Application of optimal control method based on Hurwicz criterion in intelligent power grid dispatching

Abstract

The invention relates to the technical field of intelligent power grid dispatching, in particular to application of an optimal control method based on a Hurwicz criterion in intelligent power grid dispatching. The method comprises the following steps: a pessimistic value optimal control research framework is based on an uncertain system; introducing an uncertain singular non-causal system, and outputting an equivalent transformation form; establishing an optimal control problem unification framework of the Hurwicz criterion; the Hurwiz type optimal control method aims at the Hurwiz type optimal control problem of a linear and nonlinear uncertain singular non-causal system. And the Hurwicz type optimal control obtained in the step S5 is applied to intelligent power grid dispatching. According to the application of the optimal control method based on the Hurwicz criterion in intelligent power grid dispatching, dynamic programming and uncertainty theories are combined, a recursion equation is constructed to solve the optimal control problem in intelligent power grid dispatching, an analytical expression and an optimal value of optimal control are deduced for linear and nonlinear uncertain singular non-causal systems, and the optimal control method based on the Hurwicz criterion is applied to intelligent power grid dispatching. And the practical applicability of the provided framework is verified through a calculation example.

Description

Technical Field

[0001] The present invention relates to the technical field of smart grid dispatching, and in particular to the application of an optimal control method based on the Hurwicz criterion in smart grid dispatching. Background Art

[0002] Optimal control is the cornerstone of modern cybernetics. It involves designing control strategies to optimize specific performance metrics while satisfying system constraints. Pontryagin's maximum principle, Bellman's dynamic programming, and Kalman's filtering theory laid a solid foundation for the development of optimal control.

[0003] The development of optimal control emphasized the necessity of accurate modeling for solving complex problems. Rosenbrock exposed the limitations of traditional state-space models and introduced the concept of singular systems. Luenberger and Arbel investigated the existence and uniqueness of solutions for linear and nonlinear singular systems. Cobb established controllability and observability criteria for singular systems in the context of structural system analysis, as well as their duality principle. Building on this foundation, Dai combined optimal control with singular system theory, laying a comprehensive foundation for optimal control of deterministic singular systems.

[0004] The study of systems affected by various types of uncertainty is of key importance in the fields of systems science and optimal control. When uncertainty is modeled using random variables or Wiener processes, a stochastic system results, which in turn leads to the study of random singular systems. For example, Shu and Li introduced dynamic programming to the analysis of optimal control problems for random singular systems and developed a framework for solving recursive equations based on probability expectations. Vlasenko et al. established conditions for the existence and uniqueness of solutions to random singular systems. Li and Ma solved an indeterminate linear quadratic problem in a singular Markov jump system by applying an equivalent transformation method.

[0005] The application of probability theory relies on sufficient historical data to construct a probability distribution that approximates the true frequency. However, in many complex or emerging systems, sample data is scarce or even non-existent, and expert judgment often becomes a key source of information. Due to the inherent conservatism and subjectivity of human cognition, expert assessments often deviate significantly from objective frequencies. Liu showed that using a probabilistic framework to interpret expert credibility can lead to paradoxical or counterintuitive conclusions. This insight directly led to the development of uncertainty theory, in which uncertainty is modeled as uncertain variables and expert assessments are expressed through uncertainty distributions. When system uncertainty is described using such uncertain variables, an uncertain system can be obtained.

[0006] Zhu demonstrated that understanding the optimal and recursive equations can solve the optimal control problem for uncertain systems. This foundational work inspired a series of studies, laying the theoretical foundation for analyzing uncertain singular systems. Subsequently, Shu and Zhu proposed a rigorous analytical framework for analyzing the stability of uncertain singular systems. Under the uncertain expectation criterion, Shu et al. developed a method for solving the recursive equations for uncertain singular systems by introducing regularity and impulse-free conditions. Although the expectation criterion is widely used in uncertain optimization, it has significant limitations. For example, in urban income distribution, severe polarization can lead to misleading average values. To address this issue, Shu and Zhu applied the optimistic value criterion to uncertain singular systems, obtaining several insightful results. Later, Chen et al. studied the linear quadratic optimal control problem under the pessimistic value criterion. The development of these criterion designs not only enriched the theory of uncertain optimization but also provided new tools for dealing with risk preferences and trade-offs. Recent research on the optimal control of uncertain singular systems has focused on systems that are both regular and impulse-free. Despite significant progress, these studies have encountered significant challenges in extending to non-causal systems that only satisfy the regularity condition. Such non-causal systems arise frequently in real-world applications, including problems in signal processing and circuit analysis, where methods designed for causal and pulseless systems may fail.

