Electric power system robust unit combination method for coping with wind power uncertainty

Abstract

The invention provides a power system robust unit combination method for coping with wind power uncertainty. The power system robust unit combination method specifically comprises the following steps: S1, establishing a robust unit combination model considering K uncertain sets; s2, establishing a robust unit combination model for controlling risks; s3, passing through the Camp; and constructing and solving sub-problems and main problems of the robust unit combination model considering the K uncertain sets and the robust unit combination model of the control risk, and obtaining the robust unit combination method of the power system. On the basis of a traditional two-stage robust unit commitment model, a plurality of uncertainty sets are considered, a robust unit commitment model considering K uncertainty sets and a robust unit commitment model for controlling risks are established, and a Camp for solving two new models is constructed based on a main and sub-problem framework; the CG algorithm enables the model to have robustness and economical efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of power system operation control, and in particular to a robust unit combination method for a power system to cope with wind power uncertainty. Background Art

[0002] The proposal of the "dual carbon" goal has accelerated the pace of my country's transformation to a new power system. In the future, new energy will become the main form of energy in my country. Wind power generation has become one of the fastest-growing power sources in my country due to its clean and low-carbon characteristics.

[0003] The unit combination problem of day-ahead dispatch in power system optimization dispatch is very important. The unit combination of power system refers to a dispatch method that reasonably formulates the unit start-stop plan and reasonably arranges the output of each unit within a certain dispatch cycle (usually one day) to minimize the total power generation cost of the system. The result of unit combination must meet the power needs of users and the various safety constraints of the power system.

[0004] In traditional power systems, uncertainty mainly comes from load fluctuations. The existing technology has a high accuracy in load forecasting, so the deviation between the load forecast value and the actual value is very small, and the load forecast error within one day generally does not exceed 5%. In the unit combination problem of day-ahead dispatching in traditional power systems, replacing the actual value of the load with the predicted value of the load will not cause a big impact. However, in power systems with a high proportion of wind power, in addition to the uncertainty of the load, the wind power output power has stronger randomness and volatility. Although many researchers have done a lot of research on wind power output prediction, the current short-term wind power output power prediction error is still around 20% due to the influence of climate, topography, and temperature. It can be seen that after large-scale grid connection of wind power, if the traditional dispatching method is still used, that is, the wind power output forecast value is used to replace the actual wind power output value. It will inevitably bring severe challenges to the safe operation of the power system. Therefore, it is of great significance to study the power system unit combination method considering wind power uncertainty.

[0005] Compared with random optimization, interval optimization and other methods, the robust optimization method has moderate computational complexity and the decisions made can be guaranteed to be feasible in any scenario. It has good robustness and is widely used in unit commitment problems. However, the traditional two-stage robust optimization method has the problem of being too conservative, and the decision results often lose economic efficiency or cannot ensure the safe operation of the power system.

[0006] In view of this, the present invention proposes a robust unit combination method for a power system to cope with wind power uncertainty. Summary of the invention

[0007] The purpose of the present invention is to propose a robust unit combination method for power systems that cope with wind power uncertainty. On the basis of the traditional two-stage robust unit combination model, multiple uncertainty sets are considered, and a robust unit combination model considering K uncertainty sets and a robust unit combination model for controlling risks are established. Based on the main-subproblem framework, a C&CG algorithm for solving the two new models is constructed, so that the model has both robustness and economy.

[0008] To achieve the above object, the technical solution of the present invention is: a method for robust unit combination of power system to cope with wind power uncertainty, specifically comprising the following steps:

[0009] S1. Establishing a robust unit commitment model considering K uncertainty sets, wherein the model includes a first stage and a second stage;

[0010] The goal of the first stage of the model is to minimize the unit startup and shutdown costs; the decision variables include unit operation state variables and unit startup / shutdown state variables; the constraints considered in the first stage of decision-making include unit minimum operation time constraints, unit minimum shutdown time constraints, and logical constraints between startup / shutdown state variables and operation state variables;

[0011] The goal of the second stage of the model is to minimize the sum of the fuel cost and the wind curtailment penalty cost of the system corresponding to the K uncertainty sets when the uncertain parameters of the K uncertainty sets are all taken to the worst; the decision variables include the output of each unit; the constraints considered in the second stage decision include the output constraints of thermal power units, system power balance constraints, thermal power unit ramp constraints, wind farm output constraints, transmission line transmission capacity constraints, system reserve capacity constraints and non-negative constraints of decision variables;

[0012] S2. Establishing a robust unit commitment model for controlling risk, wherein the model includes a first stage and a second stage;

[0013] The goal of the first stage of the model is to minimize the total cost under the most likely forecast scenario; the decision variables include unit operation state variables, unit start / shutdown state variables, and the corresponding thermal power unit output and wind farm optimal dispatch output under the forecast scenario; the constraints are the minimum operation / shutdown time constraints of the unit and the various safety constraints of the power system under the forecast scenario;

[0014] The goal of the second stage of the model is the performance indicators formulated by decision makers based on uncertain parameters in harsh scenarios; the decision variables are the optimal dispatch output of wind farms and thermal power units corresponding to each uncertainty set; the constraints are the various safety constraints of the power system;

[0015] S3. Through the C&CG algorithm, the sub-problems and main problems of the robust unit commitment model considering K uncertainty sets and the robust unit commitment model with controlled risks are constructed and solved respectively, and the robust unit commitment method of the power system is obtained.

