CN112819278B - Segmented affine method for solving two-stage robust optimization unit combination model - Google Patents

Segmented affine method for solving two-stage robust optimization unit combination model Download PDF

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CN112819278B
CN112819278B CN202011638305.7A CN202011638305A CN112819278B CN 112819278 B CN112819278 B CN 112819278B CN 202011638305 A CN202011638305 A CN 202011638305A CN 112819278 B CN112819278 B CN 112819278B
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黎静华
徐逸夫
周爽
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Abstract

The invention provides a piecewise affine method for solving a two-stage robust optimization unit combination model, which belongs to the field of robust optimization models of power system unit combinations and specifically comprises the following steps: based on the output adjustment capability of the thermal power generating unit, a two-stage robust optimization unit combination model in a max-min-max form is established; the thermal power generating unit output adjustment capability is used for coping with load uncertainty and wind power uncertainty of a power system; affine the uncertain set to a simplex space based on a simplex segmentation affine method to obtain a two-stage robust optimization unit combination model in the simplex space; converting the two-stage robust optimization unit combination model in the simplex space into a robust optimization unit combination model in the affine space; and solving a robust optimization unit combination model of the affine space to obtain a thermal power unit combination mode. Compared with the traditional iterative algorithm, the piecewise affine method for solving the two-stage robust optimization unit combination model is quicker and simpler.

Description

Segmented affine method for solving two-stage robust optimization unit combination model
Technical Field
The invention belongs to the field of robust optimization models of power system unit combinations, and in particular relates to a piecewise affine method for solving a two-stage robust optimization unit combination model.
Background
Power system crew combinations play an important role in coping with renewable energy power uncertainty. The essence of the unit combination is to aim at the minimum running cost of the power system, and consider the mathematical problem of economic dispatch class of corresponding constraint in actual engineering. Different from the traditional unit combination problem, the random fluctuation of the renewable energy grid-connected output is caused, so that random variables appear in the novel unit combination problem, the unit combination problem is changed from deterministic planning to uncertain planning, and the difficulty of a dispatcher in making a unit start-stop plan to cope with the uncertainty of the renewable energy output is increased.
Solving a robust optimization model is usually a difficult process, and therefore, the original model needs to be transformed to some extent, and the transformed model needs to be conveniently solved in polynomial time. The traditional static robust optimization model belongs to a max-min game pattern, and the novel two-stage robust optimization model belongs to a max-min-max game pattern. The traditional static robust optimization model can be made after uncertainty is known, and belongs to the Here-and-now optimization model. The first stage of the novel two-stage robust optimization model is the Here-and-now optimization model, and the second stage is the Wait-and-see optimization model. Therefore, the novel part of the method of the two-stage robust optimization model can be made after uncertainty is observed, and the feedback mechanism enables the novel two-stage robust optimization model to be lower in conservation than a traditional static robust optimization model.
The application of the two-stage robust optimization in the power system is disclosed in an application example of the power system robust economic dispatch (two), a two-stage robust optimization model for coping with wind power uncertainty is firstly constructed, and an uncertainty set for representing the wind power uncertainty is secondly provided. Although the above model can effectively cope with uncertainty of wind power, the solution of the model is relatively complex, and the traditional Bender decomposition method can solve the model, but the Bender decomposition method cannot guarantee the iteration times of the solution process. The literature 'solution wo-stage robust optimization problems using a column-and-constraint generation method' proposes a C & CG column and constraint generation method to solve a two-stage robust optimization model, and the method can effectively improve the Solving efficiency and reduce the iteration number of the Solving process, but the Big-M method which relies on subjective parameters to carry out linearization makes the Solving result of the model too subjective. The M value of Big-M method in the test will seriously affect the final model solving result. Research literature A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization proposes a simplex-based piecewise affine method, considers the decision variables of the second stage of two-stage robust optimization as hidden functions of uncertain variables, and realizes the problem of converting the two-stage robust optimization into single-stage robust optimization. However, the different forms of the uncertainty set in actual engineering applications will affect the accuracy of the transformed model.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a piecewise affine method for solving a two-stage robust optimization unit combination model, which aims to solve the problem that the solving is complex due to the existence of an uncertain set in the existing robust optimization model.
In order to achieve the above purpose, the invention provides a piecewise affine method for solving a two-stage robust optimization unit combination model, which comprises the following steps:
based on the output adjustment capability of the thermal power generating unit, a two-stage robust optimization unit combination model in a max-min-max form is established; the thermal power unit output adjustment capability is used for coping with load uncertainty and wind power uncertainty of a power system, and an objective function of a first-stage unit model is that the sum of the start-stop cost and the fuel cost of the thermal power unit of the power system is minimum; the objective function of the second-stage unit model is that the sum of the positive rotation standby cost and the negative rotation standby cost is minimum under the worst condition of the power system based on load uncertainty and wind power uncertainty; the constraint condition comprises the constraint of the total output adjusting capacity of the thermal power generating unit;
affine an uncertain set in a two-stage robust optimization unit combination model in a max-min-max form to a simplex space based on a simplex piecewise affine method to obtain the two-stage robust optimization unit combination model in the simplex space;
based on the polygon uncertain set, defining a scale factor in a simple space as a minimum value of scheduling time length and unit time length, and defining a dominant vector as summation of unit vectors to obtain a robust optimization unit combination model of an affine space;
and solving the robust optimization unit combination model of the affine space to obtain a thermal power unit combination mode.
