Optimization device, optimization method, and program

Abstract

To achieve robust optimization even in the case of including an equality constraint condition.SOLUTION: An optimization device for optimizing a determinable decision variable so as to minimize or maximize an objective function for evaluating a system while satisfying a constraint condition includes: a selection unit for selecting the finite number of samples as an assumed value of an uncertain variable; a calculation unit for solving a relaxation problem that relaxes a system equation and an inequality expression constraint condition in accordance with each sample to calculate a candidate for a decision variable being an optimal solution; a determination unit for determining an inequality expression constraint condition having the largest deviation amount among a plurality of original inequality expression constraint conditions in the case of applying the candidate for the decision variable; and a generation unit for adding the determined inequality expression constraint condition to the relaxation problem to generate a new relaxation problem. The calculation unit solves the new relaxation problem to calculate a candidate for a decision variable being an optimal solution.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to an optimization device, an optimization method, and a program.

[0002] 2. Description of the Related Art Conventionally, robust optimization methods that output results of robust optimization have been known (see, for example, Patent Documents 1, 2, 3, 4, and 5). Patent Document 1: International Publication WO2018 / 155319 Patent Document 2: International Publication WO2017 / 145664 Patent Document 3: JP 2009-32104 A Patent Document 4: JP 2006-293483 A Patent Document 5: JP 2013-12185 A Other known academic papers include the following: (1)I. Gilboa: "Theory of Decision under Uncertainty", Cambridge University Press (2009) (2) J. Bram: "The Lagrange Multiplier Theorem for Max-Min with Several Constraints," SIAM J. Applied Mathematics, Vol. 14, pp. 665-667 (1966) (3)VFDemyanov, and VN Malozemov:"Introduction to Minmax," John Wiley (1974) (4) K. Shimizu, and E. Aiyoshi: "Necessary Conditions for min-max Problems and Algorithms by a Relaxation Procedure", IEEE Transactions on Automatic Control, Vol. 25, No. 1 (1980) (5)A. Ben-Tal, and A. Nemirovski:"Robust Convex Optimization", Mathematics of Operations Research, Vol.23, No. 4, pp. 769-805 (1998) (6)A. Ben-Tal, L. El Ghaoui, and A. Nemirovski:"Robust Optimization", Prinston University Press (2009) (7)S. Leyffer, M. Menickelly, T. Munson, C. Vanaret, and S. M.Wild:"A Survey of Nonlinear Robust Optimization", Information Systems and Operational Research, Vol. 58, No. 2, pp. 342-373 (2020) (8)F.John:"Extremum Problems with Inequalities as Side Conditions", in Studies and Essays, Courant Anniversary Vol. K,O, Friedrichs, O.E. Neugebauer and J.J. Stoker (Eds.), Wiley-Interscience, New York (1948) (9)K.R. Gehner:"Necessary and Sufficient Optimality Conditions for the Fritz John Problem with Linear Equality Constraints", SIAM J. Control, Vol. 12, pp. 140-149 (1974) (10) JW Blankenship, and JE Falk: "Infinitely Constrained Optimization Problems", Journal of Optimization Theory and Applications, Vol. 19, pp. 261-281 (1976) (11) K. Shimizu, and E. Aiyoshi: "A New Solution to Optimiaztion-Satisfaction Problems by a Penalty Method", Automatica, Vol. 18, No. 1, pp. 37-46 (1982) (12) T. Sekine, A. Yokoyama, K. Yasuda, Y. Hayashi, T. Tanabe, H. Okamoto and Y. Tada, “Optimal Power Flow (OPF) in Electric Power Systems”, Japan Electric Association(2002). (13)Y. Zhang: "General Robust-Optimization Formulation for Nonlinear Programming", journal of Optimization Theory and Applications, Vol. 132, pp.111-124 (2007) (14)MCCampi, S. Garatti, and M. Prandini: "The Scenario Approach for Systems and Control Design", Annual Reviews in Control, Vol. 33, No. 2, pp.149-157 (2009) (15)T. Wada, R. Morita, T. Asai, I. Masubuchi, and Y. Fujisaki: "Randomized solution for robust optimal power flow", 53rd IEEE Conference on Decision and Control (2014) (16) J. Bessa, P. Pinson, and H. Morais: "Active Distribution Grid Management Based on Robust AC Optimal Power Flow", IEEE Trans. on Smart Grid, Vol. 9, No. 6, (2018) (17) Kiyotaka Shimizu and Eitaro Aiyoshi: "Mathematical Programming", Chapter 11, Shokodo (1984) (18) O. Stein: "Bi-Level Strategies in Semi-Infinite Programming", Springer (2003) (19)Ipopt (Interior Point OPTimizer), https: / / projects.coin-or.org / Ipopt (20)A. Fischer: "A special Newton-Type Optimization Method", Optimization, Vol. 24, pp. 269-284 (1992) Summary of the Invention [Problem to be solved by the invention]

[0003] In system design and planning problems, the system characteristics are described by the equality constraints of the optimization problem. When the system characteristics are directly affected by uncertain variables, the uncertain variables are directly included in the equality constraints. However, it is not easy to directly apply the idea of ​​robust optimization, which considers robustness only for inequality constraints, to such optimization problems with equality constraints. [Means for solving the problem]

[0004] In order to solve the above problem, a first aspect of the present invention provides an optimization device that optimizes determinable decision variables so as to minimize or maximize an objective function for evaluating a system while satisfying constraint conditions. The constraint conditions may include a system equation, which is an equality constraint condition including an uncertain variable, and a plurality of inequality constraint conditions including the uncertain variable. The optimization device may optimize a decision variable such that the plurality of inequality constraint conditions are satisfied regardless of any value within an expected range of the uncertain variable. The optimization device may include a selection unit. The selection unit may select a finite number of samples as expected values ​​of the uncertain variable. The optimization device may include a calculation unit. The calculation unit may calculate a candidate for a decision variable that is an optimal solution by solving a relaxation problem in which the system equation and the plurality of inequality constraint conditions are relaxed corresponding to each sample. The optimization device may include a determination unit. The determination unit may determine an inequality constraint condition that has the largest deviation amount among the original plurality of inequality constraint conditions when the candidate for the decision variable is applied. The optimization device may include a generation unit. The generating unit may add the determined inequality constraint to the relaxed problem to generate a new relaxed problem. The calculating unit may solve the new relaxed problem to calculate candidates for decision variables that are optimal solutions.

[0005] When the candidate decision variables obtained by solving the relaxation problem are applied to the original multiple inequality constraint conditions, the optimization device may repeat the calculation of candidate decision variables by the calculation unit, the judgment by the judgment unit, and the generation of a new relaxation problem by the generation unit until the deviation amount of each of the multiple inequality constraint conditions becomes equal to or less than a predetermined threshold.

[0006] The optimizer may optimize determinable decision variables to minimize or maximize a curve-smoothed objective function.

[0007] The optimization device may increase the threshold value to speed up the convergence speed in optimization, or may decrease the threshold value to improve the optimization accuracy.

[0008] The selection unit may select one sample as an assumed value of the uncertain variable. The calculation unit may apply one sample to the system equation and the multiple inequality constraint conditions to calculate a candidate for the decision variable that is an optimal solution and a state variable determined by the system equation according to the assumed value. The determination unit may have a first determination unit and a second determination unit. The first determination unit may apply the candidate for the decision variable that is an optimal solution to the original multiple inequality constraint conditions to determine whether or not any of the multiple inequality constraint conditions does not satisfy the condition. If any of the multiple inequality constraint conditions does not satisfy the condition, the second determination unit may determine the inequality constraint condition with the largest deviation. The generation unit may select, as the new sample, the assumed value of the uncertain variable that gives a result with the largest deviation among the original multiple inequality constraint conditions. The calculation unit may apply the new sample to the system equation and a plurality of inequality constraint conditions to calculate new decision variable candidates that are optimal solutions, and new state variables that are determined by the system equation according to the new sample.