[0016] Preferably, the objective function of the robust unit commitment model considering K uncertainty sets is specifically:

[0017]

[0018] In the formula, x, y, z, u, are the operating status of thermal power units, the output of thermal power units, the startup status of thermal power units, the shutdown status of thermal power units, and the output vector of wind power units; I is the total number of thermal power units; T is the total dispatch period; K is the number of uncertain sets; J is the number of wind farms; a i is the startup cost of unit i; b i is the downtime cost of unit i; c i is the fuel cost of unit i; j is the unit wind curtailment cost of wind farm j; k is the weight coefficient corresponding to the kth uncertainty set; z it is the startup status of unit i in period t; u it is the shutdown status of unit i in period t; y it is the output of unit i in period t; is the wind power output of wind farm j in period t; w jt is the optimal dispatch output of wind farm j in period t; the first term in formula (2) is the goal of the first stage, which is to minimize the start-up and shutdown costs of the units; the second term in formula (2) is the goal of the second stage, which is to minimize the sum of the fuel cost and the wind abandonment penalty cost of the system corresponding to the K uncertainty sets when the uncertainty parameters of the K uncertainty sets are all taken to the worst; where the form of the kth wind power uncertainty set is as follows:

[0019]

[0020] Where, J is the total number of wind farms; T is the total number of dispatch periods; is the upper limit of wind power output of the kth uncertain centralized wind farm j in period t, w jt,k is the lower limit of wind power output power of the kth uncertain concentrated wind farm j in period t; is the predicted wind power output power of wind farm j in period t; is an auxiliary variable introduced to control whether the wind power output of wind farm j reaches the boundary of the uncertain set. When is 1, When the value of reaches the upper limit of the interval, When is 1, The value of reaches the lower limit of the interval. When both are 0, The value of is the predicted value; Γ1 and Γ2 are conservative control parameters, which are used to control the actual wind power output to reach the boundary.

[0021] Preferably, the minimum operating time constraint of the unit, the minimum downtime constraint of the unit, and the logical constraints of the start / stop state variables and the operating state variables are as follows;

[0022] Minimum unit operating time constraints:

[0023]

[0024] In the formula, is the minimum operating time of unit i; x it is the operating status of unit i in period t;

[0025] Minimum downtime constraints for units:

[0026]

[0027] In the formula, is the minimum downtime of unit i;

[0028] Logical constraints on start / stop status variables and running status variables:

[0029]

[0030] In formula (3) to formula (6),

[0031] Preferably, the thermal power unit output constraint, system power balance constraint, thermal power unit ramp constraint, wind farm output constraint, transmission line transmission capacity constraint, system spare capacity constraint and decision variable non-negative constraint are specifically as follows;

[0032] Output constraints of thermal power units:

[0033]

[0034] Where P i + and P i _ are the upper and lower limits of the output of unit i respectively;

[0035] System power balance constraints:

[0036]

[0037] Where, J is the total number of wind farms; d t is the total load during period t, M is the total number of load nodes, d mtis the load of load node m in period t;

[0038] Thermal power unit climbing constraints:

[0039]

[0040] In the formula, is the maximum upward output of unit i; is the maximum downward output of unit i;

[0041] Wind farm output constraints:

[0042]

[0043] Transmission line transmission capacity constraints:

[0044]

[0045] Where P l-max is the maximum transmission capacity of line l; G l-i , G l-j and G l-m They are the generator output power transfer distribution factor of the node where thermal power unit i is located to line l, the generator output power transfer distribution factor of the node where wind farm j is located to line l, and the generator output power transfer distribution factor of the load node m to line l. The calculation method is:

[0046] G l-i =(X m’,i -X n’,i ) / x l , G l-j =(X m’,j -X n’,j ) / x l , G l-m =(X m’,m -X n’,m ) / x l , where x l is the reactance of line l, subscripts m' and n' are the first and end nodes of line l, X m',i , X n',i , X m',j , X n',j , X m',m , X n',m All are elements in the impedance matrix;

[0047] System spare capacity constraints:

[0048]

[0049] Where r is the spinning reserve ratio;

[0050] Decision variables are constrained to be non-negative:

[0051]

[0052] Preferably, in S3, the robust unit commitment model considering K uncertainty sets is solved respectively by C&CG algorithm, specifically:

[0053] Step 1: Construct the main problem and the sub-problems after KKT transformation, initialize the parameters of each model, set UB = Inf, LB = -Inf, and the initial number of iterations k0 = 1;

[0054] Step 2: Solve the main problem and obtain the solution x of the first stage decision variables * , and update the lower bound LB = max{LB, value(MPFunc + η)};

[0055] Step 3: Solve for x * Substitute into the subproblem SP k , solve each sub-problem;

[0056] Step 4: Determine whether all sub-problems have solutions. If so, return the constraints: To the main problem, and update the upper bound

[0057] Step 5: Return the constraints corresponding to the worst scenario found for each sub-problem to the main problem;

[0058] Step 6: Determine whether the upper bound is close to the lower bound. If so, end the process and obtain the optimal solution. If not, k0=k0+1, return to step 2, and enter the next iteration.

[0059] Preferably, the sub-problems of the robust unit commitment model considering K uncertainty sets are constructed and solved, specifically:

[0060] According to the given first-stage decision variables The definition of the kth subproblem is as follows:

[0061]

[0062]

[0063] The above sub-problems are double-layer problems with a max-min structure. The inner-layer min-structured linear programming problem is removed of the min sign through the KKT condition, and then the complementary relaxation constraints in the KKT condition are equivalent to inequality constraints through the big M method. The max-min problem is converted into a max-structured MILP problem for solution.

[0064] The main problem of the robust unit commitment model considering K uncertain sets is constructed and solved, specifically:

[0065] After solving the sub-problems, the scenarios identified by the sub-problems and their corresponding constraints are gradually added back to the main problem. The main problem is constructed as follows:

[0066]

[0067] st(3)-(6)(43)

[0068]

[0069]

[0070] Where l = 1, 2, ..., r represents r scenarios identified by sub-problems; Equations (3) to (6) are constraints related to the decision variables in the first stage; in Equation (44) is the second-stage objective function corresponding to the lth extreme point of the kth uncertainty set. The auxiliary variable η is introduced to transform the two-stage robust optimization model into a one-stage optimization problem. Equations (45) to (55) are the scenarios identified by the subproblems. The corresponding constraints are For the scene The corresponding decision variables;

[0071] Each iteration, the scenario is returned in the main problem The corresponding constraints and the scene must be generated at the same time The corresponding decision variable As the iterations proceed, the constraints returned to the main problem increase, and the number of bad scenarios considered by the main problem gradually increases, making the decision in the first stage better.

[0072] Preferably, the first-stage objective function of the risk-controlled robust unit commitment model is specifically set as follows:

[0073]

[0074] In the formula, λ j is the unit wind curtailment cost of wind farm j; is the wind power output power predicted by wind farm j in period t; 0,it For The output of unit i in period t under the scenario; w 0,jt For In the scenario, wind farm j is optimally dispatched to output in period t; Formula (17) minimizes the sum of start-up and shutdown costs, fuel costs, and wind abandonment penalty costs when the wind power output is the predicted value.