Preferably, the constraint further comprises: the method comprises the following steps of system power balance constraint, minimum start-up time and minimum shutdown time constraint of a thermal power unit, thermal power unit state transition constraint, thermal power unit output upper and lower limit constraint, thermal power unit climbing constraint, thermal power unit output adjusting range constraint and thermal power unit output adjusting capacity constraint.
Preferably, the objective function of the two-stage robust optimization crew combination model in the form of max-min-max is:
f＝f 1 +f 2
Figure BDA0002877409180000031
Figure BDA0002877409180000032
f is an objective function of the two-stage robust optimization unit combination model; f (f) 1 An objective function of the first stage unit model; t is the scheduling time; t is the scheduling time length; u (u) i,t =1 is that the thermal power unit i is turned off at t-1 and turned on at t; v i,t =1 is that the thermal power unit i is started at the time t-1 and is shut down at the time t; gamma ray i,t The state variable of the thermal power unit i at the time t is obtained;
Figure BDA0002877409180000033
the output power of the thermal power generating unit i at the time t is obtained;
Figure BDA0002877409180000034
The upward output force adjusting capability of the thermal power unit i at the time t is provided;
Figure BDA0002877409180000035
The downward output adjusting capability of the thermal power unit i at the time t is provided; deltap t For the power system at time tThermal power unit output reduction delta caused by load and wind power prediction errorp t ；
Figure BDA0002877409180000036
And deltap t Is an uncertain variable;
Figure BDA0002877409180000037
Is->
Figure BDA0002877409180000038
Is a set of uncertainty of (2);Uis deltap t Is a set of uncertainty of (2); c (C) start,i The starting cost of the unit i; c (C) shut,i The shutdown cost of the unit i; a, a i 、b i 、c i The fuel cost coefficient is the secondary, primary and constant of the thermal power unit i; c (C) up,i Positive standby cost coefficients for unit i; c (C) down,i Negative standby cost coefficient of unit i; n (N) G Is the number of thermal power generating units in the electric power system.
Preferably, the objective function in the robust optimization crew combination model of affine space is:
Figure BDA0002877409180000039
Figure BDA0002877409180000041
Figure BDA0002877409180000042
Figure BDA0002877409180000043
Figure BDA0002877409180000044
Figure BDA0002877409180000045
Figure BDA0002877409180000046
Figure BDA0002877409180000047
preferably, the total output power adjustment capability constraint of the thermal power generating unit in the two-stage robust optimization unit combination model in the form of max-min-max is as follows:
Figure BDA0002877409180000048
Figure BDA0002877409180000049
Figure BDA00028774091800000410
Figure BDA00028774091800000411
wherein the prediction error of the load at the time t is as follows
Figure BDA00028774091800000412
The prediction error of wind power of the power system is +.>
Figure BDA00028774091800000413
α t ∈[-1,0]，
Figure BDA00028774091800000414
β t ∈[-1,0]，
Figure BDA00028774091800000415
Figure BDA00028774091800000416
The method is characterized in that the method is the thermal power unit output increment caused by the load prediction error and the wind power prediction error of the power system at the time t; deltap t The method is thermal power unit output reduction caused by the prediction error of load and wind power at the time t.
Preferably, the total output adjustment capability constraint of the thermal power unit in the robust optimization unit combination model of the affine space is as follows:
Figure BDA00028774091800000417
Figure BDA00028774091800000418
wherein,,
Figure BDA00028774091800000419
a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit;e t a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit; e, e t Essentially the t-th dimensional component of βv, the physical meaning is the largest multiple of the payload fluctuation at time t.
A computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of a piecewise affine method of solving a two-stage robust optimization assembly model.
In general, the above technical solutions conceived by the present invention have the following beneficial effects compared with the prior art:
the uncertainty of the load of the power system and the wind power of the power system is intensively reflected in the net loads of the load and the wind power of the power system, and the objective function of the second-stage unit model in the built max-min-max-form two-stage robust optimization unit combination model is aimed at
Figure BDA0002877409180000051
Figure BDA0002877409180000052
In order to ensure that the original conservation of the uncertainty combination of the constructed polyhedron is not lost after affine transformation, the product form of the uncertainty variable and the uncertainty variable exists in the two-stage robust optimization unit combination model in the pure space, and the method adopts the following steps that the two-stage robust optimization unit combination model in the pure space>
Figure BDA0002877409180000053
And converting the two-stage robust optimization unit combination model in the pure space into the robust optimization unit combination model in the affine space, and solving by adopting a mature CPLEX solver. Compared with the traditional iterative algorithm, the piecewise affine method for solving the two-stage robust optimization unit combination model is quicker and simpler.