[0009] The determination unit may use a variable independent of the state variable obtained by the calculation unit as the state variable determined by the system equation according to the expected value.

[0010] The determination unit may include, as a determination target, an inequality constraint condition for which the margin of deviation from the condition when a candidate decision variable is applied in the previous determination is equal to or less than a predetermined value.

[0011] The selector may select samples in regions where the objective function varies as a function of the uncertain variables.

[0012] The selector may select the sample exhibiting the maximum value as the assumed value of the uncertain variable.

[0013] The optimization device may be a power flow optimization device for a power system in which distributed power sources are introduced. The uncertain variables may be output variables of the distributed power sources.

[0014] In a second aspect of the present invention, a program for causing a computer to function as any one of the optimization devices described above may be provided.

[0015] In a third aspect of the present invention, a computer may provide an optimization method for optimizing deterministic decision variables so as to minimize or maximize an objective function for evaluating a system while satisfying constraint conditions. The constraint conditions may include a system equation, which is an equality constraint condition including an uncertain variable, and a plurality of inequality constraint conditions including the uncertain variable. The optimization method may optimize a decision variable such that the plurality of inequality constraint conditions are satisfied regardless of any value within an assumed range of the uncertain variable. The optimization method may include a selection step of selecting a finite number of samples as assumed values ​​of the uncertain variable. The optimization method may include a calculation step of calculating a candidate for a decision variable that is an optimal solution by solving a relaxed problem in which the system equation and the plurality of inequality constraint conditions are relaxed corresponding to each sample. The optimization method may include a determination step of determining an inequality constraint condition that has the largest deviation amount among the original plurality of inequality constraint conditions when the candidate for the decision variable is applied. The optimization method may include a generation step of generating a new relaxed problem by adding the determined inequality constraint condition to the relaxed problem. In the calculation step, a new relaxed problem may be solved to calculate candidates for decision variables that are optimal solutions.

[0016] The above summary of the invention does not list all of the necessary features of the present invention. Also, subcombinations of these features may also be inventions. [Brief description of the drawings]

[0017] [Figure 1] FIG. 1 illustrates an example of an optimization device according to an embodiment of the present invention. [Diagram 2] 1 is a flowchart showing an example of an optimization method performed by the optimization device. [Diagram 3] 13 is a flowchart illustrating an example of a selection process of an initial sample. [Figure 4] 13 is a flowchart showing another example of the selection process of an initial sample. [Diagram 5] 13 is a flowchart showing another example of an optimization method performed by the optimization device. [Figure 6] FIG. 1 is a diagram showing a test system for a two-bus system. [Figure 7] FIG. 1 is a diagram showing a test system for a four-bus system. [Figure 8] 22 illustrates an example computer 2200 in which aspects of the present invention may be embodied, in whole or in part. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0018] The present invention will be described below through embodiments of the invention, but the following embodiments do not limit the scope of the invention according to the claims. Furthermore, not all of the combinations of features described in the embodiments are necessarily essential to the solution of the invention.

[0019] 1 is a diagram showing an example of an optimization device 100 according to an embodiment of the present invention. The optimization device 100 optimizes parameters of the system 200 so as to minimize or maximize an objective function for evaluating the system 200 while satisfying constraint conditions. The design and planning problem of the system 200 is formulated as an optimization problem.

[0020] When the influence of the external environment on the system 200 is uncertain, at least one of the objective function and the constraint conditions includes uncertain variables such as unknown parameters. In this case, the design and planning problem of the system 200 is formulated as an "optimization problem under uncertainty" (hereinafter referred to as an "uncertain optimization problem").

[0021] When the objective function includes uncertain variables, it is discussed as a decision-making problem under uncertainty. Decision-making problems under uncertainty are called min-max problems (or max-min problems), and their optimality conditions and calculation methods are being considered.

[0022] The optimization device 100 of this embodiment executes robust optimization that imposes so-called robustness that guarantees in advance the satisfaction of constraint conditions against fluctuations within expected ranges of uncertain variables after implementation of an optimal solution. The robust optimization in the optimization device 100 of this embodiment may be applied even when at least one of the objective function and the constraint conditions is nonlinear.

[0023] In system design and planning problems, the characteristics of the system 200 are described as system equations, which are equality constraints of the optimization problem. When the characteristics of the system 200 are directly affected by uncertain variables, the uncertain variables are directly included in the equality constraints. In one example, the system 200 is a power system, and the optimization device 100 realizes optimization of power flow for the power system in which distributed power sources are introduced. In this case, the uncertain variables are output variables of the distributed power sources. Therefore, the problem is formulated as one in which the uncertain variables are included in the nonlinear equality constraints consisting of power flow equations.

[0024] However, it is not easy to directly apply the concept of robust optimization, which considers robustness only for inequality constraints, to such equality constraint optimization problems. The concept of robust optimization has been extended only to linear programming problems with linear equality constraints, but it is difficult to extend it directly to nonlinear equality constraints. For this reason, robust optimization methods based on linearization of the nonlinear equality constraints and approximation methods called scenario approaches that determine the values ​​of uncertain variables based on random sampling have been proposed for uncertain optimization problems with nonlinear equality constraints. Approximation methods have also been applied to optimal power flow calculations that are highly nonlinear, but they are unable to calculate solutions that satisfy robustness in the strict sense. In addition, when using the scenario approach, the problem scale increases in order to improve the approximation accuracy.

[0025] The optimization device 100 of this embodiment provides a formulation of an uncertain optimization problem and a solution method thereof when the characteristics of the system 200 are described by equality constraint conditions and an objective function, and the equality constraint conditions include uncertainties. In particular, the optimization device 100 realizes application to optimal power flow calculations of a power system in which distributed power sources are introduced as the system 200.

[0026] The equation constraint conditions describing the characteristics of the system 200 have variables equal to or greater than the number of equations. Therefore, there are degrees of freedom equal to the number of variables, which exceeds the number of equations. The variables handled by the optimization device 100 include uncertain variables y and decision variables u. The decision variables u are variables with degrees of freedom whose values ​​can be determined artificially. The uncertain variables y are unknown variables that cannot be determined.

[0027] The equality constraint condition also includes a state variable x that is determined depending on an uncertain variable y. The optimization device 100 estimates in advance the value of the uncertain variable y depending on the decision variable u under a rational criterion that handles the uncertainty due to the uncertain variable y. The optimization device 100 also estimates the state variable x, which is the remaining variable that satisfies the equality constraint condition, depending on the value of the uncertain variable y and the decision variable u.

[0028] The optimization device 100 in this embodiment takes into account the characteristics of the system by distinguishing between variables that satisfy such equality constraint conditions. In this way, the optimization device 100 formulates an uncertain optimization problem with equality constraint conditions including an uncertain variable y. As a rational criterion for estimating this uncertain variable y and the associated state variable x, the optimization device 100 employs a "min-max criterion" or a "robustness criterion."

[0029] The optimization device 100 sequentially generates and increases certain assumed states (scenarios) for the characteristics of the system 200. The optimization device 100 can calculate a solution that satisfies a min-max criterion for the objective function and a robustness criterion for the inequality constraint conditions. The optimization device 100 can also apply the above formulation and solution method to an optimal power flow calculation for a power system in which distributed power sources have been introduced. However, the application of the optimization device 100 is not limited to optimal power flow calculation for a power system.

