in, These are the drift parameters for spot price and option price, respectively. These are the volatility parameters for spot prices and option prices, respectively.

[0036] Based on the spot electricity purchase cost, the option electricity purchase cost, and the secondary convex transaction friction cost related to the option purchase volume, a cumulative electricity purchase cost process is established. in, This is a quadratic convex trading friction intensity used to characterize the market characteristic of increasing marginal execution costs as the size of option purchases increases. It also avoids boundary-point strategies such as buying all spot or all options, making the output results more consistent with real-world trading scenarios. The larger the value, the worse the liquidity of the options market, and the stronger the cost-increasing effect of large-scale procurement.

[0037] 3. Construct a mean-variance power purchase optimization objective Using terminal cumulative electricity purchase cost The mean-variance form is used to describe the cost-risk dual trade-off objective of the electricity purchaser: Finding the electricity purchasing strategy that minimizes the objective function ,Right now: Among them, conditional expectation describes the average level of the cumulative electricity purchase cost at the terminal, while conditional variance describes the volatility risk of the cumulative electricity purchase cost at the terminal. This represents the risk aversion coefficient. Since the objective function does not satisfy the standard Bellman recurrence structure, an extended HJB solution framework is established using the time-consistent equilibrium approach to avoid obtaining a pre-committed optimal solution that will be actively overturned at a future time point.

[0038] 4. Establish a one-dimensional equivalent model after state dimensionality reduction. By utilizing the shared driving force between spot prices and option prices, we define... ; in, Indicates option price volatility With spot price volatility The ratio; This represents the drift term resulting from the spot price process and the option price process. Fluctuation Term The constant parameters that are jointly determined; Define invariant parameters ;then ; in, This represents the invariant constructed from the common driving relationship between spot price and option price; Indicates spot price negative Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. Indicates the current time; express; The ratio of the volatility of the price in the option channel to the volatility of the spot price; Therefore, the option prices in the original problem can be uniformly rewritten as spot prices. and parameters The function is obtained, thus yielding a one-dimensional equivalent solution model.