The two-stage robust optimization unit combination model based on the output adjustment capability of the thermal power unit and capable of coping with the load fluctuation of the power system and the random fluctuation of wind power can effectively cope with the random fluctuation of the load and the wind power of the power system in the climbing range.
Drawings
FIG. 1 is a flow chart of a piecewise affine method for solving a two-stage robust optimization unit combination model provided by the invention;
FIG. 2 is a schematic diagram of a combined output of a two-stage robust optimization unit obtained by adopting an affine method solution scheme I according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a two-stage robust optimization unit combined output obtained by adopting an affine method solution scheme II according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a combined output of a two-stage robust optimization unit obtained by adopting a C & CG solution scheme II according to an embodiment of the invention;
fig. 5 is a comparison diagram of combined output boundaries of a two-stage robust optimization unit using affine method and C & CG solution according to an embodiment of the present invention.
Detailed description of the preferred embodiments
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
The invention has the following overall invention points:
the invention provides a two-stage robust optimization unit combination model for coping with load fluctuation and wind power random fluctuation of a power system based on the output adjustment capability of a thermal power unit, and the model enables an online thermal power unit to effectively cope with the random fluctuation of the load and wind power of the power system in a climbing range.
Based on a space affine method, an uncertain variable set in the traditional two-stage robust optimization is converted into a space model in a simplex form, and space dominant points of the original uncertain variable set are deduced and given. Based on the characteristics of the polygon uncertainty set, simplex space dominant points of the power system load and wind power uncertainty are provided.
The invention provides a piecewise affine method of a two-stage robust optimization model based on pure form and space dominant points of a proposed bill. According to the method, a traditional min-max-min two-stage robust optimization unit combination model is converted into a single-stage unit combination model only containing min, so that a unit combination mode of a system can be obtained without iterative solution.
As shown in fig. 1, the invention provides a piecewise affine method for solving a two-stage robust optimization unit combination model, which comprises the following steps:
based on the output adjustment capability of the thermal power generating unit, a two-stage robust optimization unit combination model in a max-min-max form is established; the thermal power unit output adjustment capability is used for coping with load uncertainty and wind power uncertainty of a power system, and an objective function of a first-stage unit model is that the sum of the start-stop cost and the fuel cost of the thermal power unit of the power system is minimum; the objective function of the second-stage unit model is that the sum of the positive rotation standby cost and the negative rotation standby cost is minimum under the worst condition of the power system based on load uncertainty and wind power uncertainty; the constraint condition comprises the constraint of the total output adjusting capacity of the thermal power generating unit;
affine an uncertain set in a two-stage robust optimization unit combination model in a max-min-max form to a simplex space based on a simplex piecewise affine method to obtain the two-stage robust optimization unit combination model in the simplex space;
based on the polygon uncertain set, defining a scale factor in a simple space as a minimum value of scheduling time length and unit time length, and defining a dominant vector as summation of unit vectors to obtain a robust optimization unit combination model of an affine space;
and solving the robust optimization unit combination model of the affine space to obtain a thermal power unit combination mode.
Preferably, the constraint further comprises: the method comprises the following steps of system power balance constraint, minimum start-up time and minimum shutdown time constraint of a thermal power unit, thermal power unit state transition constraint, thermal power unit output upper and lower limit constraint, thermal power unit climbing constraint, thermal power unit output adjusting range constraint and thermal power unit output adjusting capacity constraint.
Preferably, the objective function of the two-stage robust optimization crew combination model in the form of max-min-max is:
f＝f 1 +f 2
Figure BDA0002877409180000071
Figure BDA0002877409180000072
f is an objective function of the two-stage robust optimization unit combination model; f (f) 1 An objective function of the first stage unit model; t is the scheduling time; t is the scheduling time length; u (u) i,t =1 is that the thermal power unit i is turned off at t-1 and turned on at t; v i,t =1 is that the thermal power unit i is started at the time t-1 and is shut down at the time t; gamma ray i,t The state variable of the thermal power unit i at the time t is obtained;
Figure BDA0002877409180000081
the output power of the thermal power generating unit i at the time t is obtained;
Figure BDA0002877409180000082
The upward output force adjusting capability of the thermal power unit i at the time t is provided;
Figure BDA0002877409180000083
The downward output adjusting capability of the thermal power unit i at the time t is provided; deltap t The method is characterized in that the thermal power unit output reduction delta caused by load and wind power prediction error at time t of the power systemp t ；
Figure BDA0002877409180000084
And deltap t Is an uncertain variable;
Figure BDA0002877409180000085
Is->
Figure BDA0002877409180000086
Is a set of uncertainty of (2);Uis deltap t Is a set of uncertainty of (2); c (C) start,i The starting cost of the unit i; c (C) shut,i The shutdown cost of the unit i; a, a i 、b i 、c i The fuel cost coefficient is the secondary, primary and constant of the thermal power unit i; c (C) up,i Positive standby cost coefficients for unit i; c (C) down,i Negative standby cost coefficient of unit i.