[0007] Smart grids, the development direction of modern power systems, aim to achieve efficient, reliable, and secure transmission and distribution of electricity. With the continued growth of electricity demand, the large-scale integration of renewable energy, and the continuous development of the electricity market, the structure and operating environment of smart grids are becoming increasingly complex. Smart grid dispatching, as a core component in ensuring stable grid operation and optimizing resource allocation, faces numerous severe challenges. Traditional grid dispatching, primarily based on deterministic models and simple control strategies, struggles to adapt to the numerous uncertainties inherent in smart grids. For example, the intermittent and volatile nature of renewable energy sources (such as solar and wind power) makes accurate prediction of their generated power difficult. Electricity loads are also affected by various factors, such as weather and economic activity, exhibiting random variations. These uncertainties place tremendous pressure on the grid's power balance, voltage stability, and frequency regulation, and traditional dispatching methods often struggle to cope with these challenges.

[0008] In smart grids, many practical problems can be abstracted into singular non-causal systems. Singular systems are dynamic systems with algebraic constraints. They can more accurately describe complex phenomena in power systems, such as power flow constraints and equipment operating limitations. Non-causal systems reflect the inherent feedback delays and complexity of information interaction within the system. For example, within the power system's communication network, data transmission and processing can experience time delays, resulting in non-causal responses.

[0009] However, relatively little research exists on singular non-causal systems, particularly when considering system uncertainty. Traditional control methods, primarily designed for conventional causal systems, cannot be directly applied to singular non-causal systems. When singular non-causal systems are subject to uncertainty, system stability and performance can be severely impacted, leading to deviations in grid scheduling and even failures.

[0010] Currently, a variety of control methods have been proposed for smart grid scheduling, such as model predictive control and robust control. Model predictive control can address system uncertainty to a certain extent by predicting the future state of the system and optimizing the control strategy in a rolling manner. However, this method requires the establishment of a precise system model and is computationally intensive, making it less efficient when dealing with complex, singular, and non-causal systems. Robust control focuses on designing controllers with a certain degree of robustness to ensure system stability under uncertain disturbances. However, robust control is typically designed based on a worst-case scenario, which may result in overly conservative control strategies and compromise some system performance. Furthermore, most existing control methods do not fully consider the Hurwicz criterion and are unable to effectively balance risk preference and decision-making benefits.

[0011] The Hurwicz criterion is a decision analysis method that uses weighting coefficients to balance the decision maker's optimism and pessimism, thereby enabling more rational decisions under uncertainty. Although the Hurwicz criterion has been widely used in fields such as economics and management, its application in smart grid scheduling is still in its infancy. Currently, no research or technology has combined the Hurwicz criterion with optimal control methods for uncertain singular noncausal systems and applied it to smart grid scheduling.

[0012] Therefore, an optimal control method based on the Hurwicz criterion is designed and applied in smart grid scheduling to provide a technical solution to the above technical problems. Summary of the Invention

[0013] Based on this, it is necessary to provide an optimal control method based on the Hurwicz criterion for application in smart grid scheduling in order to solve the technical problems raised in the above background technology.

[0014] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0015] The application of the optimal control method based on the Hurwicz criterion in smart grid scheduling is as follows:

[0016] S1: Based on uncertainty theory, define optimistic and pessimistic values;

[0017] S2: Solve the optimal control problem based on the optimal control framework based on pessimistic values;

[0018] S3: Introduce an uncertain singular non-causal system and output an equivalent transformation form;

[0019] S4: Establish a unified framework for optimal control problems based on the Hurwicz criterion, balance optimism and pessimism through weight parameters, and derive recursive equations as well as optimal control and optimal values;

[0020] S5: Provides the algorithm steps for solving the Hurwicz-type optimal control problem for linear and nonlinear uncertain singular non-causal systems;

[0021] S6: Apply the Hurwicz-type optimal control obtained in step S5 to smart grid scheduling.

[0022] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S1, optimistic values ​​and pessimistic values ​​are introduced to describe uncertain variables, and the expression is as follows:

[0023] ξ sup (α)=sup{l|M{ξ≥l}≥α}, ξ inf (α)=inf{l|M{ξ≤l}≥α}.

[0024] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S2, the optimal control framework based on the pessimistic value in the optimal control problem is expressed as follows:

[0025]

[0026] Among them, α∈(0,1] is the confidence level;

[0027] The corresponding recursive form is:

[0028]

[0029] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S3, the steps are as follows:

[0030] Construct a nonlinear uncertain singular non-causal system, the expression is as follows:

[0031] Ey(j+1)=Hy(j)+f(ω(j),ξ j ), j = 0, 1, 2, ..., J-1;

[0032] Among them, (E , H) is regular, is a vector-valued function that satisfies the Lipschitz condition.

[0033] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S3, the steps are as follows:

[0034] The Hurwicz optimal control framework balances the optimistic and pessimistic values ​​through the weight parameter ρ∈[0,1], and the expression is as follows:

[0035]

[0036] If ρ = 1, it corresponds to a completely optimistic strategy, where the decision focus is on the best outcome. If ρ = 0, the strategy is completely pessimistic, tending towards absolute risk aversion and the worst-case scenario. If (0 < ρ < 1), it represents a balance.