[0030] The formulation of the optimization problem considering the uncertainty of the system 200 will be described. The system 200 to be designed and planned is assumed to be a static system without time dependency. The system 200 is modeled by a state variable x that represents the state quantity of the elements constituting the system 200, and a system equation that represents the relationship between the elements.

[0031] When there is freedom in planning and design, the system equation h(u, y, x) which is an equality constraint condition includes the decision variable u as described above. The optimization device 100 calculates the values ​​of the decision variable u and the state variable x which minimize or maximize the objective function for evaluating the system 200, using this system equation as an equality constraint condition.

[0032] On the other hand, there exists an uncertain variable y, which is an undetermined variable or unknown parameter that cannot be determined artificially, such as the influence of the external environment of the system. Here, let us define the decision variable u as u∈R. N Let the uncertain variable y be y∈R. I Let the state variable x be x∈R. L Let function h:R N ×R I ×R L →R L Using this, the system equation can be expressed as the following equation (1).

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[0033] In addition, the uncertain influence from the external environment is considered in the objective function and the inequality constraints using the same uncertain variables. N ×R I ×R L →R 1 , and function g:R N ×R I ×R L →R M Using the above, the objective function is expressed by equation (2), and the inequality constraint is expressed by equation (3).

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[0034] The optimization problem is expressed by the following equations (4a), (4b), and (4c).

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[0035] Since the value of the uncertain variable y is undetermined, it is virtually impossible to solve this optimization problem. Therefore, the optimization device 100 assumes the value of the uncertain variable y based on some criterion. In this way, the optimization device 100 solves the optimization problems (4a), (4b), and (4c). The optimization device 100 determines the optimal value of the decision variable u. According to the optimization device 100, the real system is planned and designed in advance based on the determined optimal value. As a result, at the stage when the real system is actually affected by the external environment, that is, at the stage when the value of the uncertain variable y is determined, the value of the state variable x of the real system corresponding to this is determined for the first time.

[0036] Therefore, it is desirable to assume the value of the uncertain variable y by introducing a rational criterion in the preliminary design and planning stages. In the design and planning of the system 200 under consideration, the uncertain variable y is assumed to be in an uncertain situation where there is no probabilistic or statistical information about the uncertain variable y.

[0037] As an example, in the optimal power flow calculation of a power system with distributed power sources, in addition to so-called power sources (hereafter abbreviated as "determined power sources") where the values ​​of variables associated with the power source, such as the operating state of a normal generator, can be artificially determined, it is possible to consider an optimization problem in a power system that also has distributed power sources (hereafter abbreviated as "uncertain power sources") whose values ​​of variables representing their operating state are uncertain due to output characteristics that are difficult to predict in advance, such as because the power source is derived from natural energy or because consumers do not have all the information about the distributed power sources they own. The system equations for such a power system can be expressed as the following equation (5) using the node set N (in italics) and branch set B (in italics) as the pattern of a set of certain nodes and branches, (N, B), as the power flow equation that models the network relationship in the power system and the relationship between power and voltage based on the AC method.

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[0038] In the system equations (1) and (5), u is a decision variable that represents an operating state that can be determined in advance. In particular, u in (5) may mainly correspond to the output setpoint and voltage setpoint for the deterministic power source and the output limit value for the uncertain power source.

[0039] In the system equations expressed by equations (1) and (5), y is a variable that represents the operating state of the uncertain power source, and x is a state variable in the system equations. In one example, x corresponds to the voltage of the power system and the power line flow. After the optimal value of the decision variable u is implemented in the actual power system, the detected value y of the uncertain power source, which is an uncertain variable, is obtained, and the value of the state variable x of the power system corresponding to these is determined.

[0040] Moreover, if the value y of the uncertain power supply variable is assumed in advance according to some criterion, it is possible to assume the value of the state variable x corresponding to this and the value u of the decision variable. In general, in a situation where the value of the uncertain variable y is undetermined in the design and planning stage of the system 200, it is considered reasonable to assume the worst case scenario, for example, as one of the so-called crisis management strategies to ensure safety and security. Specifically, for equation (3), which is the physically imposed inequality constraint condition equation, a safety criterion, so-called "robustness criterion," is adopted that avoids violation of the constraint condition even if the worst case scenario occurs after the design and planning for the value of the uncertain variable for the inequality constraint condition. That is, the individual component functions g of the inequality constraint function are m Equation (6) is introduced in place of the inequality constraint (4b) so that the inequality constraint is not violated even if the uncertain variable y takes a value that maximizes each of (u, y, x).

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[0041] where g = (g1, gM). T The set Y (in italics) of uncertain real numbers y represents the possible range of values ​​that the variable can take, and is assumed to be a bounded closed set. Similarly, for the objective function f(u, y, x), we assume the occurrence of uncertain variable values ​​that make it the worst, and based on the so-called "min-max criterion" that maximizes the improvement of the situation at that time, we adopt equation (7) instead of minimizing the objective function in equation (4a).

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[0042] As long as we are prepared for the objective function value in equation (7) above, we can rest assured that whatever uncertain variable values ​​occur later, they will be improved from this value. What should be noted in the above assumptions about the uncertain variable y is that the objective function f and the multiple component functions g of the constraint function m, m = 1, , M. For example, in the case of inequality constraints, when different safety standards are imposed, it is equivalent to assuming the worst case for each safety standard. Therefore, even if the uncertain variables are physically the same, the objective function f and multiple component functions g of the constraint function m , m=1,...,M, we use the subscripted variables y0, y1,...,y M These differentiated variables are called "assumed variables."

[0043] In addition, the set of constraints specific to each of the assumptions, i.e., the assumptions y0, y1, . . . , y M The assumed range of is assumed to be common to the set Y (in italics) specific to the uncertain variable y. In addition, any value of the decision variable u and the assumed value y that is assumed independently of it are assumed m Similarly, the state variable x, which is assumed to satisfy the system equation corresponding to the objective function f and the multiple component functions of the constraint function, is regarded as a different assumed variable x. m , m = 1, . . . , M. The system equations are also differentiated by y m , m = 1, , M as individual constraints of the maximization operation. Therefore, the uncertain optimization problem assuming the worst case scenario taking into account the uncertainties in the system equations is expressed as equations (8a) and (8b).

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[0044] Here, U (in italics) is the set of constraints specific to the decision variables. Also, the set of assumptions x for the state variables x is m , m = 1, , M, we can consider a common unique constraint set X (in italics) for them. This constraint set X (in italics) is m Inequality constraints g with only one argument m (x m )≦0, g m (u, x m , y m)≦0, m=1,...,M is assumed to be incorporated into the

[0045] As described above, in the uncertain optimization method in the optimization device 100 of this embodiment, the state variable x, which is differentiated from the decision variable u, is considered, and the equality constraint (1) is positioned as a system equation that assumes this state variable x together with the uncertain variable y. This consideration is the reason why "equality constraints" including the uncertain variable y have not been handled in optimal satisfiability or robust optimization.

[0046] In addition, when assuming an uncertain variable y by considering the system equation as an equality constraint, it is necessary to limit the decision variables u to those for which there exist assumed values ​​for the uncertain variable y and state variable x that satisfy the system equation (1), that is, to consider equation (9), which is an implicit constraint on the decision variable u. However, for the sake of simplicity, we will assume here that the condition in equation (9) holds for any element of the constraint set U (in italics) specific to the decision variables.

[0047]

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[0048] Next, we introduce a variable σ that satisfies equation (10) and corresponds to the upper limit of the objective function.