Preferably, the objective function in the robust optimization crew combination model of affine space is:
Figure BDA0002877409180000087
Figure BDA0002877409180000088
Figure BDA0002877409180000089
Figure BDA00028774091800000810
Figure BDA00028774091800000811
Figure BDA00028774091800000812
Figure BDA00028774091800000813
Figure BDA00028774091800000814
preferably, the total output power adjustment capability constraint of the thermal power generating unit in the two-stage robust optimization unit combination model in the form of max-min-max is as follows:
Figure BDA00028774091800000815
Figure BDA00028774091800000816
Figure BDA00028774091800000817
Figure BDA00028774091800000818
wherein the prediction error of the load at the time t is as follows
Figure BDA00028774091800000819
The prediction error of wind power of the power system is +.>
Figure BDA00028774091800000820
α t ∈[-1,0]，
Figure BDA00028774091800000821
β t ∈[-1,0]，
Figure BDA00028774091800000822
Figure BDA00028774091800000823
The method is characterized in that the method is the thermal power unit output increment caused by the load prediction error and the wind power prediction error of the power system at the time t; deltap t The method is thermal power unit output reduction caused by the prediction error of load and wind power at the time t.
Preferably, the total output adjustment capability constraint of the thermal power unit in the robust optimization unit combination model of the affine space is as follows:
Figure BDA0002877409180000091
Figure BDA0002877409180000092
wherein,,
Figure BDA0002877409180000093
a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit;e t a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit; e, e t Essentially the t-th dimensional component of βv, the physical meaning is the largest multiple of the payload fluctuation at time t.
Specifically, the invention provides a piecewise affine method for solving a two-stage robust unit combination, which comprises the following steps:
step 1: building a two-stage robust optimization unit combination model for coping with load uncertainty and wind power uncertainty of a power system based on the output adjustment capability of a thermal power unit, wherein the two-stage robust optimization unit combination model comprises the following concrete steps:
objective function: first-stage objective function f of two-stage robust optimization unit combination model 1 The second stage objective function f is the minimum sum of the start-stop cost and the fuel cost of the thermal power unit of the power system 2 The sum of the positive rotation standby cost and the negative rotation standby cost is minimum under the worst condition of the load uncertainty and the wind power uncertainty of the power system;
f＝f 1 +f 2
Figure BDA0002877409180000094
Figure BDA0002877409180000095
f is an objective function of the two-stage robust optimization unit combination model; f (f) 1 Is a first stage objective function; t is the scheduling time; t is the scheduling time length; u (u) i,t =1 is that the thermal power unit i is turned off at t-1 and turned on at t; v i,t =1 is that the thermal power unit i is started at the time t-1 and is shut down at the time t; gamma ray i,t Is the state variable gamma of the thermal power unit i at the moment t i,t =1 power on, otherwise power off;
Figure BDA0002877409180000101
the output power of the thermal power generating unit i at the time t is obtained;
Figure BDA0002877409180000102
The upward output force adjusting capability of the thermal power unit i at the time t is provided;
Figure BDA0002877409180000103
The downward output adjusting capability of the thermal power unit i at the time t is provided; deltap t The method is characterized in that the thermal power unit output reduction delta caused by load and wind power prediction error at time t of the power systemp t ；
Figure BDA0002877409180000104
And deltap t Is an uncertain variable;
Figure BDA0002877409180000105
Is->
Figure BDA0002877409180000106
Is a set of uncertainty of (2);Uis deltap t Is a set of uncertainty of (2); c (C) start,i The starting cost of the unit i; c (C) shut,i The shutdown cost of the unit i; a, a i 、b i 、c i The fuel cost coefficient is the secondary, primary and constant of the thermal power unit i; c (C) up,i Positive standby cost coefficients for unit i; c (C) down,i Negative standby cost coefficient of unit i;
the constraint conditions are as follows:
(1) System power balance constraint:
the physical meaning of the constraint is: the sum of the output of the online thermal power generating unit and the predicted wind power output in the power system at the moment t is equal to the predicted load of the power system;
Figure BDA0002877409180000107
wherein,,
Figure BDA0002877409180000108
predicting wind power output of the power system at the time t;
Figure BDA0002877409180000109
The predicted load of the power system at the time t is obtained;
(2) Minimum start-up time and minimum shut-down time constraint of thermal power generating unit:
the physical meaning of the constraint is: the continuous on-line time and the continuous off-line time of the thermal power plant i should be at least equal to the minimum on-time and the minimum off-time of the thermal power plant i, which can be expressed as follows:
-γ i,t-1 +γ i,t -γ i,τ ≤0,τ∈{t,…,min(T on +t-1,T)},t∈{2,…,T}
γ i,t-1 -γ i,t +γ i,k ≤1,k∈{t,…,min(T off +t-1,T)},t∈{2,…,T}
wherein, tau is the period of time that the thermal power unit i must be started; k is the period of time that the thermal power unit i must be shut down;
(3) Thermal power generating unit state transition constraint:
the physical meaning of the constraint is: operating state gamma of thermal power unit i at time t i,t Must meet u i,t And v i,t Logic limitations of (2);
Figure BDA0002877409180000111
wherein u is i,t And v i,t Co-constraining gamma i,t Is a variation of (2);
(4) Thermal power generating unit output upper and lower limit constraint:
the physical meaning of the upper and lower limit constraint of the output of the thermal power unit is as follows: output of thermal power unit i at t moment
Figure BDA0002877409180000112
Should not be less than the lower limit of the output of the thermal power generating unit i multiplied by gamma i,t And is not greater than the upper limit of the output of the thermal power unit i multiplied by gamma i,t The method can be concretely represented as follows:
Figure BDA0002877409180000113
wherein,,
Figure BDA0002877409180000114
the lower limit of the output of the thermal power unit i;
Figure BDA0002877409180000115
The upper limit of the output of the thermal power unit i;
(5) Climbing constraint of thermal power generating unit:
the physical meaning of the constraint is: the output variation of the thermal power unit i at the time t and the time t-1 should meet the climbing capacity limit of the thermal power unit i;
Figure BDA0002877409180000116
Figure BDA0002877409180000117
wherein,,
Figure BDA0002877409180000118
the upward climbing capacity of the thermal power unit i;
Figure BDA0002877409180000119
The downward climbing capacity of the thermal power unit i;
(6) Thermal power generating unit output adjusting range constraint:
the physical meaning of the constraint is: the maximum power and the minimum power of the thermal power unit are between the products of the thermal power unit state and the upper and lower limits of the thermal power unit output; the method can be concretely expressed as follows:
Figure BDA00028774091800001110
Figure BDA00028774091800001111
wherein,,
Figure BDA00028774091800001112
the lower limit of the output of the thermal power unit i;
Figure BDA00028774091800001113
The upper limit of the output of the thermal power unit i; in addition, the thermal power unit i needs to meet the requirement that the thermal power unit i can climb from the lower output limit at the time t-1 to the upper output limit at the time t, and can climb from the upper output limit at the time t-1 to the lower output limit at the time t;
Figure BDA00028774091800001114
Figure BDA0002877409180000121
(7) Thermal power generating unit output adjusting capability constraint:
the physical meaning of the constraint is: the upward adjustment capability and the downward adjustment capability of the thermal power generating unit i at the time t are not greater than the climbing capability of the unit;
Figure BDA0002877409180000122
Figure BDA0002877409180000123
(8) Total output capacity adjustment capacity constraint (uncertain constraint) of thermal power unit of electric power system:
let the prediction error of the load at time t be
Figure BDA0002877409180000124
The prediction error of wind power of the power system is +.>
Figure BDA0002877409180000125
Wherein,,α t ∈[-1,0]，
Figure BDA0002877409180000126
β t ∈[-1,0]，
Figure BDA0002877409180000127
based on the above, the thermal power unit output increase amount of the power system at the time t caused by the load prediction error and the wind power prediction error is +.>
Figure BDA0002877409180000128
Can be expressed as:
Figure BDA0002877409180000129
the above expression
Figure BDA00028774091800001210
In the worst case of (2) is the increase of the load of the power system +.>
Figure BDA00028774091800001211
And wind power output of the power system is reduced +.>
Figure BDA00028774091800001212
Thermal power unit output reduction delta caused by prediction error of load and wind power at time t of power systemp t Can be expressed as:
Figure BDA00028774091800001213
the above expression deltap t In the worst case of (1) an increase in wind power output of a power system
Figure BDA00028774091800001214
While the load of the power system is reduced +.>
Figure BDA00028774091800001215
Further, the above formulas (1) and (2) can be rewritten as:
Figure BDA00028774091800001216
Figure BDA00028774091800001217
due to the need whatsoever
Figure BDA00028774091800001218
And (3) withh t How to change, the positive standby and the negative standby of the thermal power generating unit in the electric power system can be used for solving the problem of +.>
Figure BDA00028774091800001219
And deltap t Therefore, the overall capacity adjustment capability constraint of the thermal power generating unit of the power system is expressed as:
Figure BDA0002877409180000131
Figure BDA0002877409180000132
uncertainty collection
Figure BDA0002877409180000133
Rewritable->
Figure BDA0002877409180000134
Figure BDA0002877409180000135
Uncertainty collectionURewritable->
Figure BDA0002877409180000136
Step 2: two-stage robust optimization segmentation affine method based on simplex;