[0075] Preferably, the constraints related to the first-stage decision variables in the risk-controlled robust unit commitment model are specifically as follows:

[0076] st(3)-(6)(18)

[0077]

[0078] Among them, equations (3) to (6) are the minimum operation and downtime constraints of the unit and the logical constraints between the decision variables; equations (19) to (26) are the prediction scenarios The various safety constraints of the power system under test.

[0079] Preferably, the constraints related to the second-stage decision variables and the constraints for controlling risk in the risk-controlled robust unit commitment model are specifically as follows:

[0080] Constraints associated with the second-stage decision variables:

[0081]

[0082] In the formula, is the wind power output power of wind farm j corresponding to the kth uncertainty set in period t; k,it is the output of unit i corresponding to the kth uncertainty set in period t; w k,jt is the optimal dispatch output of wind farm j corresponding to the kth uncertainty set in period t; Equations (27) to (34) are the safety output constraints of each unit corresponding to the kth uncertainty set;

[0083] Constraints to control risk:

[0084]

[0085] Formula (35) is the hard constraint imposed by the decision maker on the performance indicators formulated under the severe scenario: in the kth severe scenario, the sum of the minimum operating cost of the thermal power unit and the penalty cost of wind power abandonment shall not exceed the given limit γ k .

[0086] Preferably, the robust unit commitment model for controlling risk is solved by the C&CG algorithm, as follows:

[0087] Step 1: Construct the main problem and sub-problem models and initialize various parameters;

[0088] Step 2: Solve the initial main problem and obtain the optimal solution x under the prediction scenario * ;

[0089] Step 3: Add x * Substitute SP k , solve K sub-problems;

[0090] Step 4: Return the constraints corresponding to the scenarios identified by each sub-problem to the main problem;

[0091] Step 5: Determine whether each sub-problem has a solution and whether the hard constraints corresponding to each sub-problem are satisfied; if all sub-problems have a solution and the hard constraints corresponding to each sub-problem are satisfied, the iteration ends and the final optimal solution is obtained; otherwise, the hard constraints corresponding to the scene identified by the sub-problem are returned to the main problem, and the number of iterations is increased by one, and then return to step 2 to enter the next iteration;

[0092] The construction and solution methods of the sub-problems of the robust unit commitment model for controlling risk are consistent with the construction and solution methods of the sub-problems of the robust unit commitment model considering K uncertain sets;

[0093] The main problem of the robust unit commitment model for controlling risk is constructed and solved, specifically:

[0094]

[0095] Among them, equations (3) to (6) are the constraints related to the decision variables in the first stage; equations (18) to (25) are the prediction scenarios The various security constraints of the power system under the subproblem; Equation (45)-Equation (55) are the constraints corresponding to the scenarios identified by the subproblems; Equation (57) is the hard constraint that the objective function of the subproblem needs to satisfy;

[0096] Substitute the decision of the main problem into the sub-problem, and judge whether the performance index of each sub-problem meets the requirements, which serves as the basis for whether the iteration is terminated.

[0097] Compared with the prior art, the present invention has the following beneficial effects:

[0098] Based on the traditional two-stage robust unit combination model, the present invention proposes an improved robust unit combination model that considers multiple uncertainty sets in view of the significant impact of the selection of uncertainty sets on the decision-making results. By introducing K uncertainty sets, a new robust unit combination model is established, and the influence of each uncertainty set is integrated through weight coefficients to balance the conservatism and risk of the model. This model framework is similar to the traditional two-stage robust unit combination model, but through the introduction and integration of multiple uncertainty sets, the flexibility and accuracy of decision-making are improved. At the same time, a robust unit combination model for controlling risks is constructed, and hard constraints are added on the basis of traditional prediction scenarios to ensure the control of performance indicators in the worst scenarios, further improving the safety and robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0099] Figure 1The flowchart of the C&CG algorithm of the robust unit commitment model considering K uncertainty sets of the present invention;

[0100] Figure 2 The figure is a flow chart of the C&CG algorithm of the robust unit commitment model for controlling risk of the present invention. DETAILED DESCRIPTION

[0101] The following is combined with Figure 1-2 , the technical solution of the present invention is specifically described.

[0102] (1) Based on the traditional two-stage robust unit combination model, the present invention first establishes a robust unit combination model that considers K uncertainty sets based on the idea of ​​selecting multiple uncertainty sets to balance conservatism and risk. The model is similar to the basic framework of the traditional two-stage robust unit combination model, except that the model of the present invention considers more uncertainty sets and integrates the influence of each uncertainty set through weight coefficients. Secondly, a robust unit combination model for controlling risk is constructed, which is similar to the basic framework of the traditional unit combination model. The target scenarios are all prediction scenarios, but the model of the present invention adds hard constraints to the model to limit the performance indicators under the worst scenario.

[0103] (2) The present invention constructs a column and constraint generation algorithm based on the main-subproblem framework to solve the two new models. The subproblems of the two new models are max-min structure problems and cannot be solved directly. The inner min problem is first converted into a set of equations through the KKT condition, and then the complementary relaxation constraint conditions are converted using the big M method, and finally converted into a max-structured MILP problem. The main problem of the two models is a MILP problem itself, and the MILP problem is easy to solve using the existing solver, so after the conversion, both the main problem and the subproblem can be solved.

[0104] 1. Robust unit commitment modeling considering K uncertain sets

[0105] Consider the randomness and volatility of wind power output, that is, the output of each wind farm fluctuates between the predicted value and the predicted boundary value. At the same time, consider the time correlation and spatial correlation of wind power output, that is, the number of wind farms reaching the predicted boundary value at different times does not exceed the given value and the number of wind farms reaching the predicted boundary value at the same time does not exceed the given value. Multiple uncertainty sets are considered in the model, and the form of the kth wind power uncertainty set is given as follows:

[0106]

[0107] Where, J is the total number of wind farms; T is the total number of dispatch periods; is the upper limit of wind power output of the kth uncertain centralized wind farm j in period t, w jt,kis the lower limit of wind power output power of the kth uncertain concentrated wind farm j in period t; is the predicted wind power output power of wind farm j in period t; is an auxiliary variable introduced to control whether the wind power output of wind farm j reaches the boundary of the uncertain set. When is 1, When the value of reaches the upper limit of the interval, When is 1, The value of reaches the lower limit of the interval. When both are 0, The value of is the predicted value; Γ1 and Γ2 are conservative control parameters, which are used to control the actual wind power output to reach the boundary.