[0049]

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[0050] In this case, the uncertain optimization problem expressed by equations (8a) and (8b) can be equivalently replaced by equations (11a), (11b), and (11c).

[0051]

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[0052] Furthermore, we introduce equations (12a) and (12b) as follows:

[0053]

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[0054] Using equations (12a) and (12b), the simultaneous satisfaction of equations (11b) and (11c) is equivalent to the satisfaction of equation (13).

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[0055] In the above formula (13), the outer max operation, G0(u, σ), G m (u, σ), m = 1, , M, replace the max operation contained throughout (12a) and (12b) by the latter maximization variable (x m , y m ) is made common. As a result, equations (11b) and (11c) can be combined into equation (14) as a single constraint.

[0056]

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[0057] Therefore, the uncertain optimization problem (8a) and (8b) taking into account the system equation (1) can be formulated as (15a) and (15b).

[0058]

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[0059] Next, we consider a method for solving the problem given by equations (15a) and (15b). We explain the principle and calculation procedure of the relaxation method. Equations (15a) and (15b), or the equivalent problems given by equations (11a) to (11c), can be regarded as two-level optimization problems due to their structure. Here, a two-level optimization problem is a method for satisfying constraint conditions under an assumed worst-case scenario, while at the same time imposing constraints from a higher-level perspective on the worst-case scenario and evaluating it with an objective function. However, no method for solving a two-level optimization problem that takes into account everything from the system equation corresponding to the variable y to the assumed state variable x is known.

[0060] In addition, the inequality constraint given by equation (15b) is a complex constraint on the variables (σ, u). Specifically, the inequality constraint is a complex constraint on the variables (σ, u) for multiple component functions g m , m = 1,...,M, we extract the function with the maximum component (inner maximization operation max), and require that the maximum value of the maximization operation with respect to the state variable x and the uncertain variable y (outer maximization operation max) is zero or less. However, the function on the left side of the inequality constraint (15b) may be a non-differentiable function with respect to the variables (σ, u). Therefore, it is difficult to directly use a nonlinear optimization method to solve the problems shown in (15a) and (15b).

[0061] On the other hand, the outer maximization operation for the variables (x, y) in equation (15b) can be equivalently expressed as equation (16) using universal quantification.

[0062]

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[0063] If the constraint set Y (in italics) for the uncertain variable y consists of an infinite number of elements (uncountably infinite), the number of inequality constraints will also be uncountably infinite, corresponding to all of the (x, y). Therefore, in this specification, we extend the relaxation method, which is a solution method for optimization problems with uncountably infinite inequality constraints.

[0064] 1, the optimization device 100 includes a selection unit 110, a calculation unit 120, a determination unit 130, and a generation unit 140. In this example, the optimization device 100 includes a smoothing unit 150. However, the optimization device 100 is not limited to having the smoothing unit 150. The smoothing unit 150 smoothes the objective function using a curve.

[0065] The optimization device 100 may be a computer, and the selection unit 110, calculation unit 120, determination unit 130, generation unit 140, and smoothing unit 150 may each be realized as a function of a calculation device such as a CPU.

[0066] The selection unit 110 selects a finite number of samples y as assumed values ​​of the uncertain variable y. (s) ∈Y (italics), s = 1, . . . , S. Given any decision variable u∈U (italics), select a sample y (s) The system equation (17) can be considered as a certain scenario according to the state x (s) Think about it.

[0067]

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[0068] x (s) is sample y (s) If we note that the variables satisfy the system equation shown in equation (17) corresponding to , equation (16), which is a countably infinite number of inequalities, can be relaxed to a set of S inequality constraints and system equations corresponding to each sample, as shown in equations (18a) and (18b).

[0069]

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[0070] Furthermore, equation (18a) is equivalent to M+1 inequalities, such as equation (19), being true simultaneously.

[0071]

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[0072] Therefore, the constraint relaxation problem for the uncertain optimization problem shown in equation (15) is expressed by the following equations (20a) to (20e).

[0073]

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[0074] The optimization device 100 calculates a plurality of assumed values ​​y of an uncertain variable y. (s) In response to this, multiple system equations (20d) are assumed as assumed states (scenarios).

[0075] Next, the calculation unit 120, the determination unit 130, and the generation unit 140 in the optimization device 100 will be described. (s) The relaxed problems (20a) to (20e) are solved by relaxing the system equations and the inequality constraints in accordance with the above, and the optimal solution candidates for the decision variables u (u bar (in the formula, σ is underlined) (k) and σ bar (in the formula, σ is underlined) are calculated.

[0076] The determination unit 130 may include a first determination unit 132 and a second determination unit 134. The first determination unit 132 applies decision variable candidates, which are optimal solutions obtained by the calculation unit 120 solving the relaxation problems (20a) to (20e), to the original multiple inequality constraint conditions (15b) to determine whether or not there is any inequality constraint condition (15b) that does not satisfy the condition. The second determination unit 134, when there is any inequality constraint condition (15b) that does not satisfy the condition, determines the inequality constraint condition with the largest deviation. The determination unit 130 determines the expected value y (s) As the state variable x determined by the system equation according to, a variable independent of the state variable x obtained by the calculation unit 120 is used. That is, the information on the state variable x obtained in the calculation process by the calculation unit 120 is not used in the maximization process by the judgment unit 130. This makes it possible to prevent a situation in which the optimization process cannot be completed due to the state variable x being fixed and losing the degree of freedom.

[0077] The generating unit 140 adds the determined inequality constraints to the relaxed problems (20a) to (20e) to generate new relaxed problems. The calculating unit 120 solves the new relaxed problems to calculate decision variable candidates that are optimal solutions. In particular, the generating unit 140 calculates the assumed value y(k) of the uncertain variable that gives the result with the largest deviation among the original multiple inequality constraints (15b) as a new sample y of the uncertain variable. (s(k)+1) The calculation unit 120 selects the new sample y (s(k)+1) By applying the system equation and multiple inequality constraints (15b), we obtain the optimal solution, a new decision variable candidate u and a new sample (s(k)+1) The new state variable x is determined by the system equation according to (s(k)+1) Calculate the following:

[0078] In the optimization device 100, a new inequality constraint is added, and the assumed value y of the uncertain variable y that gives the new inequality constraint is calculated. (s) A new system equation (20d) that defines the state variable u corresponding to is added as an assumed state.

[0079] The relaxation method used in the optimization device 100 uses a nonlinear optimization technique to solve the relaxation problem represented by equations (20a) to (20e). The calculation unit 120 solves the relaxation problem represented by equations (20a) to (20e) to obtain solutions σ (in the equation, an overline is added to σ) (k) and u (in the equation, an overline is added to σ) (k). The determination unit 130 then finds an inequality constraint condition that most violates the inequality constraint condition equation (15b) of the uncertain optimization problem (15) that is originally to be solved and that was ignored in the relaxation problem. Then, in the optimization device 100, the generation unit 140 adds the inequality constraint condition that most violates the inequality constraint condition equation (15b) to the relaxation problem (20a) to (20e) to generate a new relaxation problem. The calculation unit 120 re-solves the new relaxation problem.

[0080] In the optimization device 100, when the decision variable candidates obtained by solving the relaxation problem are applied to the original plurality of inequality constraint conditions (15b), the calculation unit 120 calculates the decision variable candidates (σ bar, u bar), the judgment unit 130 judges, and the generation unit 140 generates a new relaxation problem repeatedly until the deviation amount of each of the plurality of inequality constraint conditions (15b) becomes equal to or less than a predetermined threshold value ε.

[0081] Let k be the number of iterations in the iterative procedure of the relaxation method. Let S(k) be the number of samples of the uncertain variable y in the k-th iteration. Let the optimal solution of the relaxation problem (20a) to (20e) with S = S(k) be as follows.

