CN112819278B - A Piecewise Affine Method for Solving Two-Stage Robust Optimal Unit Combination Model - Google Patents

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 for power system unit combinations, and specifically includes the following steps: based on the thermal power unit output adjustment capability, establishing max-min Two-stage robust optimization unit combination model in the form of ‑max; thermal power unit output adjustment capability is used to deal with power system load uncertainty and wind power uncertainty; simplex-based piecewise affine method affine uncertain set to Simplex space, obtain the two-stage robust optimization unit combination model in the simplex space; convert the two-stage robust optimization unit combination model in the simplex space into a robust optimization unit combination model in the affine space; solve the affine space The robust optimization unit combination model of the model is used to obtain the combination mode of thermal power units. Compared with the traditional iterative algorithm, the segmented affine method for solving the two-stage robust optimization unit combination model provided by the present invention is faster and simpler.

Description

A piecewise affine method for solving two-stage robust optimization unit commitment model

Technical Field

The present invention belongs to the field of robust optimization models of power system unit combination, and more specifically, relates to a piecewise affine method for solving a two-stage robust optimization unit combination model.

Background Art

The unit combination of power systems plays an important role in dealing with the uncertainty of renewable energy output. The essence of unit combination is to minimize the operating cost of the power system while considering the corresponding constraints in actual engineering projects. Unlike the traditional unit combination problem, the random volatility of renewable energy grid-connected output leads to the emergence of random variables in the new unit combination problem, which changes the unit combination problem from deterministic planning to uncertain planning, increasing the difficulty for dispatchers to formulate unit start and stop plans to deal with the uncertainty of renewable energy output.

Solving a robust optimization model is usually a difficult process. Therefore, the original model needs to be transformed to a certain extent. The transformed model needs to be easily solved in polynomial time. The traditional static robust optimization model belongs to the max-min game pattern, while the new two-stage robust optimization model belongs to the max-min-max game pattern. The traditional static robust optimization model must be made after the uncertainty is known, which belongs to the here-and-now optimization model. The first stage of the new two-stage robust optimization model is the here-and-now optimization model, and the second stage is the wait-and-see optimization mode. Therefore, some methods of the new two-stage robust optimization model can be made after the uncertainty is observed. This feedback mechanism makes the new two-stage robust optimization model less conservative than the traditional static robust optimization model.

In the “Application Examples of Robust Economic Dispatch of Power Systems (II)”, the application of two-stage robust optimization in power systems is disclosed. First, a two-stage robust optimization model for dealing with wind power uncertainty is constructed, and then an uncertainty set that characterizes wind power uncertainty is proposed. Although the above model can effectively deal with the uncertainty of wind power, the solution of the model is relatively complex. Although the traditional Bender decomposition method can solve the model, the Bender decomposition method cannot guarantee the number of iterations in the solution process. The document “Solving two-stage robust optimization problemsusing a column-and-constraint generation method” proposes a C&CG column and constraint generation method to solve the two-stage robust optimization model. Although this method can effectively improve the solution efficiency and reduce the number of iterations in the solution process, its Big-M method that relies on subjective parameters for linearization makes the model solution results too subjective. In the experiment, the M value of the Big-M method will seriously affect the final model solution results. The research paper "A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization" proposed a piecewise affine method based on simplex, which considers the second-stage decision variables of two-stage robust optimization as implicit functions of uncertain variables, and realizes the conversion of two-stage robust optimization into a single-stage robust optimization problem. However, in actual engineering applications, different forms of uncertainty sets will affect the accuracy of the converted model.

Summary of the invention

In view of the defects of the prior art, the purpose of the present invention is to provide a piecewise affine method for solving a two-stage robust optimization unit combination model, aiming to solve the problem of complex solution caused by the existence of uncertain sets in the existing robust optimization model.