[0108] For the first stage model, the goal is to minimize the unit startup and shutdown costs. The decision variables are 0-1 variables that need to be made before the uncertain parameters are determined. They are generally the unit operation state variables and the unit startup / shutdown state variables. The constraints related to the first stage decision are the unit minimum operation / shutdown time constraints.

[0109] For the second-stage model, since the model considers a high proportion of wind power access and wind power may not be fully absorbed, it is necessary to add a penalty cost for wind abandonment. Therefore, the goal of the second stage is to minimize the sum of the fuel cost and wind abandonment penalty cost of the system corresponding to the K uncertainty sets when the uncertain parameters of the K uncertainty sets are all taken to the worst. The decision variables in the second stage are continuous variables, which are made after the decision variables in the first stage and the uncertain parameters in the second stage are determined, and are generally the output of each unit. The constraints related to the second-stage decision are the various safety constraints of the power system. Since the improved model takes into account multiple uncertainty sets, the uncertain factors are described more fully, and the decision better balances risk and conservatism.

[0110] 1.1 Objective Function

[0111]

[0112] In the formula, x, y, z, u, are the operating status of thermal power units, the output of thermal power units, the startup status of thermal power units, the shutdown status of thermal power units, and the output vector of wind power units; I is the total number of thermal power units; T is the total dispatch period; K is the number of uncertain sets; J is the number of wind farms; a i is the startup cost of unit i; b i is the downtime cost of unit i; c i is the fuel cost of unit i; j is the unit wind curtailment cost of wind farm j; k is the weight coefficient corresponding to the kth uncertainty set; z it is the startup status of unit i in period t; uit is the shutdown status of unit i in period t; y it is the output of unit i in period t; is the wind power output of wind farm j in period t; w jt The optimal dispatch output of wind farm j in period t. Formula (2) is essentially a joint optimization of the objectives of the first stage and the second stage of the model. The first term of the formula is the objective of the first stage, and the second term is the objective of the second stage, so that the total objective function is minimized when the uncertain parameters take the worst case. That is, the sum of the start-stop cost, fuel cost and wind abandonment penalty cost is minimized.

[0113] 1.2 Constraints considered in the first stage of decision-making

[0114] (a) Minimum unit operating time constraints

[0115] Due to the influence of thermal inertia, after the thermal power unit is turned on, it must be kept running for a period of time before it can be shut down. This constraint can be written as follows:

[0116]

[0117] In the formula, is the minimum operating time of unit i; x it is the operating status of unit i in period t.

[0118] (b) Minimum downtime constraints for units

[0119] Also due to the influence of thermal inertia, the thermal power unit needs to be given sufficient time to cool down after being shut down before it can be started up again. This constraint can be written as follows:

[0120]

[0121] In the formula, is the minimum downtime of unit i.

[0122] (c) Logical constraints between start / stop state variables and running state variables

[0123] These three variables do not exist independently, but are mutually determined and satisfy certain logical relationships:

[0124]

[0125] In formula (3) to formula (6),

[0126] 1.3 Constraints considered in the second stage of decision-making

[0127] (a) Output constraints of thermal power units

[0128] The output range of thermal power units is strictly limited; the upper limit is usually the rated power of the thermal power unit; the lower limit is the minimum output value that must be guaranteed when the unit is outputting, and usually the lower limit of the output is required to keep the boiler from extinguishing.

[0129]

[0130] Where P i + and P i _ are the upper and lower limits of the output of unit i respectively.

[0131] (b) System power balance constraints

[0132] When the network losses of the power system are ignored, according to Kirchhoff's current law, the sum of the output of each thermal power unit and the output of the wind turbine unit in the system at each moment is equal to the load of the system.

[0133]

[0134] Where, J is the total number of wind farms; d t is the total load during period t, M is the total number of load nodes, d mt is the load of load node m in period t.

[0135] (c) Thermal power unit climbing constraints

[0136] During the dispatching process, the output of each thermal power unit needs to be adjusted according to the load of the system at different time periods. During a certain period of time during the adjustment process, the output adjustment range of each thermal power unit shall not exceed a certain value. In addition, the output of the unit during the startup period and the output of the unit before the shutdown period shall not exceed the minimum output of the unit.

[0137]

[0138] In the formula, is the maximum upward output of unit i; is the maximum downward output of unit i.

[0139] (d) Wind farm output constraints

[0140] At any time, the optimal dispatch output of each wind farm should be greater than or equal to 0 and not greater than the wind power output power of the wind farm.

[0141]

[0142] (e) Transmission line capacity constraints

[0143] Each line in the system has its maximum active power limit allowed to pass. The Generation Transfer Distribution Factor (GTDF) is usually used to describe the change in active power flow in the line caused by the change in the active output power of the generator. When the output of each thermal power unit changes, the power flow of the line will also change, and it is necessary to consider whether the active power transmitted in the line exceeds the limit at this time.

[0144]

[0145] In the formula, G l-i , G l-j and G l-m They are the generator output power transfer distribution factor of the node where thermal power unit i is located to line l, the generator output power transfer distribution factor of the node where wind farm j is located to line l, and the generator output power transfer distribution factor of the load node m to line l, with G l-i For example, the calculation method is: G l-i =(X m,i -X n,i ) / x l , where x l is the reactance of line l, m and n are the first and last nodes of line l, X m,i and X n,i is an element in the impedance matrix; P l-max is the maximum transmission capacity of line l.

[0146] (f) System reserve capacity constraints

[0147] The spare capacity in the system ensures the system's ability to continue to supply power when a fault occurs. It is usually the difference between the sum of the maximum output of all operating thermal power units and wind turbine outputs in the system and the actual load of the system.

[0148]

[0149] Where r is the spinning reserve ratio, which is determined by operating experience.

[0150] (g) Non-negative constraints on decision variables

[0151] The output of each thermal power unit should be greater than or equal to 0 at any time.