[0082]

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[0083] The first judgment unit 132 judges whether or not this solution satisfies the inequality constraint condition equation (15b) of the original uncertain optimization problem. If this solution does not satisfy the inequality constraint condition equation (15b) of the original uncertain optimization problem, the second judgment unit 134 finds the inequality constraint condition that most violates the constraint condition among them. Specifically, by substituting (σ bar (k), u bar (k)) into the left side of equation (15b) or equation (14) and executing the maximization operation, it is possible to judge whether the maximum value of the maximization problem of the following equations (21a) and (21b) is non-positive.

[0084]

number

[0085]

number

[0086] If φ(σ bar (k), u bar (k)) in the above equation (22) is 0 or less (in implementation, it is ε or less: ε is a sufficiently small integer determined in advance), then the solution satisfies the inequality constraint condition equation (15b) (equivalent condition equation (16)) of the original uncertain optimization problem. (σ bar (k), u bar (k)) becomes the optimal solution of the original uncertain optimization problem (15a) and (15b). On the other hand, if φ(σ bar (k), u bar (k))>0 (or φ(σ bar (k), u bar (k))>ε) in equation (22), then σ bar (k) and u bar (k) most violate the inequality constraint (15b). If this is expressed using the equivalent condition equation (16), then of these countably infinite inequality constraints, the inequality constraint (23) corresponding to (x hat (k), y hat (k)) most violates it.

[0087]

number

[0088] So, y(k) is the new sample of the uncertain variable y (s(k)+1) Let the state variable u and the new sample y (s(k)+1) The state variables x satisfying the new system equation (24) (s(k)+1) Introduce the following.

[0089]

number

[0090] The generating unit 140 adds the equations (25a) and (25b) together with the equation (24) to the relaxed problem (the equations (20a) to (20e)) to generate a new relaxed problem. The calculating unit 120 then repeatedly solves the new relaxed problem.

[0091]

number

[0092] 2 is a flowchart showing an example of an optimization method by the optimization device 100. The procedure of the relaxation method taking into account the system equation will be described with reference to the flowchart. The selection unit 110 selects the initial sample y (1) Select ∈Y (italics) ∩Y′ (italics) (step S10). The selection unit 110 sets the number of samples as S(1)=1 and the number of repetitions as k=1. Here, Y′ (italics) is given by equation (26).

[0093]

number

[0094] The calculation unit 120 solves the relaxed problem (equations (20a) to (20e)) with S=S(k) to calculate the optimal solution as follows (step S20). When k=1, the calculation unit 120 applies one sample selected by the selection unit 110 to the system equation and the multiple inequality constraint conditions (equations (15a) and (15b)) to calculate a decision variable candidate (u bar) that is the optimal solution and a state variable (x (s) Calculate the value of (bar).

[0095]

number

[0096] The determination unit 130 solves the maximization problems of equations (21a) and (21b) and sets the maximum solution to (x(k), y(k)). The determination unit 130 substitutes the maximum solution (x(k), y(k)) for the maximum function of equation (22) to calculate the value φ(σ(k), u(k)).

[0097] If the maximum function value φ(σ(k), u(k))>threshold ε is true, the determination unit 130 determines that the deviation from the original inequality constraint condition (15b) is greater than threshold ε (step S30: NO). If the deviation is greater than threshold ε (step S30: NO), the generation unit 140 sets S(k+1)=S(k)+1 and y (S(k+1)) = y(k). As a result, the generation unit 140 adds the corresponding inequality constraints to (20a) to (20e) to generate a new relaxed problem. On the other hand, if the value of the maximum function φ(σ(k), u(k))≦ε (step S30: YES), S(k+1) is set to S(k) (step S60).

[0098] The reason why the threshold value ε (>0) is introduced into the judgment condition in step S30 is to allow a predetermined small error in the satisfaction of the inequality constraint condition (16) so that the relaxation method can be terminated in a finite number of iterations by generating a finite number of samples of the uncertain variable y.

[0099] If S(k+1)=S(k) (step S70: NO), the optimization device 100 ends the process, and the calculation unit 120 outputs u(k) as the optimal solution u to the problem (15) (step S80). If S(k+1)=S(k) is not true (step S70: NO), k←k+1 is set, and the process returns to step 2.

[0100] The processing of steps 30 and 40 may be performed by solving the maximization problems (21a) and (21b). In the cases of (21a) and (21b), the two max operations of the objective function can be interchanged, so that it is relatively easy to calculate the maximum solution (x(k), y(k)). In other words, the problems (21a) and (21b) are equivalent to the following equation (27).

[0101]

number

[0102] The determination unit 130 solves (28a), (28b), (28c), and (28d), which are the maximization problems with M+1 equality constraints in the square brackets of the equation (27).

[0103]

number

[0104] Then, the determination unit 130 selects the M+1 maximum solutions (x hat m (k), y hat m (k)), m = 0, 1, . . . , M, and the corresponding M + 1 function values ​​f(u(k), x(k), y(k)) - σ(k), g m (u bar(k), x hat m( k), y hat m, M. Then, the determination unit 130 calculates the maximum function value among these function values ​​as φ(σ bar(k), u bar(k)). The determination unit 130 gives this maximum function value φ(σ bar(k), u bar(k)) (x hat m (k), y hat m By rewriting (x(k), y(k)) as (x(k), y(k)), the maximum solution (x(k), y(k)) can be calculated.

[0105] In order to confirm the convergence of the optimization method in this embodiment, a simple numerical example will first be considered.

[0106]

number

[0107] In this example, the objective function f does not contain the uncertain variable y, but it does contain the state variable x, so the min-max criterion with equality constraints is min-max x {f(u,x|h(u,x,y)=0} is applied. The same applies to constraint functions g3 and g4 that do not include the decision variable u and the uncertain variable y. For example, the robustness criterion for g3 is min-max x {g3|h(u, x, y) = 0}.

[0108] From the above, the relaxation problem (20a) to (20e) for this example can be expressed as (s) , s=1,...,S is the assumed state variable x (s) , s = 1, ..., S are the arguments of functions f, g2, and g3, and are expressed by the following equations (30a) to (30h).

[0109]

number

[0110] The maximization problems (21a) and (21b) for this problem are expressed as the following (31a) and (31b).

[0111]

number

[0112] In addition, as a nonlinear optimization method applied to the problem (28) (in this example, M=4, and a maximization problem with five equality constraints) that is actually executed to solve the relaxation problem (equations (30a) to (30h)) in step S20 of the relaxation method procedure and the maximization problem (equations (31a) and (31b)) in step 30, an Interior Point OPTimizer (IPOPT) may be used as an example. In addition, in step S30, the tolerance parameter ε of the judgment condition, which is a threshold, is set to 1e-7. In this case, the initial sample of the uncertain variable is set to y (1) = 5 (the number of initial samples is S(1) = 1), and the results of applying the relaxation method are shown in Tables 1 and 2.

[0113] [Table 1]

[0114] [Table 2]

[0115] The columns of Table 1 represent, in order, the set of samples of the generated uncertain variables Y (s) (Italics) σ(k), u(k), and y (s) The corresponding {x bar (s)}. An asterisk is added to indicate that, according to Table 1, for k=3, the decision variable u = 5 and the corresponding min-max value σ = 900 were obtained.

[0116] The columns of Table 2 show the maximum values ​​of f-σ, g1, g2, g3, and g4 in the five equality-constrained maximization problems in problems (28a) to (28d) executed to solve the maximization problem (equations (31a) and (31b)), and the further maximum value of φ(22) among them. Table 2 shows that the calculation was terminated at k=3 because all of these values ​​satisfied the termination criterion, i.e., ε(=1e-7) or less.