To achieve the above object, the present invention provides a piecewise affine method for solving a two-stage robust optimization unit commitment model, comprising the following steps:

Based on the output regulation capability of thermal power units, a two-stage robust optimization unit combination model in the form of max-min-max is established; the output regulation capability of thermal power units is used to cope with the uncertainty of power system load and wind power, and the objective function of the first-stage unit model is to minimize the sum of the start-up and shutdown cost and fuel cost of the thermal power units in the power system; the objective function of the second-stage unit model is to minimize the sum of the positive spinning reserve cost and the negative spinning reserve cost under the worst case of power system load uncertainty and wind power uncertainty; the constraints include the total output regulation capability constraint of thermal power units;

Based on the simplex piecewise affine method, the uncertain set in the two-stage robust optimization unit commitment model of max-min-max form is affine-mapped to the simplex space to obtain the two-stage robust optimization unit commitment model in the simplex space;

Based on the polyhedral uncertain set, the scale factor in the simple space is defined as the minimum value of the scheduling duration and the unit duration, and the dominating vector is defined as the sum of unit vectors to obtain the robust optimization unit commitment model in the affine space.

The robust optimization unit combination model in affine space is solved to obtain the combination mode of thermal power units.

Preferably, the constraints also include: system power balance constraints, minimum startup time and minimum shutdown time constraints of thermal power units, thermal power unit state transition constraints, thermal power unit output upper and lower limit constraints, thermal power unit climbing constraints, thermal power unit output adjustment range constraints and thermal power unit output adjustment capability constraints.

Preferably, the objective function of the two-stage robust optimization unit commitment model in the form of max-min-max is:

f＝f ₁ +f ₂

为火电机组i在t时刻的输出功率；

为火电机组i在t时刻的向上出力调节能力；

为火电机组i在t时刻的向下出力调节能力；Δp _t为电力系统在时刻t因负荷与风电预测误差而引起的火电机组出力减少量Δp _t；

与Δp _t为不确定变量；

为

的不确定集合；U为Δp _t的不确定集合；C_start,i为机组i的启动成本；C_shut,i为机组i的关停成本；a_i、b_i、c_i为火电机组i的二次、一次、常数燃料成本系数；C_up,i为机组i的正备用成本系数；C_down,i机组i的负备用成本系数；N_G为电力系统中火电机组台数。Among them, f is the objective function of the two-stage robust optimization unit commitment model; f ₁ is the objective function of the first-stage unit model; t is the scheduling time; T is the scheduling duration; u _i,t = 1 means that the thermal power unit i is shut down at time t-1 and started at time t; vi _,t = 1 means that the thermal power unit i is started at time t-1 and shut down at time t; γ _i,t is the state variable of the thermal power unit i at time t;

is the output power of thermal power unit i at time t;

is the upward output regulation capability of thermal power unit i at time t;

is the downward output regulation capability of thermal power unit i at time t; Δ p _t is the output reduction of thermal power unit Δ p _t caused by the load and wind power forecast error at time t;

and Δ p _t are uncertain variables;

for

is the uncertain set of Δ p t; U is the uncertain set of Δ p _t ; C _start,i is the startup cost of unit i; C _shut,i is the shutdown cost of unit i; a _i , b _i , c _i are the secondary, primary and constant fuel cost coefficients of thermal power unit i; C _up,i is the positive standby cost coefficient of unit i; C _down,i is the negative standby cost coefficient of unit i; _NG is the number of thermal power units in the power system.

Preferably, the objective function in the robust optimization unit commitment model in affine space is:

Preferably, the total output regulation capability constraint of the thermal power units in the two-stage robust optimization unit commitment model of max-min-max form is:

电力系统风电的预测误差为

α _t∈[-1,0]，

β _t∈[-1,0]，

为电力系统在t时刻因负荷的预测误差与风电的预测误差而引起的火电机组出力增加量；Δp _t为在t时刻因负荷与风电的预测误差而引起的火电机组出力减少量。Among them, the load prediction error at time t is

The prediction error of wind power in the power system is

α _t ∈[-1,0],

β _t ∈[-1,0],

is the increase in thermal power unit output caused by the load forecast error and wind power forecast error at time t; Δ p _t is the decrease in thermal power unit output caused by the load and wind power forecast errors at time t.