the standard two-stage robust optimization model may be expressed as pi AR (U)；
Figure BDA0002877409180000137
Figure BDA0002877409180000138
Figure BDA0002877409180000139
Figure BDA00028774091800001310
Wherein,,
Figure BDA00028774091800001311
x is the first stage decisionA variable; y (h) is a second stage decision variable; h is an uncertainty variable;
for a given uncertainty set
Figure BDA00028774091800001312
Definition simplex->
Figure BDA00028774091800001313
Figure BDA00028774091800001314
Wherein β is a scale factor such that +.>
Figure BDA00028774091800001315
e j A unit vector of m dimension which is the j-th dimension being 1; v is the dominant vector of m dimension, v ε U; if->
Figure BDA00028774091800001316
For->
Figure BDA00028774091800001317
All have->
Figure BDA00028774091800001318
Then->
Figure BDA00028774091800001319
Dominating U; wherein (1)>
Figure BDA00028774091800001320
(θ) + =max {0, θ }, and there are:
Figure BDA00028774091800001321
a simplex-based piecewise affine method is given:
Figure BDA00028774091800001322
Figure BDA00028774091800001323
wherein,,
Figure BDA00028774091800001324
for optimizing model->
Figure BDA00028774091800001325
Is the optimal solution of (a):
Figure BDA0002877409180000141
Figure BDA0002877409180000142
Figure BDA0002877409180000143
Ax+By m+1 ≥βv (6)
Figure BDA0002877409180000144
to this end, a two-stage robust optimization model pi in the form of original max-min-max AR (U) adopting the approximate substitution solutions of (3) - (7);
for the shape like
Figure BDA0002877409180000145
Is given by +.>
Figure BDA0002877409180000146
Figure BDA0002877409180000147
Wherein->
Figure BDA0002877409180000148
When k=m, based on +.>
Figure BDA0002877409180000149
Figure BDA00028774091800001410
The affine method of (2) is as follows:
Figure BDA00028774091800001411
Figure BDA00028774091800001412
the dominant simplex at this time
Figure BDA00028774091800001413
Step 3: an affine method for constructing a two-stage robust optimization unit combination model for coping with load uncertainty and wind power uncertainty of a power system based on the output adjustment capability of the thermal power unit is constructed;
based on the two-stage robust optimization segmented affine method, a segmented affine method for solving a two-stage robust optimization unit combined model considering the output adjustment capability of the thermal power unit is provided, wherein the method comprises the following steps:
first stage affine variables:
Figure BDA00028774091800001414
Figure BDA00028774091800001415
Figure BDA00028774091800001416
Figure BDA00028774091800001417
second stage affine variables:
Figure BDA00028774091800001418
Figure BDA00028774091800001419
uncertain set conversion:
Figure BDA0002877409180000151
the conversion is as follows:
Figure BDA0002877409180000152
Figure BDA0002877409180000153
the conversion is as follows:
Figure BDA0002877409180000154
based on the formulas (8) to (15), the following model is solved to obtain a thermal power unit combination scheme;
Figure BDA0002877409180000155
Figure BDA0002877409180000156
Figure BDA0002877409180000157
-γ i,t-1 +γ i,t -γ i,τ ≤0,τ∈{t,…,min(T on +t-1,T)},t∈{2,…,T} (19)
γ i,t-1 -γ i,t +γ i,k ≤1,k∈{t,…,min(T off +t-1,T)},t∈{2,…,T} (20)
Figure BDA0002877409180000158
Figure BDA0002877409180000159
Figure BDA00028774091800001510
Figure BDA00028774091800001511
Figure BDA00028774091800001512
Figure BDA00028774091800001513
Figure BDA00028774091800001514
Figure BDA00028774091800001515
Figure BDA00028774091800001516
Figure BDA00028774091800001517
Figure BDA00028774091800001518
Figure BDA00028774091800001519
wherein,,
Figure BDA00028774091800001520
a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit;e t a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit; e, e t Essentially the t-th dimensional component of βv, the physical meaning is the largest multiple of the payload fluctuation at time t.
A two-stage robust optimization unit combination model for coping with load uncertainty and wind power uncertainty of a power system based on output adjustment capability of a thermal power unit is composed of the formulas (16) - (32), and the model can be solved by adopting a mature CPLEX solver.
A computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of a piecewise affine method of solving a two-stage robust optimization assembly model.
In summary, in the two-stage robust optimization unit combination model based on the output adjustment capability of the thermal power unit, the uncertainty of the load of the power system and the wind power of the power system is concentrated in the two net loads. In the established method, a polyhedral uncertain set is established, and the uncertain variable set of the traditional two-stage robust optimization is affined to a simplex space based on a simplex method, so that the established polyhedral uncertain set is ensured not to lose the original conservation after affine transformation; finally, the method converts the affine dominant simplex into the two-stage robust optimization unit combination model into the affine space robust optimization unit combination model, and compared with the traditional iterative algorithm, the method can quickly acquire the thermal power unit combination mode.