[0037] The optimal value depends on the initial state and the final state, which is expressed as

[0038] For each k=J-1, J-2, ..., 1, 0, the recursive equation is expressed as follows:

[0039]

[0040] Among them, ω * (k+1),…,ω * (J-1) is the optimal control,

[0041] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S4, the linear uncertain singular non-causal Hurwicz type optimal control problem is as follows:

[0042] For a given uncertain singular system, check the regularity of the conditions (E, H) and D2 = 0, continue if the conditions are met, and exit if the conditions are not met;

[0043] Convert the optimal control problem into an equivalent problem;

[0044] Calculate l in reverse order j 、 r j 、s j , t j ;

[0045] Computational optimal control;

[0046] for and Assignment, obtain all states through the state equation, the expression is as follows:

[0047]

[0048] Calculate the optimal value.

[0049] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S4, the Hurwicz type optimal control problem of a nonlinear uncertain singular non-causal system is as follows:

[0050] For the uncertain singular system given in the problem, check the regularity of the conditions (E, H) and D2 = 0, continue if the conditions are met, and exit if the conditions are not met;

[0051] Convert the optimal control problem into an equivalent problem;

[0052] by Calculate α in reverse order j , β j , r j , s j , σ j , t j , j, j=0,1,2,…,J-1;

[0053] Computational optimal control;

[0054] Assign values ​​to relevant variables and obtain all states through state equations. The expressions are as follows:

[0055]

[0056] Calculate the optimal value.

[0057] As a preferred embodiment of the application of the optimal control method based on the Hurwicz criterion provided by the present invention in smart grid scheduling, in step S6, the steps are as follows:

[0058] Based on the power system control scenario, such as the matrix E captures the system singular dynamics, the uncertainty vector ξ j Modeling uncertain external inputs, the state weight vector is given;

[0059] By setting the weight parameter ρ and confidence level α, using the Hurwicz criterion to balance risks and benefits, solve the optimal control problem in a physically driven numerical example inspired by power system control, including determining the type of system, performing coordinate transformation, applying the algorithm to obtain the optimal solution steps, and giving the corresponding state and control variable trajectories and the optimal value changes as the parameters change.

[0060] It can be seen without a doubt that the technical solution described above in this application can definitely solve the technical problem to be solved in this application.

[0061] At the same time, through the above technical solutions, the present invention has at least the following beneficial effects:

[0062] The present invention provides an optimal control method based on the Hurwicz criterion for application in smart grid scheduling. Combining dynamic programming with uncertainty theory, a recursive equation is constructed to solve the optimal control problem in smart grid scheduling. Analytical expressions and optimal values ​​of the optimal control are derived for linear and nonlinear uncertain singular non-causal systems, and the practical applicability of the proposed framework is verified through numerical examples. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following is a brief introduction to the drawings required for use in the description of the embodiments. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0064] Figure 1 A schematic diagram of a method for developing a Hurwicz-type optimal control problem for an uncertain singular non-causal system according to the present invention;

[0065] Figure 2 A schematic diagram of the trajectory of the control variables of the present invention;

[0066] Figure 3 A schematic diagram of the trajectory of the state of the present invention;

[0067] Figure 4 Schematic diagram of the trajectory of the optimal value of the present invention. DETAILED DESCRIPTION

[0068] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific implementation cases described herein are only used to explain the present invention and are not intended to limit the present invention.

[0069] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0070] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features and technical solutions therein may be combined with each other.

[0071] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, it does not require further definition or explanation in subsequent drawings.

[0072] Example 1

[0073] Reference Figure 1 , Application of optimal control method based on Hurwicz criterion in smart grid scheduling.

[0074] 1. Basics

[0075] 1.1. Uncertainty theory

[0076] The uncertain variable ξ defined on the uncertainty space (Γ, L, M) is associated with the uncertainty distribution Φ(x) = M{ξ≤x}, x∈R. The uncertain expected value of ξ is given by:

[0077]

[0078] This expectation is well defined as long as at least one of the two integrals is finite.

[0079] In addition to expectations, Liu [Liu B.Uncertainty Theory,2nd,Berlin:Springer,2007] Two further indicators are introduced to describe the uncertain variables:

[0080] Definition 1.1(Liu [Liu B.Uncertainty Theory,2nd,Berlin:Springer,2007] ) Given an uncertain variable ξ and a confidence level α∈(0,1], the α-optimistic and α-pessimistic values ​​of ξ are defined as:

[0081] ξ sup (α)=sup{l|M{ξ≥l}≥α}, ξ inf (α)=inf{l|M{ξ≤l}≥α};

[0082] Lemma 1.1 (Liu [Liu B.UncertaintyTheory,2nd,Berlin:Springer,2007] ) Let ξ be an uncertain variable and ρ∈R, then the following properties exist:

[0083] (ρ·ξ) sup (α)=ρ·(ξ) sup (α),(ρ·ξ) inf (α)=ρ·(ξ) inf (α);

[0084] If ρ ≥ 0;

[0085] (ρ·ξ) sup (α)=ρ·(ξ) inf (α),(ρ·ξ) inf (α)=ρ·(ξ) sup (α);

[0086] If ρ<0. For two independent uncertain variables ξ and η, the expression is as follows:

[0087] (ξ+η) sup (α)=(ξ) sup (α)+(η) sup (α),(ξ+η) inf (α)=(ξ) inf (α)+(η) inf (α);

[0088] Under the uncertainty metric M, the uncertainty expectation reflects the average tendency of ξ. Furthermore, ξ exceeds its optimistic value α with a confidence level of at least α and is less than or equal to its pessimistic value α at the same confidence level. According to the theory that for mutually independent uncertain variables, the optimistic and pessimistic values ​​are linearly additive.