[0117] The process leading up to the end of the calculation is as follows. First, the solution (σ bar(k) and u bar(k)) of the relaxation problem in row k=1 in Table 1 is determined to be the one that deviates most from the constraint (30b) related to the objective function by the maximum value 1.75e+2 = φ(σ bar(1), u bar(1)) = f(u bar(1, x bar(1)) - σ bar(1) in row k=1 in Table 2. Then, the maximum solution y bar(1) = 10 that gives this is the solution for the new sample y of the uncertain variable y. (2) = 10, and the corresponding constraint is added to the relaxation problem. The solution to this relaxation problem is the row k = 2 in Table 1, and the constraint from which this solution deviates the most is determined by the maximum value 5 = φ(σ(2),u(2)) = g4(x(2)) in the row k = 2 in Table 2. The maximum solution y(2) = 0 that gives this is the new sample y of the uncertain variable y. (3) = 0, and the corresponding constraint is added to the relaxation problem, resulting in k = 3. Note that similar results to those above were obtained when the positions of the initial samples of the uncertain variables were randomly changed 100 times, confirming the global convergence of the relaxation method for the selection of the initial samples and its robustness to uncertain variables.

[0118] 3 is a flowchart showing an example of the selection process of the initial sample. The selection unit 110 extracts a region in which the objective function changes according to the uncertain variable y (step S100). The selection unit 110 selects the sample y in the region in which the objective function changes according to the uncertain variable y. (1) (Step S102). In the region where the objective function does not change depending on the uncertain variable y, the initial sample y (1)If you select , the optimization process may not proceed. Therefore, in the region where the objective function changes depending on the uncertain variable y, the sample y (1) By selecting this option, the optimization process can proceed reliably.

[0119] 4 is a flowchart showing another example of the selection process of the initial sample. The selection unit 110 selects the sample y (1) (Step S110) The process in Fig. 4 also makes it possible to prevent the optimization process from stalling and ensure that the optimization process proceeds reliably.

[0120] Fig. 5 is a flowchart showing another example of the optimization method by the optimization device 100. The optimization method shown in Fig. 5 adds the processing of step S65 to the optimization method shown in Fig. 2. The processing contents other than step S65 in the optimization method shown in Fig. 5 are similar to the processing contents of the optimization method shown in Fig. 2, so detailed explanations will be omitted.

[0121] The judgment unit 130 acquires the margin of error until the condition is deviated (deviation amount>threshold value ε) when the decision variable candidate u is applied in the previous judgment. The judgment unit 130 includes the inequality constraint condition whose margin is equal to or less than a predetermined value as a judgment target (step S65). In this way, the inequality constraint condition whose margin is equal to or less than a predetermined value affects the optimization result. Such an inequality constraint condition is called active.

[0122] When the margin is greater than a predetermined value, the determination unit 130 may not include the corresponding inequality constraint in the determination of whether the optimal solution satisfies the inequality constraint. In this way, an inequality constraint with a sufficient margin before deviation (deviation amount > threshold ε) is unlikely to affect the optimization result. Such an inequality constraint is called inactive. However, a parameter that requires a high safety factor may always be included in the determination of whether the optimal solution satisfies the inequality constraint.

[0123] Next, a case where the optimization method of the present invention is applied to an optimal power flow calculation for a power system in which distributed power sources are introduced will be described. Consider the problem of optimally operating a power system under an uncertain power source introduced into the power system. The decision maker in this problem may be a system operator or the like. The system operator formulates an optimal power flow calculation based on the system information held by the system operator. The system operator manages and maintains the voltage and power flow of the transmission line based on the optimal power flow calculation. The uncertain power sources are assumed to be distributed power sources held by power generation companies or consumers. The uncertain power sources are connected to the power system by inverter interconnection. The active power output of these uncertain power sources has uncertainty due to various factors. The range in which the output value of the active power of the uncertain power sources can take is explicitly given. The system operator can limit the output of the uncertain power sources by any set value and issue a reactive power output command within the output capacity of each inverter. Here, the following variables are introduced.

[0124]

number

[0125] p G , v G , p D , q D are the decision variables u for system operators, etc. Only y is the uncertain variable y for system operators, etc., and p G0 , q Q , p, q, v, and δ are the state variable x that satisfies the load flow equation (equivalence constraint). For simplicity, the variable that combines the decision variable u and its direct product set are expressed as equation (32).

[0126]

number

[0127] Similarly, the variable obtained by combining the state variables is expressed by equation (33).

[0128]

number

[0129] The optimal power flow calculation considering distributed power sources can be formulated as equations (34a) to (34G).

[0130]

number

[0131] Here, the first term f0 of the objective function (34a) is the transmission loss of the entire power system, and is a function that mainly depends on the active power. The second term of the objective function (34a) is the active power y i The limit value of p Di The objective function ρ(>0) is a weighting coefficient for the objective function. Equations (34b) to (34e) express the power system state variable p G0 , q Q , p, q, and v are inequality constraints. Specifically, equation (34b) is the upper and lower limit constraint on the active power of the slack bus. Equation (34c) is the upper and lower limit constraint on the reactive power of the deterministic power source. Equation (34d) is the upper and lower limit constraint on the voltage. Equation (34e) is the line capacity constraint. Furthermore, equation (34g) is the inverter output capacity constraint for each uncertain power source i. The inverter output capacity s Di is known to the system operator. Furthermore, equation (34g) is the load flow equation for the entire power system. Here, equations (34a) to (34g) can be rewritten as the following equations (35a), (35b), and (35c).

[0132]

number

[0133] A min-max criterion is applied to the objective function (35a), and a robustness criterion is applied to the inequality constraint function (35b). This makes it possible to directly apply the formulation of problems (8) and (15), and the calculation procedure of the relaxation method that takes into account the system equations for these problems.

[0134] In order to verify the usefulness of the proposed method, we applied the relaxation method to two simple test systems. In this paper, we show two types of test systems, a two-bus system and a four-bus system, as the simple test systems.

[0135] Fig. 6 shows a test system for a two-bus system. Fig. 7 shows a test system for a four-bus system. In Fig. 6 and Fig. 7, an optimization device 100 is an optimization device for power flow in a power system 102 in which a distributed power source (uncertain power source) 10 has been introduced. An uncertain variable y is an output variable of the distributed power source 10.

[0136] The settings of the constraints and the like are shown in Table 3. In Fig. 6, the determined power source 20 is connected to bus 0, and the distributed power source 10 is connected to bus 1. In Fig. 7, the determined power source 20-1 is connected to bus 0, and the determined power source 20-2 is connected to bus 3. The distributed power source (DER1) 10-1 is connected to bus 1, and the distributed power source (DER2) 10-2 is connected to bus 2. In addition, a tap 30 is provided between bus 0 and bus 3.

[0137] [Table 3]

[0138] Note that all values ​​in the figures and tables are in units of PU. The relaxation method was set with a tolerance parameter of ε = 1e-4, and the nonlinear optimization method IPOPT was used. The equations containing the max operation in problem (35a) and (35b) contain non-differentiable points. Therefore, nonlinear optimization methods cannot be applied to this max operation as is. Therefore, the non-differentiability can be avoided by replacing it with the Fischer-Burmeister function, which is a smoothing function.

[0139] Optimal solution u and state variables x obtained by applying the relaxation method to a two-bus system (s ), the uncertain variable y (s) are shown in Table 4. The corresponding inequality constraint function values ​​f-σ, g are shown in Table 5.