[0039] After transforming the original problem into a one-dimensional model by utilizing the shared driving relationship between spot prices and option prices, two functions to be solved are introduced: where, As an auxiliary function, it is used to characterize the parameters. Given the conditions, from time 1 At the terminal time The expected cumulative electricity purchase cost conditions; The equilibrium cost function is used to characterize the time-consistent equilibrium cost under the same conditions and in the mean-variance sense. Neither of these are pre-given known functions. Instead, the unknown function is obtained by solving a system of one-dimensional equations and numerical algorithms. The one-dimensional system is then written as... in, ; ; in, Represents the state variable regarding spot prices. One-dimensional differential operators; This represents an auxiliary function used to characterize the parameters. Given the conditions, from time 1 At the terminal time The expected cumulative electricity purchase cost conditions; Indicates the current time; Indicates the spot price; Indicates in the parameter One-dimensional instantaneous electricity purchase cost rate under given conditions; Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; Indicates terminal time; Indicates the spot price for end-users; This represents the equilibrium cost function, used to characterize the parameters. Given conditions, the cost of time-consistent equilibrium in the mean-variance sense; Indicates the risk aversion coefficient; express Regarding spot prices The first-order partial differential; Indicates about time and spot prices A sufficiently smooth function; express Regarding time The partial derivatives; The drift parameter represents the price movement in the spot market. express Regarding spot prices The first-order partial derivative; express Regarding spot prices The second-order partial derivative; A parameter representing the volatility of spot prices; Indicates the power purchase control strategy; Indicates time User load requirements; This represents the constant value form of the invariant parameter constructed from the common driving relationship between spot price and option price; Indicates spot price of Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. The value represents the intensity of friction in a quadratic convex transaction, used to characterize the market characteristic of increasing marginal execution costs as the size of option purchases increases; 5. Calculation-based balanced feedback control Under this model, the explicit expression for balanced feedback control is written as: in The corresponding spot electricity purchase ratio is in, Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; This means projecting real numbers onto an interval. Projection operator on; Indicates the intensity of friction in a quadratic convex transaction; Indicates time User load requirements; Indicates the real number Projected onto interval Projection operator on; Represent any real number; 6. Output discrete solutions and discrete control through a fully implicit finite difference algorithm. like Figure 3 As shown, Figure 3 The flowchart for solving the fully implicit finite difference algorithm is as follows: First, logarithmic price and time inversion transformations are performed on the one-dimensional equivalent model, and the grid and boundary conditions are set. Then, at each time level, the tridiagonal linear equations corresponding to the auxiliary function are solved sequentially, the risk correction term is calculated, and the tridiagonal linear equations corresponding to the equilibrium cost function are solved again. The Thomas algorithm is used for efficient solution at each step. Finally, the discrete solution and discrete control results are output, as detailed below: To obtain numerical solutions usable for engineering implementation, a logarithmic price transformation and a time inversion transformation are introduced into the one-dimensional equivalent model: ; in, This represents the logarithmic price variable, i.e., the spot price. The natural logarithm; Indicates the time variable after inversion; Indicates the end time of the trading cycle; Indicates the current time; definition here Auxiliary function The representation after variable transformation Equilibrium cost function The representation after variable transformation. And the initial conditions after transformation are... remember The transformed one-dimensional system is written as Among them, auxiliary function The conditional expectation of the cumulative cost of the corresponding terminal, the equilibrium cost function The corresponding equilibrium cost in the mean-variance sense, source term Defined as Truncate the unbounded interval into a finite logarithmic price interval. Homogeneous Neumann boundary conditions are set at both ends. A uniform grid is assumed to be used in the spatial direction. , in, Take a uniform grid in the time direction. , in . remember Initialize to For the auxiliary function equation, discretization at the interior points using backward time difference and central spatial difference yields the following results: in . remember , The above equation can then be rearranged into a system of tridiagonal linear equations. in, Describing auxiliary functions Discrete value at the nth time level and the i-th spatial grid node; Indicates the time grid step size; Represents the convection term coefficient; Represents the diffusion term coefficient; Indicates the spatial grid step size; This represents the discrete value of the source term at the (n+1)th time level and the ith spatial grid node; Represents the discrete diffusion coefficient; Represents the discrete convection coefficient; Indicates the spatial grid node index; Indicates the number of spatial grid divisions; In seeking Then, the first derivative is approximated using the central difference to calculate the risk correction term. ; in, This represents the discrete value of the risk correction term at the (n+1)th time level and the ith spatial grid node; Indicates the risk aversion coefficient; A parameter representing the volatility of spot prices; Discretizing the equilibrium cost function equation in the same way yields the tridiagonal linear equation system corresponding to the equilibrium cost function. in, Represents the equilibrium cost function Discrete value at the nth time level and the i-th spatial grid node; This represents the discrete value of the risk correction term at the (n+1)th time level and the ith spatial grid node; At each time level, the process proceeds in the order of solving the auxiliary function equation, calculating the risk correction term, and finally solving the equilibrium cost function equation. The Thomas algorithm is used to solve the tridiagonal linear equation system corresponding to each time level.

[0040] After completing the numerical solution, output the discrete auxiliary function solution. Discrete equilibrium cost function solution and discrete control ; Based on this, the option-based electricity purchase ratio, spot electricity purchase ratio, and cumulative electricity purchase cost for each discrete node within the trading period are further obtained. To verify the effectiveness of the method of this invention, the dynamic equilibrium strategy, pure spot strategy, fixed ratio strategy, and pure option strategy can be compared on the same batch of simulated spot price paths. For each strategy, the terminal cumulative electricity purchase cost is progressively accumulated according to the immediate cost expression, and the sample mean, sample variance, and mean-variance combined target value of the terminal cumulative electricity purchase cost are calculated: ; ; ; in, To simulate the number of paths; Indicates the cumulative cost of electricity purchased by the end user. The sample mean; Indicates the simulated path index; Indicates the first Terminal time under simulated path The cumulative cost of electricity purchases; Indicates the cumulative cost of electricity purchased by the end user. The sample variance; This represents the mean-variance combined target value estimate of the cumulative electricity purchase cost at the terminal. This represents the risk aversion coefficient. Example 4