[0089] 1.2 Optimal Control Problem

[0090] The core goal of uncertain optimal control is to determine the control strategy that optimizes the performance indicators of uncertain systems. Due to the existence of uncertain variables, their numerical characteristics are usually used to evaluate these performance indicators.

[0091] Zhu [Zhu Y.Uncertain Optimal Control,Singapore:Springer Nature,2019] An optimal control framework based on uncertain expectations is proposed:

[0092]

[0093] Where y(j)∈R m It's a state. is control, j ∈R m ξ represents the uncertainty vector, is a vector-valued function, is a real-valued function, for each K=J-1 , J-2,…,1,0, let V(y(k),k) be the optimal value function from the kth stage to the final stage. Then we have the following equation:

[0094]

[0095] Among them, ω * (k+1),…,ω * (J-1) is the optimal control, V(y(J),J)=ψ(y(J),J).

[0096] The expected value is an average value in the sense of uncertainty measure and usually reflects the characteristics of uncertain variables well. However, in scenarios where the distribution is highly skewed or extremely polarized, relying on the expected value may lead to misleading or suboptimal decisions. To address this limitation, Chen et al. [Chen X,Cao Z,Zhang Z.Linear quadratic optimal control and zero-sum game for uncertain time-delay systems based o n pessimistic value.European Journal of Control,2025,84:101224.] Introduced pessimistic value-based expressions:

[0097]

[0098] Where α∈(0,1] is the confidence level. The corresponding recursive form is

[0099]

[0100] Find the optimal control method for this problem.

[0101] The optimal control problem, Equations (1) and (2), is formulated within the framework of standard uncertain systems. In contrast, the well-known Leontief dynamic input-output model belongs to the class of singular systems. The emergence of such non-causal singular models can be attributed to a variety of physical considerations. For example, in advanced signal processing, reconstructing complete signal information often requires considering future values, thus requiring non-causal models. Similarly, in circuit design, the output of components with memory effects depends not only on past and present inputs but also on potential future inputs, which has prompted the development of non-causal system representations.

[0102] 2. Uncertain singular non-causal systems

[0103] Dai [Dai L.Singular Control Systems.Berlin:Springer-Verlag,1989] A class of linear singular systems is introduced, which is expressed as follows:

[0104] Ey(j+1)=Hy(j)+Πω(j),j=0,1,2,…,J-1 (5)

[0105] In this system, the state is represented by y(j)∈R m Indicates that control is Represents. Matrix E, H∈R m*m 、 It is deterministic, assuming rankE = q <m。

[0106] Definition (Dai [Dai L.Singular Control Systems.Berlin:Springer-Verlag,1989] )The system (5) is called regular if det(xE-H)≡0, where x is a complex variable and det(·) represents the determinant of the matrix. The system (5) is called impulse-free if deg(det(xE-H))=rankE, where deg(·) represents the degree of the resulting polynomial.

[0107] From a physical perspective, the regularity condition ensures that the system model is mathematically well-posed and physically meaningful. For singular systems like (5), regularity guarantees that the matrix pair (E, A) does not produce a purely algebraic equation, i.e., det(xE-H)≡0. This avoids situations where the system dynamics are ill-defined. For example, consider a circuit of parallel capacitors and inductors, which is modeled by a singular matrix due to algebraic constraints (such as Kirchhoff's laws). The regularity condition ensures that the circuit equations have a unique solution for the voltage and current in time, preventing contradictory constraints (e.g., infinite current). If det(xE-H)≡0, the circuit will have no dynamic behavior, making it physically irrelevant. The pulse-free condition ensures that the system does not produce an impulsive response (e.g., an infinite spike in the state or control) at the initial time. This is crucial for practical systems because pulses cannot be implemented in hardware and may damage components. For example, if a robotic arm undergoes constrained motion with singular dynamics, a pulse-free system can avoid instantaneous velocity jumps that would violate physical constraints.

[0108] Considering that the system (5) is subject to the uncertainty vector The perturbation of Shu and Zhu[ Shu Y ，ZhuY.Optimistic value based optimal control for uncertain linear singular systems and application to a dynamic inpu t-output model.ISA Transactions,2017,71:235-251.] The following linear uncertain singular causal system is proposed within the framework of uncertainty theory:

[0109] Ey(j+1)=Hy(j)+Πω(j)+Dξ j ,j=0,1,2,…,J-1 (6)

[0110] Here, (E, H) satisfies both the regularity and the pulse-free condition. In contrast, for an uncertain singular noncausal system, only the regularity condition needs to be satisfied. Regularity is usually more relaxed than the pulse-free condition. It is worth noting that when an uncertain singular noncausal system satisfies both the pulse-free condition, it is simplified to a causal system, as shown in Equation (6).