[0152]

[0153] 1.4. Further explanation of the model

[0154] First, we only consider the case of two uncertainty sets. Same as the basic lemma mentioned in this section, let the uncertainty set U1 be included in the uncertainty set U2, let the weight coefficient corresponding to U1 be ρ1, let the weight coefficient corresponding to U2 be ρ2, and let ρ1+ρ2=1, α((U1,U2),(ρ1,ρ2)) be the optimal value of the model. From the basic lemma, we can get:

[0155] α(U1)≤α((U1,U2),(ρ1,ρ2))≤α(U2) (15)

[0156] In the formula, the equal sign is taken when ρ1 or ρ2 is 0.

[0157] From this formula, we can get that the decision made by the robust unit commitment model considering two uncertainty sets is more reliable than the decision made when only U1 is considered. The reason is that U1 with a small range may not contain the actual worst scenario, but after considering two uncertainty sets, this deficiency is compensated by the influence of U2. At the same time, the decision made by the new model is less conservative than the decision made when only U2 is considered. When the range of U2 is too large, the uncertainty increases, but after considering two uncertainty sets, this impact can be reduced, that is, the conservatism of the model is reduced and the economy of the system is improved.

[0158] Of course, the above conclusions can be extended to the case of K uncertain sets.

[0159] Suppose there are K uncertain sets, among which the kth uncertain set U k The corresponding weight coefficient is ρ k , and there are is the optimal value of the model. Then:

[0160]

[0161] In the robust unit commitment model considering K uncertainty sets, the selection of weight coefficients is crucial. It reflects the risk preference of decision makers. If the probability that the wind power output power falls within a certain uncertainty set is greater, a larger weight coefficient can be set for the uncertainty set.

[0162] 2. Robust unit commitment model for risk control

[0163] The robust unit commitment model for controlling risks also considers the impact of multiple uncertainty sets. However, unlike the robust unit commitment model that considers K uncertainty sets, this model solves the optimal value in a certain scenario that the decision maker wants to consider, and adds restrictions to the performance indicators that need to be considered in the worst scenario and adds them to the model as constraints.

[0164] For the first stage of the model, the decision variables include unit operation state variables, unit start / shutdown state variables, and the corresponding thermal power unit output and wind farm optimal dispatch output under the forecast scenario. The objective function considers the lowest total cost under the most likely scenario (forecast scenario), which is also the overall objective function of the model; the constraints are the minimum operation / shutdown time constraints of the unit and the various safety constraints of the power system under the forecast scenario.

[0165] For the second stage of the model, the decision variables are the optimal dispatch output of the wind farm and the output of the thermal power unit corresponding to each uncertainty set. The objective function is the performance indicator that the decision maker wants to consider in the worst scenario, which can be the amount of wind abandonment, the penalty cost of wind abandonment, the amount of load reduction, etc., which are determined by the decision maker based on the uncertain parameters. The constraints are the various safety constraints of the power system.

[0166] The decision made by the model minimizes the total cost in the most likely scenario (prediction scenario), and is also feasible in the worst scenario, ensuring the robustness of the model. Since the model imposes hard constraints on the performance indicators in the worst scenario, as long as the solution of the model exists, the decision made can ensure that the expected performance indicators in the worst scenario do not exceed the set value.

[0167] 2.1 Objective Function

[0168]

[0169] In the formula, λ j is the unit wind curtailment cost of wind farm j; is the wind power output power predicted by wind farm j in period t; 0,it For The output of unit i in period t under the scenario; w 0,jt For In the scenario, wind farm j is optimally dispatched to output power in period t. Formula (17) is essentially to minimize the total objective function when the wind power output is the predicted value. That is, the sum of the start-stop cost, fuel cost and wind abandonment penalty cost is minimized.

[0170] 2.2 Constraints related to the first-stage decision variables

[0171] st(3)-(6)(18)

[0172]

[0173] Among them, equations (3) to (6) are the minimum operation and downtime constraints of the unit and the logical constraints between the decision variables; equations (19) to (26) are the prediction scenarios The above constraints ensure the feasibility of the given decision under the prediction scenario.

[0174] 2.3 Constraints related to the second-stage decision variables

[0175]

[0176] In the formula, is the wind power output power of wind farm j corresponding to the kth uncertainty set in period t; k,it is the output of unit i corresponding to the kth uncertainty set in period t; w k,jt is the optimal dispatch output of wind farm j corresponding to the kth uncertainty set in period t. Equations (27) to (34) are the safety output constraints of each unit corresponding to the kth uncertainty set, which ensures that the given decision is feasible in the worst scenario.

[0177] 2.4. Constraints for controlling risks

[0178]

[0179] Formula (35) is the hard constraint imposed by the decision maker on the performance indicators that he wants to consider in the worst scenario. The hard constraint considered by the present invention is that in the kth worst scenario, the sum of the operating cost of the minimum thermal power unit and the penalty cost of wind power abandonment shall not exceed the given limit γ k After adding this constraint, as long as the model has a solution, it ensures that the cost in the worst scenario will not be too high.

[0180] 2.5. Further explanation of the model

[0181] The robust unit commitment model for risk control follows the idea of ​​considering multiple uncertainty sets. The model does not contain weight coefficients, and each uncertainty set has a corresponding hard constraint coefficient γ k The hard constraint coefficient also reflects the decision maker's judgment on the selected uncertainty set. Generally, a larger hard constraint coefficient should be set for a larger uncertainty set. If a larger hard constraint coefficient is set for a smaller uncertainty set, the constraint actually loses its meaning. Because the model imposes stricter constraints on more severe scenarios, looser constraints can be omitted.

[0182] The core goal of the model has changed. It is not to find the optimal decision in the worst scenario, but to find the optimal decision in the most likely scenario, which is consistent with the traditional unit combination model. However, the model imposes hard constraints on a certain performance indicator in the worst scenario, so that the model has good robustness, and the decision it gives can meet the expected requirements of the decision maker. In addition, the model can also detect whether the system can fully cope with uncertainty. When the decision maker hopes that a certain performance indicator in the worst scenario will meet certain requirements, but the model has no solution, it means that the system can no longer fully cope with the challenge of uncertainty, which provides good support for decision makers to consider whether to expand or modify the system to cope with uncertainty.

[0183] In summary, the risk-controlled robust unit commitment model is applicable when decision makers focus on the benefits under the forecast scenario and can accept that a certain performance indicator under the worst scenario does not exceed the preset value. The model is also suitable for systems with a low probability of uncertain events, such as the occurrence of unexpected events such as unit failure and shutdown. If the decision maker is more concerned about the amount of load shedding when the unit fails and shuts down, the load shedding amount in the event of an emergency can be imposed with hard constraints and added to the model.