Preferably, the total output regulation capability constraint of the thermal power units in the robust optimization unit commitment model in the affine space is:

为火电机组向上出力调节能力约束中电力系统负荷预测误差与风电预测误差的净剩值的波动系数对应的第t维为1的单位向量；e _t为火电机组向上出力调节能力约束中电力系统负荷预测误差与风电预测误差的净剩值的波动系数对应的第t维为1的单位向量；e_t实质是βv的第t维分量，物理含义为第t时刻的净负荷波动的最大倍数。in,

It is the unit vector with the t-th dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is the unit vector with the t-th dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is essentially the t-th dimension component of βv, and its physical meaning is the maximum multiple of the net load fluctuation at the t-th moment.

A computer-readable storage medium stores a computer program, which, when executed by a processor, implements the steps of a piecewise affine method for solving a two-stage robust optimization unit commitment model.

In general, the above technical solution conceived by the present invention has the following beneficial effects compared with the prior art:

为含有不确定集合的max-min形式，以及火电机组总出力调节能力约束中也含有不确定集合，存在求解极度困难的问题，采用单纯形的分段仿射方法，将max-min-max形式的两阶段鲁棒优化机组组合模型转换成单纯空间中的两阶段鲁棒优化机组组合模型，保证构建的多面体不确定结合在仿射变换后不失去原有的保守性，由于单纯空间中的两阶段鲁棒优化机组组合模型中存在不确定变量与不确定变量的乘积形式，采用

将单纯空间中的两阶段鲁棒优化机组组合模型转换为仿射空间的鲁棒优化机组组合模型，采用成熟的CPLEX求解器即可求解。相比于传统的迭代算法，本发明提供的求解两阶段鲁棒优化机组组合模型的分段仿射方法更为快速简单。In the present invention, the uncertainty of the power system load and the power system wind power is concentrated in the net load of the two. The objective function of the second stage unit model in the established max-min-max form two-stage robust optimization unit combination model is

The max-min form contains uncertain sets, and the total output regulation capacity constraints of thermal power units also contain uncertain sets, which makes it extremely difficult to solve. The simplex piecewise affine method is used to transform the two-stage robust optimization unit combination model in the max-min-max form into a two-stage robust optimization unit combination model in a simple space, ensuring that the constructed polyhedron uncertain combination does not lose its original conservatism after affine transformation. Since there are uncertain variables and uncertain variables in the two-stage robust optimization unit combination model in the simple space, the

The two-stage robust optimization unit commitment model in the simple space is converted into a robust optimization unit commitment model in the affine space, and the solution can be obtained by using the mature CPLEX solver. Compared with the traditional iterative algorithm, the piecewise affine method for solving the two-stage robust optimization unit commitment model provided by the present invention is faster and simpler.

The two-stage robust optimization unit combination model proposed in the present invention for coping with the load volatility of the power system and the random volatility of wind power based on the output regulation capability of the thermal power units includes the thermal power unit climbing constraints and the output regulation constraint range of the thermal power units, which can ensure that the random volatility of the power system load and wind power can be effectively coped with within the climbing range.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a flow chart of a piecewise affine method for solving a two-stage robust optimization unit commitment model provided by the present invention;

2 is a schematic diagram of a two-stage robust optimization unit combination output obtained by solving solution 1 using an affine method provided by an embodiment of the present invention;

3 is a schematic diagram of a two-stage robust optimization unit combination output obtained by solving the second solution using an affine method provided by an embodiment of the present invention;

FIG4 is a schematic diagram of a two-stage robust optimization unit combination output obtained by using C&CG solution scheme 2 provided in an embodiment of the present invention;

FIG5 is a comparison diagram of the output boundaries of the two-stage robust optimization unit combination using the affine method and C&CG solution provided by an embodiment of the present invention.