Examples
Based on the piecewise affine method for solving the two-stage robust optimization unit combination model provided by the invention, the embodiment provides two simulation schemes:
(1) 10 unit scheduling schemes: according to the scheme, the IEEE-39 node system unit data is adopted, the prediction error of the load of the electric power system is set to be 10%, the prediction error of wind power of the electric power system is set to be 10%, the segmented affine method is compared with the C & CG column and constraint generation method, and the solving speed of the segmented affine method is higher than that of the C & CG column and constraint generation method.
(2) 33 unit scheduling schemes: according to the scheme, the IEEE-39 node system unit data is adopted, the prediction error of the load of the electric power system is set to be 10%, the prediction error of wind power of the electric power system is set to be 20%, the segmented affine method is compared with the C & CG column and constraint generation method, and the solving speed of the segmented affine method is higher than that of the C & CG column and constraint generation method.
The present invention will be described in further detail with reference to examples.
In order to verify the correctness of the invention, the example is simulated based on MATLAB combined with CPLEX solver. Setting random energy to wind power in simulation, wherein the prediction error of wind power in a 10-unit scheduling scheme is 10%, namely the actual wind power fluctuates in a range of 0.9 to 1.1 times of the predicted wind power; the prediction error of the load in the 10-unit scheduling scheme is 10%, namely the actual load fluctuates in the interval of 0.9 times to 1.1 times of the predicted wind power; the prediction error of wind power in a 33-unit scheduling scheme is 20%, namely the actual wind power fluctuates in an interval of 0.8-1.2 times of the predicted wind power; the prediction error of the load in the 10-unit scheduling scheme is 10%, namely the actual load fluctuates in the interval of 0.9 times to 1.1 times of the predicted wind power. The specific simulation method is shown in table 1. Further, table 1 gives the corresponding solving method of the set scheme.
TABLE 1
Figure BDA0002877409180000171
Fig. 2 shows a schematic diagram of thermal power generating unit output based on the proposed affine method in scheme one. As can be seen from the figure, 10 thermal power generating units in the first scheme all have output, wherein the units G1, G4 and G8 bear most of the load of the power system. In addition, thermal power generating units G3, G6, G7, G9 and G10 have relatively low output. The main reason for the above phenomenon is caused by the function difference of the fuel cost of the thermal power unit. A better economical unit will exert relatively more force, while a less economical unit will exert relatively less force. In order to form a pair with the existing method, fig. 3 shows the output result of the thermal power generating unit in the second scheme, and the results of the two methods of the affine method and the C & CG are shared. It should be noted that the number of the system units of the example system of the scheme II is more, the variables in the scheme II are more, the constraint is more, the dimension is higher, and the comparison result of the two methods is more convincing and representative.
Fig. 3 compares affine mode with fig. 4, and two methods of C & CG solve the result of the two-stage robust optimization unit combination of 33 units. As can be seen from fig. 3 and 4, the thermal power generating units of the two methods have different output magnitudes. And because of the limit of climbing capacity of the thermal power generating unit, the actual upper limit and the actual lower limit of the output of the thermal power generating unit are different due to different output of the thermal power generating unit. In addition, the affine method provided by the embodiment does not need to set subjective parameters, the C & CG algorithm needs to carry out sub-problem complementary dual constraint linearization based on the Big-M method, and the value of M can seriously influence the final optimization result. The analysis shows that the output boundary of the thermal power unit is different due to the different output results of the thermal power unit in the two methods, so that the coping system load of the thermal power unit is different from the wind power uncertainty capability.
Fig. 5 provides the comparison result of the output boundary of the thermal power generating unit of the two methods in the second scheme. It can be seen from the figure that the upper output bound of the thermal power unit of the piecewise affine method is larger than that of the thermal power unit of C & CG, namely the thermal power unit of the piecewise affine method has more positive standby capacity. It can be seen from the figure that the lower thermal power unit output bound of the piecewise affine method is very close to the lower thermal power unit output bound of C & CG, however, at 1:00, the piecewise affine method has lower thermal power unit output bound, that is, the thermal power unit of the piecewise affine method has more negative rotation preparation for coping with uncertainty of system load and system wind power.
Table 2 shows the comparison result of the model solving time of the piecewise affine method and the C & CG in the second solution and the system running cost. The scheme II is selected because the thermal power generating unit in the scheme II is more, and the solving effect of the two-stage robust optimization unit combination problem of the large-scale system can be reflected. As can be seen from table 2, the proposed piecewise affine method has a faster solution efficiency with a model solution time of about one fourth of C & CG. Furthermore, the piecewise affine method system running cost is higher than that of C & CG, consistent with the situation reflected in fig. 3, 4, 5. As having more spare capacity would mean that the running cost of the system is higher.
TABLE 2
Figure BDA0002877409180000181
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.