[0184] 3. C&CG algorithm solves the robust unit commitment model considering K uncertain sets

[0185] 3.1. Construction and solution of sub-problems

[0186] In the robust unit commitment model considering K uncertain sets, it is assumed that the first-stage decision variables in the subproblem are given. The goal of the subproblem is to arrange the output of thermal power units and the optimal dispatch output of each wind farm under a given unit combination mode, so that the sum of the unit operation cost and the wind abandonment penalty cost is minimized in the worst scenario. K uncertain sets are considered in the model, with a total of K subproblems. Given the first-stage decision variables The definition of the kth subproblem is as follows:

[0187]

[0188]

[0189] It can be seen that solving the kth subproblem corresponds to the second stage of solving the model. In the second stage, the decision variables of the first stage have been determined, and the x in the constraint condition it In the subproblem, the known value is The second stage is a max-min structure problem, and the result is the optimal value in the worst scenario under a given unit combination. However, this "worst scenario" is not necessarily the worst scenario for the entire problem, because a given unit combination is not necessarily the optimal solution to the entire problem. And a given unit combination may have sub-problems that have no solution.

[0190] Although this "worst scenario" is not necessarily the worst scenario for the entire problem, we can add its corresponding constraints back to the main problem. As the number of bad scenarios considered in the main problem increases and the constraints become more stringent, the given unit combination will be improved, and the next time the "worst scenario" is identified, it is more likely to be closer to the worst scenario for the entire problem.

[0191] Observe the above formula, this subproblem is a double-layer problem of max-min structure, and the inner layer is a linear programming problem of min structure. First, the min symbol is removed from the inner layer linear programming problem of min structure through KKT conditions. Secondly, the complementary relaxation constraints (nonlinear constraints) in the KKT conditions are equivalent to inequality constraints through the big M method. Finally, the max-min problem is transformed into a MILP problem of max structure.

[0192] 3.2. Construction and solution of the main problem

[0193] In the main problem, the influence of the uncertainty set is not considered first. After solving the sub-problems, the scenarios identified by the sub-problems and their corresponding constraints are gradually added back to the main problem. The specific main problem is constructed as follows:

[0194]

[0195] st(3)-(6)(43)

[0196]

[0197]

[0198] Where l = 1, 2, ..., r represents r scenarios identified by sub-problems; Equations (3) to (6) are constraints related to the decision variables in the first stage; in Equation (44) is the second-stage objective function corresponding to the lth extreme point of the kth uncertainty set. The auxiliary variable η is introduced to transform the two-stage robust optimization model into a one-stage optimization problem. Equations (45) to (55) are the scenarios identified by the subproblems. The corresponding constraints are For the scene The corresponding decision variable.

[0199] It can be seen that each time the main problem is iterated, the scene will be returned The corresponding constraints and the scene must be generated at the same time The corresponding decision variable Therefore, as the number of iterations increases, the number of constraints and decision variables in the main problem will gradually increase, that is, the rows and columns of the main problem increase with the iterations, so the algorithm is called the column and constraint generation algorithm.

[0200] As the iterations proceed, the constraints returned to the main problem increase, the number of bad scenarios considered by the main problem gradually increases, and the decision of the first stage will be better. At the same time, the main problem is obviously a MILP problem, which is very easy to solve.

[0201] 4. C&CG algorithm solves the robust unit commitment model for control risk

[0202] 4.1. Construction and solution of sub-problems

[0203] In the risk-controlled robust unit commitment model, since the hard constraint considered in the second stage is that in the kth worst scenario, the sum of the operating cost and wind curtailment penalty cost of the smallest thermal power unit shall not exceed the given limit γ k Therefore, the objective function and constraint conditions in the second stage are consistent with those in model 1. The construction and solution methods of the sub-problems in model 1 have been described in the previous article and will not be repeated here.

[0204] It is worth noting that the above results are obtained only when the objective functions of the second stage of the two models of the present invention are consistent. As mentioned above, in the robust unit commitment model for risk control, the decision maker determines the performance indicators that need to be controlled in the worst scenario based on the form of uncertainty sets. If this performance indicator is replaced, when constructing the sub-problem of model 2, it is only necessary to modify SP k The objective function and equations (36)-(37) are the same, and the other constraints do not need to be modified.

[0205] 4.2. Construction and solution of the main problem

[0206] The main problem of the robust unit commitment model for controlling risk is constructed as follows:

[0207]

[0208] Among them, equations (3) to (6) are the constraints related to the decision variables in the first stage; equations (18) to (25) are the prediction scenarios The various safety constraints of the power system under the sub-problem; Equations (45)-(55) are the constraints corresponding to the scenarios identified by the sub-problems; Equation (57) is the hard constraint that the objective function of the sub-problem must satisfy.

[0209] In summary, the objective function of the main problem is consistent with the objective function of the first stage, so there is no need to construct auxiliary variables. At the same time, the model focuses on whether a certain performance indicator meets the requirements in the worst scenario. Since the purpose of solving the subproblem is to determine whether the performance indicator meets the requirements, it does not focus on the specific value of the performance indicator. The entire model does not need to update the upper and lower bounds to achieve convergence. It only needs to substitute the decision of the main problem into the subproblem to determine whether the performance indicator of each subproblem meets the requirements as the basis for whether the iteration is terminated. In addition, it is worth noting that the main problem is also a MILP problem, which is easy to solve using solvers such as CPLEX.

[0210] The C&CG algorithm flow chart of the robust unit commitment model considering K uncertain sets is described in detail as follows: Figure 1 As shown:

[0211] Step 1: Construct the main problem mentioned above and the sub-problems transformed by KKT, initialize the parameters of each model, set UB = Inf, LB = -Inf, and the initial number of iterations k = 1;

[0212] Step 2: Solve the main problem and obtain the solution x of the first stage decision variables * , and update the lower bound LB = max{LB, value(MPFunc + η)};

[0213] Step 3: Solve for x * Substitute SP k , solve each sub-problem;

[0214] Step 4: Determine whether all sub-problems have solutions. If so, return the constraints: To the main problem, and update the upper bound

[0215] Step 5: Return the constraints corresponding to the "worst scenario" found in each sub-problem to the main problem;

[0216] Step 6: Determine whether the upper bound is close to the lower bound. If so, end the process and obtain the optimal solution. If not, k = k + 1, return to step 2, and enter the next iteration.