Specific implementation plan

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

The present invention as a whole has the following inventive points:

The present invention proposes a two-stage robust optimization unit combination model for coping with power system load volatility and wind power random volatility based on the output regulation capability of thermal power units. This model enables online thermal power units to effectively cope with the random volatility of power system load and wind power within the climbing range.

Based on the spatial affine method, the uncertain variable set in the traditional two-stage robust optimization is transformed into a simplex spatial model, and the spatial dominance points of the original uncertain variable set are derived. Based on the characteristics of the polyhedral uncertain set, the simplex spatial dominance points of the power system load and wind power uncertainty are proposed.

The present invention proposes a piecewise affine method based on the proposed two-stage robust optimization model of simplex and spatial dominance points. The method converts the traditional min-max-min two-stage robust optimization unit combination model into a single-stage unit combination model containing only min, so that the unit combination mode of the system can be obtained without iterative solution.

As shown in FIG1 , the present invention provides a piecewise affine method for solving a two-stage robust optimization unit commitment model, comprising the following steps:

Based on the output regulation capability of thermal power units, a two-stage robust optimization unit combination model in the form of max-min-max is established; the output regulation capability of thermal power units is used to cope with the uncertainty of power system load and wind power, and the objective function of the first-stage unit model is to minimize the sum of the start-up and shutdown cost and fuel cost of the thermal power units in the power system; the objective function of the second-stage unit model is to minimize the sum of the positive spinning reserve cost and the negative spinning reserve cost under the worst case of power system load uncertainty and wind power uncertainty; the constraints include the total output regulation capability constraint of thermal power units;

Based on the simplex piecewise affine method, the uncertain set in the two-stage robust optimization unit commitment model of max-min-max form is affine-mapped to the simplex space to obtain the two-stage robust optimization unit commitment model in the simplex space;

Based on the polyhedral uncertain set, the scale factor in the simple space is defined as the minimum value of the scheduling duration and the unit duration, and the dominating vector is defined as the sum of unit vectors to obtain the robust optimization unit commitment model in the affine space.

The robust optimization unit combination model in affine space is solved to obtain the combination mode of thermal power units.

Preferably, the constraints also include: system power balance constraints, minimum startup time and minimum shutdown time constraints of thermal power units, thermal power unit state transition constraints, thermal power unit output upper and lower limit constraints, thermal power unit climbing constraints, thermal power unit output adjustment range constraints and thermal power unit output adjustment capability constraints.

Preferably, the objective function of the two-stage robust optimization unit commitment model in the form of max-min-max is:

f＝f ₁ +f ₂

为火电机组i在t时刻的输出功率；

为火电机组i在t时刻的向上出力调节能力；

为火电机组i在t时刻的向下出力调节能力；Δp _t为电力系统在时刻t因负荷与风电预测误差而引起的火电机组出力减少量Δp _t；

与Δp _t为不确定变量；

为

的不确定集合；U为Δp _t的不确定集合；C_start,i为机组i的启动成本；C_shut,i为机组i的关停成本；a_i、b_i、c_i为火电机组i的二次、一次、常数燃料成本系数；C_up,i为机组i的正备用成本系数；C_down,i机组i的负备用成本系数。Among them, f is the objective function of the two-stage robust optimization unit commitment model; f ₁ is the objective function of the first-stage unit model; t is the scheduling time; T is the scheduling duration; u _i,t = 1 means that the thermal power unit i is shut down at time t-1 and started at time t; vi _,t = 1 means that the thermal power unit i is started at time t-1 and shut down at time t; γ _i,t is the state variable of the thermal power unit i at time t;

is the output power of thermal power unit i at time t;

is the upward output regulation capability of thermal power unit i at time t;

is the downward output regulation capability of thermal power unit i at time t; Δ p _t is the output reduction of thermal power unit Δ p _t caused by the load and wind power forecast error at time t;

and Δ p _t are uncertain variables;

for

is the uncertain set; U is the uncertain set of Δ p _t ; C _start,i is the startup cost of unit i; C _shut,i is the shutdown cost of unit i; a _i , b _i , c _i are the secondary, primary and constant fuel cost coefficients of thermal power unit i; C _up,i is the positive standby cost coefficient of unit i; C _down,i is the negative standby cost coefficient of unit i.