Claims (7)

1. The piecewise affine method for solving the two-stage robust optimization unit combination model is characterized by comprising the following steps of:
based on the output adjustment capability of the thermal power generating unit, a two-stage robust optimization unit combination model in a max-min-max form is established; the thermal power unit output adjustment capability is used for coping with load uncertainty and wind power uncertainty of a power system, and an objective function of a first-stage unit model is that the sum of the start-stop cost and the fuel cost of the thermal power unit of the power system is minimum; the objective function of the second-stage unit model is that the sum of the positive rotation standby cost and the negative rotation standby cost is minimum under the worst condition of the power system based on load uncertainty and wind power uncertainty; the constraint condition comprises the constraint of the total output adjusting capacity of the thermal power generating unit;
affine an uncertain set in a two-stage robust optimization unit combination model in a max-min-max form to a simplex space based on a simplex piecewise affine method to obtain the two-stage robust optimization unit combination model in the simplex space;
based on the polygon uncertain set, defining a scale factor in a simple space as a minimum value of scheduling time length and unit time length, and defining a dominant vector as summation of unit vectors to obtain a robust optimization unit combination model of an affine space;
and solving the robust optimization unit combination model of the affine space to obtain a thermal power unit combination mode.
2. The piecewise affine method of claim 1, wherein the constraints further comprise: the method comprises the following steps of system power balance constraint, minimum start-up time and minimum shutdown time constraint of a thermal power unit, thermal power unit state transition constraint, thermal power unit output upper and lower limit constraint, thermal power unit climbing constraint, thermal power unit output adjusting range constraint and thermal power unit output adjusting capacity constraint.
3. The piecewise affine method of claim 1, wherein the objective function of the two-stage robust optimization crew combination model in the form of max-min-max is:
f＝f 1 +f 2
Figure FDA0002877409170000021
Figure FDA0002877409170000022
f is an objective function of the two-stage robust optimization unit combination model; f (f) 1 An objective function of the first stage unit model; t is the scheduling time; t is the scheduling time length; u (u) i，t =1 is that the thermal power unit i is turned off at t-1 and turned on at t; v i，t =1 is that the thermal power unit i is started at the time t-1 and is shut down at the time t; gamma ray i，t The state variable of the thermal power unit i at the time t is obtained;
Figure FDA0002877409170000023
the output power of the thermal power generating unit i at the time t is obtained;
Figure FDA0002877409170000024
The upward output force adjusting capability of the thermal power unit i at the time t is provided;
Figure FDA0002877409170000025
The downward output adjusting capability of the thermal power unit i at the time t is provided; deltap t The method is characterized in that the thermal power unit output reduction delta caused by load and wind power prediction error at time t of the power systemp t ；
Figure FDA0002877409170000026
And deltap t Is an uncertain variable;
Figure FDA0002877409170000027
Is->
Figure FDA0002877409170000028
Is a set of uncertainty of (2);Uis deltap t Is a set of uncertainty of (2); c (C) start，i The starting cost of the unit i; c (C) shut，i The shutdown cost of the unit i; a, a i 、b i 、c i The fuel cost coefficient is the secondary, primary and constant of the thermal power unit i; c (C) up，i Positive standby cost coefficients for unit i; c (C) down，i Negative standby cost coefficient of unit i; n (N) G Is the number of thermal power generating units in the electric power system.
4. A segmented affine method according to claim 3, wherein the objective function in the robust optimization block combination model of affine space is:
Figure FDA0002877409170000029
Figure FDA00028774091700000210
Figure FDA00028774091700000211
Figure FDA00028774091700000212
Figure FDA00028774091700000213
Figure FDA00028774091700000214
Figure FDA00028774091700000215
Figure FDA00028774091700000216
5. a segmented affine method according to claim 3, wherein the thermal power plant total output power adjustment capability constraint in the two-stage robust optimization plant combination model in the form of max-min-max is:
Figure FDA0002877409170000031
Figure FDA0002877409170000032
Figure FDA0002877409170000033
Figure FDA0002877409170000034
wherein the prediction error of the load at the time t is as follows
Figure FDA0002877409170000035
The prediction error of wind power of the power system is +.>
Figure FDA0002877409170000036
α t ∈[-1,0]，
Figure FDA0002877409170000037
β t ∈[-1,0]，
Figure FDA0002877409170000038
Figure FDA0002877409170000039
The method is characterized in that the method is the thermal power unit output increment caused by the load prediction error and the wind power prediction error of the power system at the time t; deltap t The method is thermal power unit output reduction caused by the prediction error of load and wind power at the time t.
6. The piecewise affine method of claim 5, wherein the thermal power plant total output power adjustment capability constraint in the robust optimization unit combination model of affine space is:
Figure FDA00028774091700000310
Figure FDA00028774091700000311
wherein,,
Figure FDA00028774091700000312
a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjustment capability constraint of the thermal power unit;e t and a unit vector with a t dimension of 1 corresponding to a fluctuation coefficient of a net residual value of a load prediction error and a wind power prediction error of the power system in the upward output power adjusting capacity constraint of the thermal power unit.
7. A computer readable storage medium, on which a computer program is stored, characterized in that the computer program, when being executed by a processor, implements the steps of the method of any of claims 1 to 6.
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