[0041] This embodiment illustrates the specific process of this method based on a specific electricity purchase scenario: 1. Set up electricity purchase scenarios In this embodiment, the length of the electricity purchase period is taken as... (This can be considered as one trading cycle), initial spot price Initial option price Spot price drift coefficient Volatility Option price drift coefficient Volatility , Secondary convex transaction friction intensity Risk aversion coefficient From the above parameters, we can obtain... , , , And obtain the invariant parameters from the initial value relationship. The user load demand function is represented by the following smoothed periodic function: 2. Construction of Price and Cumulative Electricity Purchase Cost Process Under the above parameters, the pricing process is as follows: in, These are the drift parameters, These are volatility parameters, Indicates time The spot price; Represents standard Brownian motion; Indicates the initial spot price; Indicates time The option price; The drift parameter represents the price movement of an option. A volatility parameter representing the price movement of an option; Indicates the initial option price; The process of accumulating electricity purchase costs is written as follows: ; in, It is a quadratic convex transaction friction term introduced regarding the electricity purchased through options. Indicates the end time The cumulative cost of electricity purchases; Indicates time Electricity purchase control strategy; Indicates time User load requirements; The term represents the intensity of quadratic convex transaction friction, used to characterize the market characteristic of increasing marginal execution costs as the scale of option purchases expands. Without this term, overly aggressive boundary control can easily be achieved when the price difference is large.

[0042] In the program implementation, parameters are first input, and then a dynamic electricity purchase model for the electricity purchaser is established based on the above-mentioned price process and cost process.

[0043] 3. Construct a mean-variance objective Establish terminal mean-variance electricity purchase target An extended HJB solution framework is established based on the time-consistent equilibrium concept. Because this problem exhibits time inconsistency, this embodiment does not employ traditional dynamic programming for direct backtracking, but instead uses an equilibrium solution method to ensure the strategy can be continuously executed throughout the entire trading cycle.

[0044] 4. Establish a one-dimensional equivalence model By utilizing the shared driving force between spot prices and option prices, we define... Define invariant parameters then Therefore, the option prices in the original problem can be uniformly rewritten as spot prices. and parameters The function is obtained, thus yielding a one-dimensional equivalent solution model.

[0045] After transforming the original problem into a one-dimensional model by utilizing the shared driving relationship between spot prices and option prices, two functions to be solved are introduced: where, As an auxiliary function, it is used to characterize the parameters. Given the conditions, from time 1 At the terminal time The expected cumulative electricity purchase cost conditions; Let be the equilibrium cost function, used to characterize the time-consistent equilibrium cost under the same conditions and in the mean-variance sense. Neither is a pre-given known function, but rather an unknown function obtained by solving a subsequently established system of one-dimensional equations and numerical algorithms. The one-dimensional system can then be written as: in, ; 5. Calculation-based balanced feedback control In the one-dimensional model, the optimal option-based electricity purchase ratio is calculated point by point using the following formula: The corresponding spot electricity purchase ratio is In the program implementation, the discrete control at each time point and at each price state is directly calculated from this definition.

[0046] 6. The fully implicit finite difference algorithm outputs discrete solutions and discrete control. To perform numerical solutions, this embodiment employs logarithmic price transformation and time inversion transformation: definition In this embodiment, the spot price calculation range is taken as follows: The corresponding logarithmic price range is: To facilitate program implementation, the number of spatial grids is taken. Number of time grids The data is discretized uniformly. Homogeneous Neumann boundary conditions are used.

[0047] remember Initialize to At each time level, first apply the explicit expression of the balanced feedback control. Calculate discrete control at the current time level Press again Calculate the source term. Then, solve the tridiagonal linear equations corresponding to the auxiliary function, taking into account the boundary conditions, to obtain... And then according to Calculate the risk correction term ; Finally, by combining the boundary conditions, the tridiagonal linear equations corresponding to the equilibrium cost function are solved, resulting in... The tridiagonal linear system at each time level is solved using the Thomas algorithm until the initial time is reached, yielding discrete solutions for each discrete time point throughout the entire trading cycle. and discrete control .