Preferably, the objective function in the robust optimization unit commitment model in affine space is:

Preferably, the total output regulation capability constraint of the thermal power units in the two-stage robust optimization unit commitment model of max-min-max form is:

电力系统风电的预测误差为

α _t∈[-1,0]，

β _t∈[-1,0]，

为电力系统在t时刻因负荷的预测误差与风电的预测误差而引起的火电机组出力增加量；Δp _t为在t时刻因负荷与风电的预测误差而引起的火电机组出力减少量。Among them, the load prediction error at time t is

The prediction error of wind power in the power system is

α _t ∈[-1,0],

β _t ∈[-1,0],

is the increase in thermal power unit output caused by the load forecast error and wind power forecast error at time t; Δ p _t is the decrease in thermal power unit output caused by the load and wind power forecast errors at time t.

Preferably, the total output regulation capability constraint of the thermal power units in the robust optimization unit commitment model in the affine space is:

为火电机组向上出力调节能力约束中电力系统负荷预测误差与风电预测误差的净剩值的波动系数对应的第t维为1的单位向量；e _t为火电机组向上出力调节能力约束中电力系统负荷预测误差与风电预测误差的净剩值的波动系数对应的第t维为1的单位向量；e_t实质是βv的第t维分量，物理含义为第t时刻的净负荷波动的最大倍数。in,

It is the unit vector with the t-th dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is the unit vector with the t-th dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is essentially the t-th dimension component of βv, and its physical meaning is the maximum multiple of the net load fluctuation at the t-th moment.

Specifically, the present invention provides a piecewise affine method for solving two-stage robust unit commitment, comprising the following steps:

Step 1: Construct a two-stage robust optimization unit commitment model based on the output regulation capability of thermal power units to cope with the uncertainty of power system load and wind power uncertainty, as follows:

Objective function: The first-stage objective function _f1 of the two-stage robust optimization unit commitment model is to minimize the sum of the start-up and shutdown costs and fuel costs of the thermal power units in the power system. The second-stage objective function _f2 is to minimize the sum of the positive spinning reserve and negative spinning reserve costs under the worst conditions of load uncertainty and wind power uncertainty in the power system.

f＝f ₁ +f ₂

为火电机组i在t时刻的输出功率；

为火电机组i在t时刻的向上出力调节能力；

为火电机组i在t时刻的向下出力调节能力；Δp _t为电力系统在时刻t因负荷与风电预测误差而引起的火电机组出力减少量Δp _t；

与Δp _t为不确定变量；

为

的不确定集合；U为Δp _t的不确定集合；C_start,i为机组i的启动成本；C_shut,i为机组i的关停成本；a_i、b_i、c_i为火电机组i的二次、一次、常数燃料成本系数；C_up,i为机组i的正备用成本系数；C_down,i机组i的负备用成本系数；Among them, f is the objective function of the two-stage robust optimization unit commitment model; f ₁ is the first stage objective function; t is the scheduling time; T is the scheduling duration; u _i,t = 1 means that the thermal power unit i is shut down at time t-1 and started at time t; vi _,t = 1 means that the thermal power unit i is started at time t-1 and shut down at time t; γ _i,t is the state variable of the thermal power unit i at time t, γ _i,t = 1 means it is turned on, otherwise it is shut down;

is the output power of thermal power unit i at time t;

is the upward output regulation capability of thermal power unit i at time t;

is the downward output regulation capability of thermal power unit i at time t; Δ p _t is the output reduction of thermal power unit Δ p _t caused by the load and wind power forecast error at time t;

and Δ p _t are uncertain variables;

for

U is the uncertain set of Δ p _t ; C _start,i is the startup cost of unit i; C _shut,i is the shutdown cost of unit i; a _i , b _i , c _i are the secondary, primary and constant fuel cost coefficients of thermal power unit i; C _up,i is the positive standby cost coefficient of unit i; C _{down,i is} the negative standby cost coefficient of unit i;