[0111] Different from the previous research on linear uncertain singular causal systems, we proposed nonlinear uncertain singular non-causal systems, which are generalized systems worth studying:

[0112] Ey(j+1)=Hy(j)+f(ω(j),ξ j ), j=0,1,2,…,J-1 (7)

[0113] Among them, (E, H) is regular, is a vector-valued function.

[0114] For the uncertain singular non-causal system (7), there are non-singular matrices F and M, which can be expressed by the coordinate transformation

[0115]

[0116] and

[0117]

[0118] This makes the system equivalent to the following decoupled form:

[0119]

[0120] Where H1∈R q*q ,f1(ω(j),ξ j )∈R q ,f2(ω(j),ξ j )∈R (m-q) , E2∈R (m-q)*(m-q) is a nilpotent matrix. The nilpotent index is denoted by

[0121] Proof: As Dai [Dai L.Singular Control Systems.Berlin:Springer-Verlag,1989] As shown, there are non-singular matrices F, M∈R m*m , such that:

[0122]

[0123] Applying this transformation to system (7) yields the equivalent expression (8).

[0124] It is worth noting that system (7) and system (8) have the same control. This transformation makes it possible to analyze the related optimal control problem through the simplified and decoupled system (8), which enables a more effective analysis of the system behavior under uncertain conditions.

[0125] 3. Optimal control problem based on Hurwicz criterion

[0126] set up and is a given measurable real-valued function, and α∈(0,1] represents the confidence level.

[0127]

[0128] The optimistic and pessimistic values ​​of α are

[0129]

[0130] and

[0131]

[0132] To unify these two evaluation criteria, a Hurwicz-type optimal control framework for uncertain singular non-causal systems (7) is introduced. This framework balances the α-optimism and α-pessimism through a weight parameter ρ∈[0,1]:

[0133]

[0134] Remarks: In problem (9), the integration of the two criteria forms a hybrid decision framework that can adapt to the subjective preferences of decision makers. Its basic principle is based on the classic Hurwicz criterion, which achieves quantitative adjustment between optimism and pessimism by introducing a risk preference coefficient ρ∈[0,1]. Specifically, when ρ=1, it corresponds to a completely optimistic strategy, where the decision focus is on the best outcome, which is suitable for scenarios with sufficient resource redundancy and high risk tolerance. On the contrary, when ρ=0, the strategy is completely pessimistic, tending to absolute risk aversion and worst-case system robustness, which is particularly common in high-risk environments. The intermediate value (0<ρ<1) represents a balanced strategy, in which the linear weighting of the two extremes enables the decision process to flexibly reconcile performance and robustness, thereby meeting the multi-dimensional requirements of practical engineering applications.

[0135] Problem (9) is equivalent to the following decomposition formula:

[0136]

[0137] in,

[0138] M=[M1M2],M1∈R m*q) ,M2∈R m*(m-q) ,and

[0139]

[0140]

[0141] Since the optimal value in (10) depends on the initial state and the final state, we use Indicate it.

[0142] For any k∈{1,..., J —1}, according to and Define the value function as follows:

[0143]

[0144] The optimal solution can be obtained through the following recurrence relation:

[0145] Theorem 3.1, for each k = J-1, J-2, ..., 1, 0, the value function satisfies the following recursive equation:

[0146]

[0147]

[0148] Among them, ω * (k+1),…,ω * (J-1) is the optimal control,

[0149] Proof: Assume Represents the right side of the recursive equation. According to the value function The definition is as follows:

[0150]

[0151] Taking the maximum value of ω(j), j=K+1,K+2,…,J-1 in (12), and then taking the maximum value of w(k) in (12), we get:

[0152]

[0153] On the other hand, there are:

[0154]

[0155] Combining (13) and (14), it is concluded that equation (11) holds. In addition, for the terminal case j = J, it is based on The definition of This completes the proof.

[0156] It is worth noting that when the matrix pair (E, A) satisfies the regularity and pulse-free conditions, problem (9) is simplified to Shu and Sheng [Shu Y ，Sheng L.Hurwicz criterion based optimal control model for uncertain descriptor systems with an application to indus trial management..《Journal of Industrial and Management Optimization,2023,19(8)：6054-6081] The optimal control problem of uncertain singular causal systems is studied. In addition, if the matrix E is not singular, problem (9) will degenerate into an optimal control problem in the standard system framework:

[0157]

[0158] In particular, when ρ = 0, the problem is completely consistent with the optimal control problem under pessimism.

[0159] The value criteria studied by Chen et al. [Chen X,Cao Z,Zhang Z.Linear quadratic optimal control and zero-sum game for uncertain time-delay systems based o n pessimistic value.European Journal of Control,2025,84:101224.] :

[0160]

[0161] Therefore, the recurrence equation (11) given in Theorem 3.1 simplifies to the form of equation (4). These results together show that the problem (9) developed in this paper has a more general structural form, including several established models as special cases.