[0048] Calculations performed under the aforementioned benchmark parameters show that the equilibrium option electricity purchase ratio exhibits a continuous and smooth feedback structure with respect to time and spot prices. The equilibrium control results under the benchmark parameters are plotted as follows: Figure 4 As shown, Figure 4 The three-dimensional surface plot of the equalization control under the reference parameters is generated by... Figure 4 It is evident that the equilibrium option-based electricity purchase ratio increases with rising spot prices. That is, when spot prices are high, electricity purchasers are more inclined to increase the option-based electricity purchase ratio to lock in future electricity costs. Conversely, when demand is high, the control becomes more conservative due to the presence of a quadratic convex transaction friction term. This result demonstrates that the present invention outputs a continuous equilibrium feedback control with clear economic implications, rather than a simple static ratio rule.

[0049] Furthermore, to more intuitively illustrate the dynamic response mechanism of equilibrium control to changes in spot prices, this embodiment constructs two representative spot price paths based on the geometric Brownian motion simulation of the spot price process. The simulation step count is set to 250, with corresponding random seeds of 123 and 456, respectively. On each price path, the optimal option electricity purchase ratio is calculated hourly according to the equilibrium feedback control formula, thereby obtaining... Figure 5 The price path and corresponding control trajectory are shown. Figure 5 To simulate the spot price path and the corresponding equilibrium control trajectory, by Figure 5 It is evident that when spot prices rise, the proportion of electricity purchased via options increases accordingly; however, when demand is at a high level, even with the same price difference, the optimal control will still contract appropriately due to the increased secondary convexity trading friction. This result demonstrates that the electricity purchase strategy output by this invention can simultaneously reflect changes in market prices, changes in demand scale, and the effect of secondary convexity trading friction.

[0050] To verify the overall effectiveness of this method, this embodiment further employs the Monte Carlo method to compare the terminal cumulative cost results of different electricity purchase strategies on the same batch of simulated spot price paths. To ensure a fair comparison, all strategies use the same batch of spot price paths, with 20,000 paths, 250 deviations, and a random seed of 123; under a single Brownian drive condition, the invariant relationship is then used... The corresponding option price path is restored. The strategies compared include: dynamic equilibrium strategy, pure spot strategy, fixed ratio 0.25 strategy, fixed ratio 0.50 strategy, fixed ratio 0.75 strategy, and pure option strategy. For each strategy, the terminal cumulative electricity purchase cost is progressively accumulated according to the immediate cost expression, and the calculation is performed. The results are listed in Table 1.

[0051] Table 1. Comparison of terminal costs for different strategies The comparison results show that the dynamic equilibrium strategy proposed in this invention performs better in the mean-variance sense and can achieve a better balance between expected electricity purchase costs and risk exposure.

[0052] As demonstrated in this embodiment, the time-consistent dynamic power purchase method proposed in this invention can output a sustainably executable dynamic power purchase strategy in scenarios involving a combination of spot and options power purchases. This strategy possesses a continuous, smooth, and interpretable feedback structure, outperforming benchmark strategies such as pure spot, fixed-ratio, and pure options strategies in terms of cost-risk trade-offs. Furthermore, it effectively reflects the significant impact of quadratic convex transaction frictions on the optimal power purchase structure. Therefore, this invention effectively achieves the aforementioned objectives, providing power purchase entities with a directly calculable and deployable dynamic power purchase decision-making tool.

[0053] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A time-consistent dynamic power purchase combination optimization method, characterized in that, Includes the following steps: Set up the electricity purchase scenario: obtain the load demand of the electricity purchaser, the spot price process, and the option price process; Construct the process of price and cumulative electricity purchase cost, and establish a stochastic process of price and dynamic cost; Constructing a mean-variance power purchase optimization objective: Based on the aforementioned spot price process, option price process, and cumulative power purchase cost process, construct a power purchase optimization objective function with the mean and variance of the terminal cumulative power purchase cost as the objective; Introducing a time-consistent equilibrium strategy (hereinafter referred to as the equilibrium strategy): The decision-making process at different times is regarded as a dynamic game between the same electricity purchaser and its continuous self in continuous time. The equilibrium strategy is defined as follows: For any given time, a small deviation is made from the original strategy within any sufficiently short local time interval. This local deviation will not improve the objective function in the first order sense. Based on the equilibrium strategy, the electricity purchase optimization objective function is transformed into the extended HJB equation. Establish a one-dimensional equivalent model after state dimensionality reduction: Utilize the common driving relationship between spot price and option price and the translation invariance of cumulative electricity purchase cost to reduce the original state system to a one-dimensional equivalent model; Calculate the explicit expression of the equilibrium feedback control: Based on the one-dimensional equivalent model, calculate the explicit expression of the equilibrium feedback control and determine the optimal option power purchase ratio and the optimal spot power purchase ratio. as well as Discrete solutions and discrete control are output through a fully implicit finite difference algorithm: The one-dimensional equivalent model is numerically solved using a fully implicit finite difference algorithm, and the power purchase strategy at discrete time nodes is output.

2. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The cumulative electricity purchase cost process introduces a quadratic convex transaction friction term related to the amount of electricity purchased through options, which is used to characterize the increasing marginal execution cost effect when the scale of option purchases expands.

3. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The objective function for optimizing electricity purchase is: in, Indicates at time Status is The conditional expectation of the cumulative electricity purchase cost at the terminal reflects the average level of the electricity purchase cost; This represents the corresponding conditional variance, reflecting the volatility risk of electricity purchase costs; Risk aversion coefficient; This indicates the power purchase control strategy, specifically the proportion of load demand covered by options; Indicates the terminal time under the power purchase control strategy. The cumulative cost of electricity purchases; Indicates the current decision-making moment; Indicates time The state variable.

4. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The step of reducing the dimension to a one-dimensional equivalent model further includes: By leveraging the shared driving force between spot prices and option prices, we define invariant parameters: ； ； This represents the invariant constructed from the common driving relationship between spot price and option price; Indicates spot price negative Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. Indicates the current time; Indicates time The option price; The ratio of the volatility of the price in the option channel to the volatility of the spot price; And establish a functional relationship between option prices and spot prices; After reducing the original problem to an equivalent one-dimensional model, two functions to be solved are introduced: an auxiliary function. and equilibrium cost function Constructing a one-dimensional system: Instead, the unknown function is obtained by solving a system of one-dimensional equations and numerical algorithms. Therefore, the one-dimensional system can be written as: ； ； in, ； ； in, Represents the state variable regarding spot prices. One-dimensional differential operators; This represents an auxiliary function used to characterize the parameters. Given the conditions, from time 1 At the terminal time The expected cumulative electricity purchase cost conditions; Indicates the current time; Indicates the spot price; Indicates in the parameter One-dimensional instantaneous electricity purchase cost rate under given conditions; Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; Indicates terminal time; Indicates the spot price for end-users; This represents the equilibrium cost function, used to characterize the parameters. Given conditions, the cost of time-consistent equilibrium in the mean-variance sense; Indicates the risk aversion coefficient; express Regarding spot prices The first-order partial differential; Indicates about time and spot prices A sufficiently smooth function; express Regarding time The partial derivatives; The drift parameter represents the price movement in the spot market. express Regarding spot prices The first-order partial derivative; express Regarding spot prices The second-order partial derivative; A parameter representing the volatility of spot prices; Indicates the power purchase control strategy; Indicates time User load requirements; This represents the constant value form of the invariant parameter constructed from the common driving relationship between spot price and option price; Indicates spot price of Power of; This represents a constant parameter determined by the drift and volatility terms of both the spot price process and the option price process. This represents the intensity of friction in a quadratic convex transaction, used to characterize the market characteristic of increasing marginal execution costs as the size of option purchases increases.

5. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The balanced feedback control is calculated using the following explicit formula: in, ； in, Indicates in the parameter Equilibrium feedback control under given conditions, i.e., time... Spot price status The optimal option-based electricity purchase ratio; This means projecting real numbers onto an interval. Projection operator on; Indicates the intensity of friction in a quadratic convex transaction; Indicates time User load requirements; Indicates the real number Projected onto interval Projection operator on; Represents any real number.

6. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The spot price process and the option price process are governed by the same standard Brownian motion. Driven by, respectively, geometric Brownian motion: ; ; in, The drift parameter represents the price movement in the spot market. The volatility parameter represents the price movement of spot prices. Indicates time The spot price; Represents standard Brownian motion; Indicates the initial spot price; Indicates time The option price; The drift parameter represents the price movement of an option. A volatility parameter representing the price movement of an option; This represents the initial option price.

7. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The fully implicit finite difference algorithm includes the following steps: Introducing a logarithmic price transformation into a one-dimensional equivalence model and time inverse transformation ; in, This represents the logarithmic price variable, i.e., the spot price. The natural logarithm; Indicates the time variable after inversion; Indicates the end time of the trading cycle; Indicates the current time; The unbounded interval is truncated into a finite logarithmic price interval, and homogeneous Neumann boundary conditions are set. Uniform grids are used for discretization in both time and space directions; At each time level, first solve the tridiagonal linear equation system corresponding to the auxiliary function equation, then calculate the risk correction term, and finally solve the tridiagonal linear equation system corresponding to the equilibrium cost function equation; and The Thomas algorithm is used to solve the tridiagonal linear equations at each time level.

8. The time-consistent dynamic power purchase combination optimization method as described in claim 1, characterized in that, The auxiliary function equation and the equilibrium cost function equation are discretized at the interior points using backward time difference and central spatial difference, respectively forming the following tridiagonal linear equation system: The discrete format corresponding to the auxiliary function equation is: ; Organized into ; in, Describing auxiliary functions Discrete value at the nth time level and the i-th spatial grid node; Indicates the time grid step size; Represents the convection term coefficient; Represents the diffusion term coefficient; Indicates the spatial grid step size; This represents the discrete value of the source term at the (n+1)th time level and the ith spatial grid node; Represents the discrete diffusion coefficient; Represents the discrete convection coefficient; Indicates the spatial grid node index; Indicates the number of spatial grid divisions; The discrete format corresponding to the equilibrium cost function equation is: ; in, Represents the equilibrium cost function Discrete value at the nth time level and the i-th spatial grid node; This represents the discrete value of the risk correction term at the (n+1)th time level and the ith spatial grid node.

9. A time-consistent dynamic power purchase combination optimization system for the electricity market, used to implement the method of any one of claims 1-8, characterized in that, include: The data acquisition module is used to acquire the load demand, spot price process, option price process, and cumulative electricity purchase cost process of the electricity purchaser. The model building module, connected to the data acquisition module, is used to construct an electricity purchase optimization objective function based on the data acquired by the data acquisition module, with the mean and variance of the cumulative electricity purchase cost of the terminal as the objective. The equilibrium strategy solution module, connected to the model construction module, is used to introduce an equilibrium strategy. It treats the decision-making process at different times as a dynamic game between the same electricity purchaser and its continuous self over continuous time. The equilibrium strategy is defined as follows: for any given moment, a small deviation is made from the original strategy within any sufficiently short local time interval, such a local deviation does not improve the objective function in the first-order sense. Based on the equilibrium strategy, the electricity purchase optimization objective function is transformed into an extended HJB equation. The dimensionality reduction module, connected to the equilibrium strategy solution module, is used to reduce the original state system to a one-dimensional equivalent model by utilizing the common driving relationship between spot prices and option prices and the translation invariance of cumulative electricity purchase costs. The feedback control calculation module, connected to the dimensionality reduction processing module, is used to calculate the equilibrium feedback control based on the one-dimensional equivalence model, and determine the optimal option power purchase ratio and the optimal spot power purchase ratio. as well as The numerical solution module, connected to the feedback control calculation module, is used to numerically solve the one-dimensional equivalent model using a fully implicit finite difference algorithm and output the electricity purchase strategy at discrete time nodes.

10. The time-consistent dynamic power purchase combination optimization system as described in claim 9, characterized in that, The numerical solution module further includes: Transformation unit, used to perform logarithmic price transformation and time inversion transformation on a one-dimensional equivalent model; Discretization unit, used to discretize the transformed model into a uniform grid in both time and space; The equation system construction unit is used to construct a tridiagonal linear equation system corresponding to the auxiliary function equation and the equilibrium cost function equation at each time level; and The solution unit is used to solve the tridiagonal linear equations sequentially using the Thomas algorithm to obtain discrete equilibrium feedback control solutions.

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