The constraints are:

(1) System power balance constraints:

The physical meaning of this constraint is: at time t, the sum of the output of the online thermal power units in the power system and the predicted output of wind power is equal to the predicted load of the power system;

为电力系统在t时刻的预测风电出力；

为电力系统在t时刻的预测负荷；in,

is the predicted wind power output of the power system at time t;

is the predicted load of the power system at time t;

(2) Constraints on the minimum startup time and shutdown time of thermal power units:

The physical meaning of this constraint is that the continuous online time and continuous offline time of thermal power unit i should be at least equal to the minimum start-up time and minimum shutdown time of thermal power unit i, which can be specifically expressed as:

-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}

Among them, τ is the time period during which thermal power unit i must be turned on; k is the time period during which thermal power unit i must be turned off;

(3) Constraints on state transition of thermal power units:

The physical meaning of this constraint is: the operating state γ _{i,t of thermal power unit i at time t} must satisfy the logical constraints of u _i,t and v _i,t ;

Among them, ui _,t and vi _,t jointly constrain the change of γi _,t ;

(4) Upper and lower limits of thermal power unit output:

应该不小于火电机组i的出力下限乘γ_i,t且不大于火电机组i的出力上限乘γ_i,t，具体可表示如下：The physical meaning of the upper and lower limits of the output of thermal power units is: the output of thermal power unit i at time t

It should be no less than the output lower limit of thermal power unit i multiplied by γ _i,t and no greater than the output upper limit of thermal power unit i multiplied by γ _i,t , which can be specifically expressed as follows:

为火电机组i的出力下限；

为火电机组i的出力上限；in,

is the lower limit of the output of thermal power unit i;

is the output upper limit of thermal power unit i;

(5) Climbing constraints of thermal power units:

The physical meaning of this constraint is: the output change of thermal power unit i at time t and time t-1 should meet the climbing capacity limit of thermal power unit i;

为火电机组i的向上爬坡能力；

为火电机组i的向下爬坡能力；in,

is the upward climbing capability of thermal power unit i;

is the downward climbing capability of thermal power unit i;

(6) Constraints on the output regulation range of thermal power units:

The physical meaning of this constraint is that the maximum and minimum power that can be generated by the thermal power unit should be between the product of the thermal power unit state and the upper and lower limits of the thermal power unit output; it can be specifically expressed as:

为火电机组i的出力下限；

为火电机组i的出力上限；此外，需满足火电机组i可以从t-1时刻的出力下限爬坡至t时刻的出力上限，且可以从t-1时刻的出力上限爬坡至t时刻的出力下限；in,

is the lower limit of the output of thermal power unit i;

is the output upper limit of thermal power unit i; in addition, it is required that thermal power unit i can climb from the output lower limit at time t-1 to the output upper limit at time t, and can climb from the output upper limit at time t-1 to the output lower limit at time t;

(7) Constraints on the output regulation capability of thermal power units:

The physical meaning of this constraint is: the upward adjustment capability and downward adjustment capability of thermal power unit i at time t shall not be greater than the climbing capability of the unit;

(8) Constraints on the total output regulation capability of thermal power units in the power system (uncertain constraints):

电力系统风电的预测误差为

其中，α _t∈[-1,0]，

β _t∈[-1,0]，

基于此，电力系统在t时刻因负荷的预测误差与风电的预测误差而引起的火电机组出力增加量

可表示为：Assume that the load prediction error at time t is

The prediction error of wind power in the power system is

Among them, α _t ∈[-1,0],

β _t ∈[-1,0],

Based on this, the increase in thermal power generation unit output caused by the load forecast error and wind power forecast error at time t is

It can be expressed as:

的最恶劣情况为电力系统负荷增加

而电力系统风电出力减少

The above formula indicates

The worst case scenario is an increase in power system load

The wind power output of the power system has decreased

The output reduction of thermal power units Δ pt caused by the forecast error of load and wind power in the power system at time _t can be expressed as:

而电力系统的负荷减少

The above formula shows that the worst case of Δ p _t is when the wind power output of the power system increases

The load on the power system is reduced

Furthermore, the above equations (1) and (2) can be rewritten as:

与h _t如何变化，电力系统中火电机组的正备用与负备用均能够应对

与Δp _t的随机性，因此，电力系统火电机组总出力调节能力约束表示为：Because of the need

Regardless of the change of h _t , the positive and negative reserves of thermal power units in the power system can cope with

and the randomness of Δ p _t , therefore, the total output regulation capacity constraint of thermal power units in the power system is expressed as:

可改写为

不确定集合U可改写为

Uncertain Set

Can be rewritten as

The uncertain set U can be rewritten as

Step 2: Two-stage robust optimization piecewise affine method based on simplex;

The standard two-stage robust optimization model can be expressed as Π _AR (U);

x为第一阶段决策变量；y(h)为第二阶段决策变量；h为不确定变量；in,

x is the first-stage decision variable; y(h) is the second-stage decision variable; h is an uncertain variable;

定义单纯形

其中，β为规模因子，使得

e_j为第j维为1的m维的单位向量；v为m维的支配向量，v∈U；如果

对于

都有

则

支配U；其中，

(θ)⁺＝max{0,θ}，并且有：For a given uncertain set

Defining a simplex

Where β is the scale factor, making

e _j is an m-dimensional unit vector with the jth dimension being 1; v is an m-dimensional dominant vector, v∈U; if

for

Both

but

Dominate U; where,

(θ) ⁺ =max{0,θ}, and we have:

This gives a simplex-based piecewise affine method:

为优化模型

的最优解：in,

To optimize the model

The optimal solution is:

Ax+By _m+1 ≥βv (6)

At this point, the original two-stage robust optimization model Π _AR (U) in the form of max-min-max is solved by approximating (3) to (7);

的多面体不确定集合，给出了

其中

当k＝m时，基于

的仿射方法为：For the shape

The polyhedron uncertain set is given by

in

When k = m, based on

The affine method is:

The dominating simplex

Step 3: Construct an affine method for a two-stage robust optimization unit commitment model based on the output regulation capability of thermal power units to cope with the uncertainty of power system load and wind power uncertainty;

Based on the above two-stage robust optimization piecewise affine method, the following piecewise affine method is given to solve the two-stage robust optimization unit commitment model considering the output regulation capability of thermal power units:

First stage affine variables:

Second stage affine variables:

Uncertain set conversion:

Translates to:

Translates to:

Based on the above equations (8) to (15), the following model can be solved to obtain the combination scheme of thermal power units:

-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 )

为火电机组向上出力调节能力约束中电力系统负荷预测误差与风电预测误差的净剩值的波动系数对应的第t维为1的单位向量；e _t为火电机组向上出力调节能力约束中电力系统负荷预测误差与风电预测误差的净剩值的波动系数对应的第t维为1的单位向量；e_t实质是βv的第t维分量，物理含义为第t时刻的净负荷波动的最大倍数。in,

It is the unit vector with the t-th dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is the unit vector with the t-th dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is essentially the t-th dimension component of βv, and its physical meaning is the maximum multiple of the net load fluctuation at the t-th moment.

The two-stage robust optimization unit commitment model for coping with power system load uncertainty and wind power uncertainty based on the output regulation capability of thermal power units is composed of equations (16) to (32). The above model can be solved using the mature CPLEX solver.

A computer-readable storage medium stores a computer program, which, when executed by a processor, implements the steps of a piecewise affine method for solving a two-stage robust optimization unit commitment model.