[0162] This transformation simplifies the original optimal control problem (9), which involves an uncertain singular non-causal system, by reformulating it into an equivalent form (10). In this reformulation, the system is decomposed into a combination of forward and backward uncertain subsystems. Using the recursion equation, the problem (10) can be analyzed in a backward recursive manner, starting from the terminal stage and gradually moving towards the initial stage. In this process, it is assumed that ξ0,ξ1,···,ξ J-1 are independent of each other. This assumption is made to exploit the linear additivity of optimistic and pessimistic values, thereby simplifying the recursive computation. Furthermore, this recursive approach is able to derive analytical solutions to certain optimal control problems involving uncertain singular non-causal systems.

[0163] 4. Special circumstances

[0164] First, consider the following Hurwicz-type optimal control problem for a linear uncertain singular non-causal system:

[0165]

[0166] in, make

[0167]

[0168] If we assume that D2=0, then problem (17) is equivalent to the following conversion problem:

[0169]

[0170] Theorem 4.1, optimal control of problem (18) ω * (j)

[0171]

[0172] get:

[0173]

[0174] and

[0175]

[0176] for, The optimal value is

[0177]

[0178]

[0179] get:

[0180]

[0181] in,

[0182] Proof: According to Theorem 4.1, we can get in When j = J-1, Theorem 3.1 applies as follows:

[0183]

[0184] make For each i w =1 , 2,…,n w get:

[0185]

[0186] in, Represents the i-th element of the vector, denoted by vector l J-1 The norm is ||l J-1 ||1, then

[0187]

[0188] get:

[0189]

[0190]

[0191] Now, considering the case of j=J-2, we can use Theorem 3.1 to obtain the following results:

[0192]

[0193] remember Then we have:

[0194]

[0195] The goal is to determine the expression Since it is necessary to determine the ω that was not calculated in the previous step * (J-1) elements, so the update process is needed:

[0196]

[0197] in The corresponding optimal value is:

[0198]

[0199] get:

[0200]

[0201] This theorem has been successfully proved by induction.

[0202] Next, consider the Hurwicz-type optimal control problem for nonlinear uncertain singular noncausal systems:

[0203]

[0204] where ω(j)∈[-1,1],λ,c∈R m .remember:

[0205]

[0206] Then, under the condition D2=0, problem (25) is equivalent to the following formula:

[0207]

[0208] Theorem 4.2, the optimal control w of problem (26) * (j) is:

[0209]

[0210] If (α j ,β j )≠(0,0);

[0211]

[0212] If (α j ,β j )≠(0,0),

[0213]

[0214] if

[0215]

[0216] if in:

[0217]

[0218] and

[0219]

[0220] in and The optimal solution is:

[0221]

[0222]

[0223] get:

[0224]

[0225] and

[0226]

[0227] For k=j,j+1…, J -1,j=0 , 1 , 2,…, J -1, where

[0228] Proof: According to Theorem 3.1, we can get in, Where j = J-1, according to Theorem 3.1, we get:

[0229]

[0230] get:

[0231]

[0232] ω * (J-1) is the maximum point, and there are the following situations:

[0233] (i) In α J-1 =β J-1 = 0, the relevant expression is simplified to:

[0234]

[0235] (ii)α J-1 ≠0, and β J-1 =0, then we get:

[0236]

[0237] and

[0238]

[0239] (iii) If α J-1 ≥0 and β J-1>0, then:

[0240]

[0241] and

[0242]

[0243] (iv) If α J-1 <0,β J-1 >0, then:

[0244]

[0245] and

[0246]

[0247] (v) If 2β J-1 ≤α J-1 ≤-2β J-1 , and β J-1 <0, then:

[0248]

[0249] and

[0250]

[0251] (vi) If α J-1 ≥-2β J-1 , and β J-1 <0, then:

[0252]

[0253] and

[0254]

[0255] (vii) If α J-1 ≤2β J-1 , and β J-1 <0, then:

[0256]

[0257] and

[0258]

[0259] Optimal control ω * (J-1) can be summarized as follows:

[0260]

[0261] remember but:

[0262]

[0263] and

[0264]

[0265] get:

[0266]

[0267]

[0268] For j = J - 2, according to Theorem 3.1, we obtain:

[0269]

[0270] in:

[0271]

[0272] The proof process is similar to the previous one, so we can conclude that:

[0273]

[0274] And, α J-2 =α J-2 ω * (J-2)+β J-2 ω(J-2) 2 , in order to determine the formula The maximum value of It is necessary to update the ω that was not calculated in the J-1 step. * (J-1) These elements are invalidated:

[0275]

[0276] If (α J-1 ,β J-1 )≠(0,0); then:

[0277]

[0278] If (α J-1 ,β J-1 )=(0,0), then:

[0279]

[0280] get:

[0281]

[0282] In summary, the theorem is successfully proved by induction.