[0140] [Table 4]

[0141] [Table 5]

[0142] g1 in Table 5 is p G0 is the lower bound of the value of p G0 g3 is the upper limit constraint value of q G0 g4 is the lower bound of q G0 g5 is the value of the upper limit constraint of v1, g6 is the value of the upper limit constraint of v1, and g7 is the value of the inverter output capacitance constraint. The relaxation method was completed after two iterations. The initial sample y (1) and the sample y obtained by solving the maximization problem (21) (2) Two assumed states (scenarios) were assumed. As far as the results are concerned, solutions that satisfy the constraints in each scenario were obtained. Specifically, in the scenario y, where the active power y of the distributed power source, which is an uncertain variable, is maximized, (1)When p = 1, the voltage upper limit constraint (g6 in Table 5) is active, and the active power limit value p D On the other hand, the scenario y (2) At q = 0, the reactive power q D The output shows that the operation takes into account all possible ranges [0, 1] of the uncertain variable y.

[0143] Next, the optimal solution u obtained by applying the relaxation method to the four-bus system and the state variables x (s) , the uncertain variable y (s) are shown in Table 6. The corresponding inequality constraint function values ​​f-σ, g are shown in Table 7.

[0144] [Table 6]

[0145] [Table 7]

[0146] Here, g1 in Table 7 is p G0 is the lower bound of the value of p G0 g3 and g4 are the upper bounds of p G g5 and g6 are the lower bounds of p G g7 is the value of the lower bound constraint on v1, g8 is the value of the lower bound constraint on v2, g9 is the value of the upper bound constraint on v1, and g 10 is the upper limit constraint value of v2. Furthermore, g 11 From g 16 is the value of the line capacity constraint. g 17 and g 18is the value of the inverter output capacity constraint. In this example, there are two uncertain variables. The relaxation method completes after two iterations. Two assumed states (scenarios) are generated based on two samples y(1) and y(2) of the uncertain variable y. In particular, the node of distributed power source 2 (DER2 in Figure 2), which is farther away from the slack bus, is considered to have a relatively large voltage fluctuation. In fact, the voltage upper limit constraint (g in Table 7) 10 ) and the lower voltage limit constraint (g8 in Table 7) are active. The active power limit value p D2 and reactive power setting value q D2 has been obtained.

[0147] In this problem, the set Y (in italics) of the active power of the uncertain power sources, which are uncertain variables, is given by upper and lower bound constraints. The samples of that variable generated by the relaxation method are likely to be selected as the endpoints of that set. In fact, that endpoint is selected in both Table 4 and Table 6. In particular, in the four-bus system, there are two uncertain variables and four endpoints. It can be seen that the method converges by generating samples of only two of the endpoints, that is, in two iterations of the relaxation method. From this, when the range of the active power of the uncertain variables is assumed by upper and lower bound constraints, it is expected that the relaxation method will converge in significantly fewer iterations compared to the number of endpoints.

[0148] Furthermore, to verify the validity of the decision variables u* obtained by the relaxation method, the values ​​of the uncertain variables y were randomly generated within the set Y (in italics) under the fixed decision variables u*, and the state variables x determined for them, their objective function values, and inequality constraint function values ​​were calculated. The number of generated points was 10 for the two-bus system and 100 for the four-bus system. The calculated objective function values ​​in each case were smaller than the min-max value σ* and furthermore satisfied all the inequality constraint conditions. Therefore, it was confirmed that the solutions obtained by the relaxation method guaranteed the min-max criterion and the robustness criterion.

[0149] As described above, in this specification, an uncertain optimization problem including equality constraint conditions describing the characteristics of the system 200 is formulated by applying a min-max criterion to the objective function and a robustness criterion to the inequality constraint conditions. The state of the system 200 is assumed as an assumed state (scenario). By successively increasing the assumed states (scenarios), a new relaxation method for calculating a solution to the uncertain optimization problem is realized. In addition, the usefulness of this method for optimal power flow calculation for a power system 102 in which a distributed power source 10 has been introduced is also confirmed.

[0150] Incidentally, the above explanation has been about a static system with no time dependency, but when the controlled object has a storage function such as a storage battery, system optimization is no longer an independent optimization problem for each time slice. This system optimization problem becomes an optimal control problem in which the time transition of the state is described by a difference equation that also includes uncertain variables. The idea of ​​differentiating between decision variables and state variables explained in this specification can be extended to such so-called "uncertain optimal control problems."

[0151] For larger-scale power system optimization problems and the above-mentioned optimal control problems with time series variables, it is expected that the computational efficiency of the relaxation method will deteriorate as the number of variables increases. For such challenges, various measures can be used, such as considering only constraints that are likely to be active as a maximization problem in the relaxation method, as shown in Figure 5.

[0152] 8 illustrates an example of a computer 2200 in which aspects of the present invention may be embodied in whole or in part. Programs installed on the computer 2200 may cause the computer 2200 to function as or perform operations associated with an apparatus or one or more sections of the apparatus according to an embodiment of the present invention and / or to perform a method or steps of a method according to an embodiment of the present invention. Such programs may be executed by the CPU 2212 to cause the computer 2200 to perform specific operations associated with some or all of the blocks of the flowcharts and block diagrams described herein.

[0153] The computer 2200 according to this embodiment includes a CPU 2212, a RAM 2214, a graphic controller 2216, and a display device 2218, which are connected to each other by a host controller 2210. The computer 2200 also includes input / output units such as a communication interface 2222, a hard disk drive 2224, a DVD-ROM drive 2226, and an IC card drive, which are connected to the host controller 2210 via an input / output controller 2220. The computer also includes legacy input / output units such as a ROM 2230 and a keyboard 2242, which are connected to the input / output controller 2220 via an input / output chip 2240.

[0154] The CPU 2212 operates according to a program stored in the ROM 2230 and the RAM 2214, thereby controlling each unit. The graphic controller 2216 retrieves image data generated by the CPU 2212 into a frame buffer or the like provided in the RAM 2214 or into the graphic controller 2216 itself, and causes the image data to be displayed on the display device 2218.

[0155] The communication interface 2222 communicates with other electronic devices via a network. The hard disk drive 2224 stores programs and data used by the CPU 2212 in the computer 2200. The DVD-ROM drive 2226 reads programs or data from the DVD-ROM 2201 and provides the programs or data to the hard disk drive 2224 via the RAM 2214. The IC card drive reads programs and data from an IC card and / or writes programs and data to an IC card.

[0156] The ROM 2230 stores therein a boot program or the like executed by the computer 2200 upon activation, and / or a program that depends on the hardware of the computer 2200. The input / output chip 2240 may also connect various input / output units to the input / output controller 2220 via a parallel port, a serial port, a keyboard port, a mouse port, and the like.

[0157] The programs are provided by a computer-readable medium such as a DVD-ROM 2201 or an IC card. The programs are read from the computer-readable medium, installed in the hard disk drive 2224, the RAM 2214, or the ROM 2230, which are also examples of computer-readable media, and executed by the CPU 2212. Information processing described in these programs is read by the computer 2200, and brings about cooperation between the programs and the various types of hardware resources described above. An apparatus or method may be constructed by realizing information manipulation or processing according to the use of the computer 2200.

[0158] For example, when communication is performed between the computer 2200 and an external device, the CPU 2212 may execute a communication program loaded in the RAM 2214 and instruct the communication interface 2222 to perform communication processing based on the processing described in the communication program. Under the control of the CPU 2212, the communication interface 2222 reads transmission data stored in a transmission buffer processing area provided in the RAM 2214, the hard disk drive 2224, the DVD-ROM 2201, or a recording medium such as an IC card, and transmits the read transmission data to the network, or writes reception data received from the network to a reception buffer processing area or the like provided on the recording medium.