[0217] The detailed description of the C&CG algorithm flow chart of the robust unit commitment model for controlling risk is as follows: Figure 2 As shown:

[0218] Step 1: Construct the main problem and sub-problem models and initialize various parameters;

[0219] Step 2: Solve the initial main problem and obtain the optimal solution x under the prediction scenario * ;

[0220] Step 3: Add x* Substitute SP k , solve K sub-problems;

[0221] Step 4: Return the constraints corresponding to the scenarios identified by each sub-problem to the main problem;

[0222] Step 5: Determine whether each subproblem has a solution and whether the hard constraints corresponding to each subproblem are satisfied. If all subproblems have a solution and the hard constraints corresponding to each subproblem are satisfied, the iteration ends and the final optimal solution is obtained. Otherwise, the hard constraints corresponding to the scene identified by the subproblem are returned to the main problem, and the number of iterations is increased by one, and then return to step 2 to enter the next iteration.

[0223] The above are preferred embodiments of the present invention. Any changes made according to the technical solution of the present invention, as long as the resulting functions do not exceed the scope of the technical solution of the present invention, belong to the protection scope of the present invention.

Claims

1. A robust unit combination method for a power system to cope with wind power uncertainty, characterized in that: The specific steps include: S1. Establishing a robust unit commitment model considering K uncertainty sets, wherein the model includes a first stage and a second stage; The goal of the first stage of the model is to minimize the unit startup and shutdown costs; the decision variables include unit operation state variables and unit startup / shutdown state variables; the constraints considered in the first stage of decision-making include unit minimum operation time constraints, unit minimum shutdown time constraints, and logical constraints between startup / shutdown state variables and operation state variables; The goal of the second stage of the model is to minimize the sum of the fuel cost and the wind curtailment penalty cost of the system corresponding to the K uncertainty sets when the uncertain parameters of the K uncertainty sets are all taken to the worst; the decision variables include the output of each unit; the constraints considered in the second stage decision include the output constraints of thermal power units, system power balance constraints, thermal power unit ramp constraints, wind farm output constraints, transmission line transmission capacity constraints, system reserve capacity constraints and non-negative constraints of decision variables; S2. Establishing a robust unit commitment model for controlling risk, wherein the model includes a first stage and a second stage; The goal of the first stage of the model is to minimize the total cost under the most likely forecast scenario; the decision variables include unit operation state variables, unit start / shutdown state variables, and the corresponding thermal power unit output and wind farm optimal dispatch output under the forecast scenario; the constraints are the minimum operation / shutdown time constraints of the unit and the various safety constraints of the power system under the forecast scenario; The goal of the second stage of the model is the performance indicators formulated by decision makers based on uncertain parameters in harsh scenarios; the decision variables are the optimal dispatch output of wind farms and thermal power units corresponding to each uncertainty set; the constraints are the various safety constraints of the power system; S3. The robust unit commitment model considering K uncertainty sets and the robust unit commitment model controlling risk are solved respectively by the C&CG algorithm to obtain a robust unit commitment method for the power system.

2. A method for robust unit combination of power systems to cope with wind power uncertainty according to claim 1, characterized in that: The objective function of the robust unit commitment model considering K uncertainty sets is specifically: In the formula, x, y, z, u, are the operating status of thermal power units, the output of thermal power units, the startup status of thermal power units, the shutdown status of thermal power units, and the output vector of wind power units; I is the total number of thermal power units; T is the total dispatch period; K is the number of uncertain sets; J is the number of wind farms; a i is the startup cost of unit i; b i is the downtime cost of unit i; c i is the fuel cost of unit i; j is the unit wind curtailment cost of wind farm j; k is the weight coefficient corresponding to the kth uncertainty set; z it is the startup status of unit i in period t; u it is the shutdown status of unit i in period t; y it is the output of unit i in period t; is the wind power output of wind farm j in period t; w jt is the optimal dispatch output of wind farm j in period t; the first term in formula (2) is the goal of the first stage, which is to minimize the start-up and shutdown costs of the units; the second term in formula (2) is the goal of the second stage, which is to minimize the sum of the fuel cost and the wind abandonment penalty cost of the system corresponding to the K uncertainty sets when the uncertainty parameters of the K uncertainty sets are all taken to the worst; where the form of the kth wind power uncertainty set is as follows: Where, J is the total number of wind farms; T is the total number of dispatch periods; is the upper limit of wind power output of the kth uncertain centralized wind farm j in period t, w jt,k is the lower limit of wind power output power of the kth uncertain concentrated wind farm j in period t; is the predicted wind power output power of wind farm j in period t; is an auxiliary variable introduced to control whether the wind power output of wind farm j reaches the boundary of the uncertain set. When is 1, When the value of reaches the upper limit of the interval, When is 1, The value of reaches the lower limit of the interval. When both are 0, The value of is the predicted value; Γ1 and Γ2 are conservative control parameters, which are used to control the actual wind power output to reach the boundary.

3. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 2, characterized in that: The minimum operating time constraint of the unit, the minimum shutdown time constraint of the unit, and the logical constraints of the start / shutdown state variables and the operating state variables are as follows; Minimum unit operating time constraints: In the formula, is the minimum operating time of unit i; x it is the operating status of unit i in period t; Minimum downtime constraints for units: In the formula, is the minimum downtime of unit i; Logical constraints on start / stop status variables and running status variables: In formula (3) to formula (6), 4. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 3, characterized in that: The thermal power unit output constraint, system power balance constraint, thermal power unit ramp constraint, wind farm output constraint, transmission line transmission capacity constraint, system reserve capacity constraint and decision variable non-negative constraint are as follows; Output constraints of thermal power units: Where P i + and P i _ are the upper and lower limits of the output of unit i respectively; System power balance constraints: Where, J is the total number of wind farms; d t is the total load during period t, M is the total number of load nodes, d mt is the load of load node m in period t; Thermal power unit climbing constraints: In the formula, is the maximum upward output of unit i; is the maximum downward output of unit i; Wind farm output constraints: Transmission line transmission capacity constraints: Where P l-max is the maximum transmission capacity of line l; G l-i , G l-j and G l-m They are the generator output power transfer distribution factor of the node where thermal power unit i is located to line l, the generator output power transfer distribution factor of the node where wind farm j is located to line l, and the generator output power transfer distribution factor of the load node m to line l. The calculation method is: G l-i =(X m’,i -X n’,i ) / x l , G l-j =(X m’,j -X n’,j ) / x l , G l-m =(X m’,m -X n’,m ) / x l , where x l is the reactance of line l, subscripts m' and n' are the first and end nodes of line l, X m',i , X n',i , X m',j , X n',j , X m',m , X n',m All are elements in the impedance matrix; System spare capacity constraints: Where r is the spinning reserve ratio; Decision variables are constrained to be non-negative:

5. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 4, characterized in that: In S3, the robust unit commitment model considering K uncertainty sets is solved by the C&CG algorithm, specifically: Step 1: Construct the main problem and the sub-problems transformed by KKT, initialize the parameters of each model, set the update upper bound UB = Inf, the update lower bound LB = -Inf, and the initial number of iterations k0 = 1; Step 2: Solve the main problem and obtain the solution x of the first stage decision variables * , and update the lower bound LB = max{LB, value(MPFunc + η)}; Step 3: Solve for x * Substitute into the subproblem SP k , solve each sub-problem; Step 4: Determine whether all sub-problems have solutions. If so, return the constraints: To the main problem, and update the upper bound Step 5: Return the constraints corresponding to the worst scenario found for each sub-problem to the main problem; Step 6: Determine whether the upper bound is close to the lower bound. If so, end the process and obtain the optimal solution. If not, k0=k0+1, return to step 2, and enter the next iteration.

6. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 5, characterized in that: The sub-problems and main problems of the robust unit commitment model considering K uncertain sets are constructed and solved, specifically: According to the given first-stage decision variables The definition of the kth subproblem is as follows: The above sub-problems are double-layer problems with a max-min structure. The inner-layer min-structured linear programming problem is removed of the min sign through the KKT condition, and then the complementary relaxation constraints in the KKT condition are equivalent to inequality constraints through the big M method. The max-min problem is converted into a max-structured MILP problem for solution. The main problem of the robust unit commitment model considering K uncertain sets is constructed and solved, specifically: After solving the sub-problems, the scenarios identified by the sub-problems and their corresponding constraints are gradually added back to the main problem. The main problem is constructed as follows: st(3)-(6)(43) Where l = 1, 2, ..., r represents r scenarios identified by sub-problems; Equations (3) to (6) are constraints related to the decision variables in the first stage; in Equation (44) is the second-stage objective function corresponding to the lth extreme point of the kth uncertainty set. The auxiliary variable η is introduced to transform the two-stage robust optimization model into a one-stage optimization problem. Equations (45) to (55) are the scenarios identified by the subproblems. The corresponding constraints are For the scene The corresponding decision variables; Each iteration, the scenario is returned in the main problem The corresponding constraints and the scene must be generated at the same time The corresponding decision variable As the iterations proceed, the constraints returned to the main problem increase, and the number of bad scenarios considered by the main problem gradually increases, making the decision in the first stage better.

7. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 6, characterized in that: The first-stage objective function of the risk-controlled robust unit commitment model is specifically set as follows: In the formula, λ j is the unit wind curtailment cost of wind farm j; is the wind power output power predicted by wind farm j in period t; 0,it For The output of unit i in period t under the scenario; w 0,jt For In the scenario, wind farm j is optimally dispatched to output in period t; Formula (17) minimizes the sum of the start-stop cost, fuel cost and wind abandonment penalty cost when the wind power output is the predicted value.

8. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 7, characterized in that: The constraints related to the first-stage decision variables in the risk-controlled robust unit commitment model are as follows: st(3)-(6)(18) Among them, equations (3) to (6) are the minimum operation and downtime constraints of the unit and the logical constraints between the decision variables; equations (19) to (26) are the prediction scenarios The various safety constraints of the power system under test.

9. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 8, characterized in that: The constraints related to the second-stage decision variables and the constraints for controlling risk in the risk-controlled robust unit commitment model are specifically as follows: Constraints associated with the second-stage decision variables: In the formula, is the wind power output power of wind farm j corresponding to the kth uncertainty set in period t; k,it is the output of unit i corresponding to the kth uncertainty set in period t; w k,jt is the optimal dispatch output of wind farm j corresponding to the kth uncertainty set in period t; Equations (27) to (34) are the safety output constraints of each unit corresponding to the kth uncertainty set; Constraints to control risk: Formula (35) is the hard constraint imposed by the decision maker on the performance indicators formulated under the severe scenario: in the kth severe scenario, the sum of the minimum operating cost of the thermal power unit and the penalty cost of wind power abandonment shall not exceed the given limit γ k .

10. A method for robust unit combination of power system to cope with wind power uncertainty according to claim 9, characterized in that: The robust unit commitment model for control risk is solved by C&CG algorithm, as follows: Step 1: Construct the main problem and sub-problem models and initialize various parameters; Step 2: Solve the initial main problem and obtain the optimal solution x under the prediction scenario * ; Step 3: Add x * Substitute SP k , solve K sub-problems; Step 4: Return the constraints corresponding to the scenarios identified by each sub-problem to the main problem; Step 5: Determine whether each sub-problem has a solution and whether the hard constraints corresponding to each sub-problem are satisfied; if all sub-problems have a solution and the hard constraints corresponding to each sub-problem are satisfied, the iteration ends and the final optimal solution is obtained; otherwise, the hard constraints corresponding to the scene identified by the sub-problem are returned to the main problem, and the number of iterations is increased by one, and then return to step 2 to enter the next iteration; The construction and solution methods of the sub-problems of the robust unit commitment model for controlling risk are consistent with the construction and solution methods of the sub-problems of the robust unit commitment model considering K uncertain sets; The main problem of the robust unit commitment model for controlling risk is constructed and solved, specifically: Among them, equations (3) to (6) are the constraints related to the decision variables in the first stage; equations (18) to (25) are the prediction scenarios The various security constraints of the power system under the subproblem; Equation (45)-Equation (55) are the constraints corresponding to the scenarios identified by the subproblems; Equation (57) is the hard constraint that the objective function of the subproblem needs to satisfy; Substitute the decision of the main problem into the sub-problem, and judge whether the performance index of each sub-problem meets the requirements, which serves as the basis for whether the iteration is terminated.

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