[0283] The main steps of the unified framework for solving Hurwicz-type optimal control problems are systematically discussed. In order to present the proposed solution more clearly, Figure 1 A schematic diagram is given in , as described above.

[0284] For problems (17) and (25), the optimal control of each stage and the analytical expression of its corresponding optimal value are derived. By utilizing the recursive relationship between the jth stage and the (j+1)th stage, the following algorithm is proposed to calculate the optimal solution of problems (17) and (25).

[0285]

[0286]

[0287] In the context of systems, the condition D2 = 0 reflects a physical simplification with a specific meaning. For example, in an electrical circuit, if D2 = 0 corresponds to the impedance of a particular branch, then this condition effectively simulates a short circuit in that branch. Such a structural change alters the distribution of current and voltage in the circuit, thereby affecting the overall dynamic behavior of the system. Similarly, in a mechanical system, if D2 = 0 represents the damping coefficient associated with a motion-restricting component, then this condition implies that the component is undamped. As a result, the system may exhibit increased oscillation amplitude, prolonged vibration duration, or delayed settling time, thereby changing its response to external disturbances.

[0288] Example 2

[0289] refer to Figure 2-Figure 4 Based on the above-mentioned first embodiment, in order to demonstrate the practical relevance and effectiveness of the theoretical framework of the present invention, a physical driven numerical example inspired by power system control is proposed:

[0290]

[0291] get:

[0292]

[0293] matrix E It captures the singular dynamics of the system and can, for example, represent the transient voltage drop caused by a fault in the power network. Uncertain external inputs such as fluctuating loads or component tolerances are simulated. is an independent linear uncertain variable, following the uncertainty distribution The state weight vector is:

[0294]

[0295]

[0296] First, there is:

[0297]

[0298] And deg det(xE - H) = 2 < rank(E) = 3, that is, the considered system belongs to the category of uncertain singular non-causal systems. There exists a non-singular matrix:

[0299]

[0300] such that

[0301]

[0302] We get:

[0303]

[0304] and

[0305] <着

[0306] Under the condition D2 = 0, problem (37) corresponds to:

[0307]

[0308] The objective function combines energy loss and terminal cost, reflecting the voltage stability at the final stage J = 7. The Hurwicz criterion plays a core role in it, by allowing the operator to balance risk and return, which is crucial for effective power grid management. By setting the weight parameter ρ = 0.6 and the confidence level α = 0.7, this framework emphasizes maximizing power supply under moderate risk, thus enhancing system resilience without taking on excessive uncertainty risks. Given the initial state and the terminal state The optimal solution of problem (38) is obtained by applying Algorithm 1. The obtained values are summarized in Table 1.

[0309] Table 1: Optimal solution of problem (38)

[0310]

[0311] Since the nilpotency of matrix E2 is 2, we can infer that This is obvious from the third column of Table 1. In addition, the optimal control The corresponding trajectory is as Figure 2 shown (for the control variable , where and Represent reactive power regulation, active power regulation and energy storage output respectively. The values ​​are listed in the fourth column of Table 1. During stage j = 0 to 2, the control strategy operates as follows: when ω1 = 1, reactive power is quickly injected to alleviate the voltage drop that occurs during the closing process. At the same time, as ω2 = 1, -1, 1 changes, the active power is adjusted in sequence to effectively cope with sudden load changes. At the same time, when ω3 = -1, the energy storage system discharges to compensate for the initial energy shortage of the system. In stages j = 3 to 6, the control logic enters a new stage: according to the voltage feedback, ω1 switches to -1, 1, -1, -1 in sequence, reversing the reactive power injection direction.

[0312] To prevent overvoltage. Meanwhile, active power is continuously supplied, ω2 = 1, 1, 1, -1, supporting generator acceleration. Meanwhile, ω3 = 1, -1, -1, 1, the energy storage system charges to absorb excess energy and suppress overfrequency events, ensuring stable system operation.

[0313] state and The trajectory of Figure 3 Shown is the status The corresponding numerical data are listed in the 5th and 6th columns of Table 1. It is worth noting that the state There is a significant change. This is because the state is affected by the optimal control, and the specific relationship is

[0314] According to the simulation results, the optimal value As ρ increases, it shows a linear growth trend, as shown in Table 2 and Figure 4 As shown, Figure 4 is the optimal value trajectory.

[0315] As shown in Table 2, when ρ = 0, the strategy only considers the pessimistic scenario (i.e., the worst case);

[0316] Table 2: Optimal values ​​for different ρ levels

[0317]

[0318] This leads to a minimum value for the objective function, reflecting a conservative control approach. In contrast, when ρ = 1, the policy focuses only on optimistic scenarios, resulting in a maximum objective value, corresponding to an aggressive control policy. For intermediate values ​​of ρ (e.g., ρ = 0.6), the policy strikes a balance between risk and performance. The objective value increases linearly with the optimism weight, consistent with the convex combination property of the Hurwicz criterion.