[0159] Furthermore, the CPU 2212 may cause all or a necessary portion of a file or database stored in an external recording medium such as the hard disk drive 2224, the DVD-ROM drive 2226 (DVD-ROM 2201), an IC card, etc. to be read into the RAM 2214, and perform various types of processing on the data on the RAM 2214. The CPU 2212 then writes back the processed data to the external recording medium.

[0160] Various types of information, such as various types of programs, data, tables, and databases, may be stored in the recording medium and undergo information processing. The CPU 2212 may perform various types of processing on the data read from the RAM 2214, including various types of operations, information processing, conditional judgment, conditional branching, unconditional branching, information search / replacement, etc., as described throughout this disclosure and specified by the instruction sequence of the program, and write back the results to the RAM 2214. The CPU 2212 may also search for information in a file, database, etc. in the recording medium. For example, when a plurality of entries each having an attribute value of a first attribute associated with an attribute value of a second attribute are stored in the recording medium, the CPU 2212 may search for an entry that matches a condition, in which an attribute value of the first attribute is specified, from among the plurality of entries, read the attribute value of the second attribute stored in the entry, and thereby obtain the attribute value of the second attribute associated with the first attribute that satisfies a predetermined condition.

[0161] The above-described programs or software modules may be stored in a computer-readable medium on the computer 2200 or in the vicinity of the computer 2200. Also, a recording medium such as a hard disk or a RAM provided in a server system connected to a dedicated communication network or the Internet can be used as a computer-readable medium, thereby providing the programs to the computer 2200 via the network.

[0162] Although the present invention has been described above using the embodiments, the technical scope of the present invention is not limited to the scope described in the above embodiments. It is clear to those skilled in the art that various modifications and improvements can be made to the above embodiments. It is clear from the description of the claims that such modifications and improvements can also be included in the technical scope of the present invention.

[0163] It should be noted that the order of execution of each process, such as operations, procedures, steps, and stages, in the devices, systems, programs, and methods shown in the claims, specifications, and drawings is not specifically indicated as "before," "prior to," etc., and may be realized in any order unless the output of a previous process is used in a later process. Even if the operational flow in the claims, specifications, and drawings is described using "first," "next," etc. for convenience, it does not mean that it is essential to perform the process in this order. [Explanation of symbols]

[0164] 10 distributed power source, 20 deterministic power source, 30 tap, 100 optimizer, 102 power system, 110 selection unit, 120 calculation unit, 130 judgment unit, 132 first judgment unit, 134 second judgment unit, 140 generation unit, 150 smoothing unit, 200 system, 2200 computer, 2201 DVD-ROM, 2210 host controller, 2212···CPU, 2214···RAM, 2216···graphics controller, 2218···display device, 2220···input / output controller, 2222···communication interface, 2224···hard disk drive, 2226···DVD-ROM drive, 2230···ROM, 2240···input / output chip, 2242···keyboard

Claims

1. An optimization apparatus for optimizing determinable decision variables so as to minimize or maximize an objective function for evaluating a system while satisfying a constraint condition, comprising: the constraints include a system equation that is an equality constraint including an uncertain variable, and a plurality of inequality constraints including the uncertain variables; optimizing the decision variables such that the plurality of inequality constraints are satisfied for all possible values ​​of the uncertain variables; The optimization device comprises: a selection unit for selecting a finite number of samples as assumed values ​​of the uncertain variables; a calculation unit that calculates candidates of the decision variables that are optimal solutions by solving a relaxation problem in which the system equation and the plurality of inequality constraint conditions are relaxed corresponding to each sample; a determination unit that determines an inequality constraint condition that has the largest deviation amount among the original inequality constraint conditions when the candidate decision variable is applied; a generating unit that generates a new relaxed problem by adding the determined inequality constraint condition to the relaxed problem, the calculation unit solves the new relaxed problem to calculate candidates for decision variables that are optimal solutions; Optimizer.

2. the calculation of the decision variable candidates by the calculation unit, the determination by the determination unit, and the generation of a new relaxation problem by the generation unit are repeated until a deviation amount of each of the plurality of inequality constraint conditions becomes equal to or less than a predetermined threshold when the candidate decision variables obtained by solving the relaxation problem are applied to the original plurality of inequality constraint conditions; The optimization device according to claim 1 .

3. the optimization device optimizes determinable decision variables to minimize or maximize the curved smoothed objective function; the optimization device adjusts the approximation accuracy of the objective function according to a degree of smoothing. The optimization device according to claim 2 .

4. Adjusting the threshold value depending on the convergence speed in the optimization. The optimization device according to claim 3 .

5. The selection unit selects one sample as an assumed value of the uncertain variable; the calculation unit applies the one sample to the system equation and the plurality of inequality constraint conditions to calculate a candidate for the decision variable that is an optimal solution and a state variable determined by the system equation according to the assumed value; The determination unit includes a first determination unit and a second determination unit, the first determination unit applies the candidate for the decision variable, which is the optimal solution, to a plurality of original inequality constraint conditions to determine whether or not there is any inequality constraint condition that does not satisfy the plurality of inequality constraint conditions; the second determination unit, when any of the plurality of inequality constraint conditions is not satisfied, determines an inequality constraint condition with a largest deviation amount; the generating unit selects, as a new sample, an assumed value of the uncertain variable that gives a result with the largest deviation from among the original plurality of inequality constraint conditions; the calculation unit applies the new sample to the system equation and the plurality of inequality constraint conditions to calculate new decision variable candidates that are an optimal solution, and new state variables determined by the system equation in accordance with the new sample. Optimization device according to any one of claims 1 to 4.

6. the determination unit uses a variable independent of the state variable obtained by the calculation unit as the state variable determined by the system equation according to the assumed value. The optimization device according to claim 5 .

7. the determination unit includes, as a determination target, an inequality constraint condition for which a margin of deviation from the condition when the candidate decision variable is applied is equal to or less than a predetermined value in the previous determination.

7. An optimization device according to any one of claims 1 to 6.

8. The selection unit selects the sample in a region where the objective function varies depending on the uncertain variables.

8. An optimization device according to any one of claims 1 to 7.

9. the selection unit selects the sample exhibiting a maximum value as an assumed value of the uncertain variable; 8. An optimization device according to any one of claims 1 to 7.

10. The optimization device is an optimization device for power flow in a power system into which a distributed power source is introduced, The uncertain variables are output variables of the distributed power sources. Optimization device according to any one of claims 1 to 9.

11. A program for causing a computer to function as the optimization device according to any one of claims 1 to 10.

12. 1. A method for optimizing, by a computer, determinable decision variables so as to minimize or maximize an objective function for evaluating a system while satisfying constraints, comprising the steps of: the constraints include a system equation that is an equality constraint including an uncertain variable, and a plurality of inequality constraints including the uncertain variables; optimizing the decision variables such that the plurality of inequality constraints are satisfied for all possible values ​​of the uncertain variables; The optimization method comprises: a selection step for selecting a finite number of samples as assumptions for said uncertain variables; a calculation step of calculating candidates of the decision variables which are optimal solutions by solving a relaxation problem obtained by relaxing the system equation and the plurality of inequality constraint conditions corresponding to each sample; a determination step of determining an inequality constraint condition that has the largest deviation amount among the original inequality constraint conditions when the candidate decision variable is applied; and a generating step of generating a new relaxed problem by adding the determined inequality constraint to the relaxed problem. In the calculation step, the new relaxed problem is solved to calculate candidates for decision variables that are optimal solutions. Optimization methods.

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