[0319] in conclusion

[0320] Combining dynamic programming and uncertainty theory, a recursive equation is carefully constructed specifically for solving such problems. Through this recursive equation, the problem can be systematically and step-by-step decomposed and solved. As a special case in the present invention, special attention is paid to the optimal control problem of linear and nonlinear uncertain singular non-causal systems. For these complex scenarios, based on the established recursive equation, the exact analytical expressions for the optimal control of these problems are successfully derived, and the related optimal values ​​are determined. In order to verify and illustrate the practical applicability of the proposed framework, a carefully designed numerical example is provided. This example clearly and comprehensively shows how to apply the proposed scheme to solve uncertain singular non-causal control problems under the Hurwicz criterion.

[0321] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to the specific embodiments described. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. Application of the optimal control method based on Hurwicz criterion in smart grid dispatching, characterized by: Here are the steps: S1: Based on uncertainty theory, define optimistic and pessimistic values; S2: Solve the optimal control problem based on the optimal control framework based on pessimistic values; S3: Introduce an uncertain singular non-causal system and output an equivalent transformation form; S4: Establish a unified framework for optimal control problems based on the Hurwicz criterion, balance optimism and pessimism through weight parameters, and derive recursive equations as well as optimal control and optimal values; S5: Provides the algorithm steps for solving the Hurwicz-type optimal control problem for linear and nonlinear uncertain singular non-causal systems; S6: Apply the Hurwicz-type optimal control obtained in step S5 to smart grid scheduling.

2. Application of the optimal control method based on Hurwicz criterion in smart grid scheduling according to claim 1, characterized in that: In step S1, optimistic and pessimistic values ​​are introduced to describe uncertain variables. The expressions are as follows: x sup (α)=sup{l|M{ξ≥l}≥α},ξ inf (a)=inf{l|M{ξ≤l}≥a}。 3. Application of the optimal control method based on Hurwicz criterion in smart grid scheduling according to claim 1, characterized in that: In step S2, the optimal control framework based on the pessimistic value in the optimal control problem is expressed as follows: Among them, α∈(0,1] is the confidence level; The corresponding recursive form is:

4. Application of the optimal control method based on Hurwicz criterion in smart grid scheduling according to claim 1, characterized in that: In step S3, the steps are as follows: Construct a nonlinear uncertain singular non-causal system, the expression is as follows: Ey(j+1)=Hy(j)+f(ω(j),ξ j ),j=0,1,2,...,J-1; Among them, (E , H) is regular, f: is a vector-valued function that satisfies the Lipschitz condition.

5. Application of the optimal control method based on Hurwicz criterion in smart grid scheduling according to claim 1, characterized in that: In step S3, the steps are as follows: The Hurwicz type optimal control framework is transformed into ρ ∈[0 , 1] Balancing optimistic and pessimistic values, the expression is as follows: like ρ =1, it corresponds to a completely optimistic strategy, where the decision focus is on the best outcome. If ρ = 0, the strategy is completely pessimistic, tending to absolute risk aversion and the worst-case scenario. If (0 < ρ < 1), it represents a balance. The optimal value depends on the initial state and the final state, which is expressed as For each k=J-1, J-2, ..., 1, 0, the recursive equation is expressed as follows: Among them, ω * (k+1),…, ω ( J -1) is the optimal control, 6. Application of the optimal control method based on Hurwicz criterion in smart grid dispatching according to claim 1, characterized in that: In step S4, the Hurwicz-type optimal control problem for linear uncertain singular non-causal systems is solved as follows: For a given uncertain singular system, corresponding to the problem, check the regularity of the condition (E, H) and D2 = 0, if the condition is met, continue, if the condition is not met, exit; Convert the optimal control problem into an equivalent problem; Calculate the coefficient l in reverse order j 、 r j 、s j , t j ; Computational optimal control; for and Assignment, obtain all states through the state equation, the expression is as follows: Calculate the optimal value.

7. Application of the optimal control method based on Hurwicz criterion in smart grid dispatching according to claim 1, characterized in that: In step S4, the Hurwicz-type optimal control problem for nonlinear uncertain singular non-causal systems is solved as follows: For the uncertain singular system given in the problem, check the regularity of the conditions (E, H) and D2 = 0, continue if the conditions are met, and exit if the conditions are not met; Convert the optimal control problem into an equivalent problem; by Calculate α in reverse order j , β j , r j , s j ,σ j , t j , j, j=0,1,2,…,J-1; Computational optimal control; Assign values ​​to relevant variables and obtain all states through state equations. The expressions are as follows: Calculate the optimal value.

8. Application of the optimal control method based on Hurwicz criterion in smart grid dispatching according to claim 1, characterized in that: In step S6, the steps are as follows: Based on the power system control scenario, such as the matrix E captures the system singular dynamics, the uncertainty vector ξ j Modeling uncertain external inputs, the state weight vector is given; By setting the weight parameter ρ and confidence level α , using the Hurwicz criterion to balance risks and benefits, solve the optimal control problem in a physically driven numerical example inspired by power system control, including determining the type of system, performing coordinate transformation, applying the algorithm to obtain the optimal solution steps, and giving the corresponding state and control variable trajectories and the optimal value changes as the parameters change.

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