In summary, in the two-stage robust optimization unit combination model based on the output regulation capability of thermal power units provided by the present invention, the uncertainty of the power system load and the wind power of the power system are concentrated in the net load of the two. In the established method, a polyhedral uncertainty set is constructed, and the uncertain variable set of the traditional two-stage robust optimization is affine-transformed to the simplex space based on the simplex method, and it is ensured that the constructed polyhedral uncertainty set does not lose its original conservatism after the affine transformation; finally, the present invention converts the two-stage robust optimization unit combination model into a robust optimization unit combination model in an affine space by using the dominant simplex after affine transformation. Compared with the traditional iterative algorithm, the present invention can quickly obtain the combination mode of thermal power units.

Example

Based on the piecewise affine method for solving the two-stage robust optimization unit commitment model provided by the present invention, this embodiment provides two simulation schemes:

(1) 10-unit dispatching scheme: This scheme adopts IEEE-39 node system unit data, sets the prediction error of power system load to 10%, and the prediction error of power system wind power to 10%. The piecewise affine method proposed in this scheme is compared with the C&CG column and constraint generation method, highlighting that the solution speed of the piecewise affine method provided by the present invention is higher than that of the C&CG column and constraint generation method.

(2) 33-unit dispatching scheme: This scheme adopts IEEE-39 node system unit data, sets the prediction error of power system load to 10%, and the prediction error of power system wind power to 20%. The piecewise affine method proposed in this scheme is compared with the C&CG column and constraint generation method, highlighting that the solution speed of the piecewise affine method provided by the present invention is higher than that of the C&CG column and constraint generation method.

The present invention is further described in detail below in conjunction with the embodiments.

In order to verify the correctness of the present invention, this example is simulated based on MATLAB combined with CPLEX solver. In the simulation, the random energy is set as wind power, and the prediction error of wind power in the 10-unit scheduling scheme is 10%, that is, the actual wind power fluctuates in the interval of 0.9 times to 1.1 times the predicted wind power; the prediction error of load in the 10-unit scheduling scheme is 10%, that is, the actual load fluctuates in the interval of 0.9 times to 1.1 times the predicted wind power; the prediction error of wind power in the 33-unit scheduling scheme is 20%, that is, the actual wind power fluctuates in the interval of 0.8 times to 1.2 times the predicted wind power; the prediction error of load in the 10-unit scheduling scheme is 10%, that is, the actual load fluctuates in the interval of 0.9 times to 1.1 times the predicted wind power. The specific simulation method is shown in Table 1. In addition, Table 1 gives the corresponding solution method of the set scheme.

Table 1

Figure 2 shows the output diagram of the thermal power units in Scheme 1 based on the proposed affine method. As can be seen from the figure, all 10 thermal power units in Scheme 1 have output, among which units G1, G4, and G8 bear most of the load of the power system. In addition, the output of thermal power units G3, G6, G7, G9, and G10 is relatively small. The main reason for the above phenomenon is the functional difference of the fuel cost of the thermal power units. The units with better economic performance will have relatively more output, while the units with poor economic performance will have relatively less output. In order to form a contrast with the existing methods, Figure 3 shows the output results of the thermal power units in Scheme 2, including the results of the proposed affine method and the C&CG method. It should be noted that the example system of Scheme 2 has more units, more variables, more constraints, and higher dimensions in Scheme 2. The comparison results of the two methods are more convincing and representative.

Figures 3 and 4 compare the results of solving the two-stage robust optimization unit combination of 33 units by the affine method and the C&CG method. It can be seen from Figures 3 and 4 that the output of the thermal power units of the two methods is different. Due to the limitation of the climbing ability of the thermal power units, the different outputs of the thermal power units will lead to different upper and lower limits of the actual output of the thermal power units. In addition, the affine method proposed in this embodiment does not need to set subjective parameters, while the C&CG algorithm needs to linearize the complementary dual constraints of the sub-problems based on the Big-M method, and the value of M will seriously affect the final optimization results. The above analysis shows that the different output results of the thermal power units of the two methods will lead to different output boundaries of the thermal power units, resulting in different capabilities of the thermal power units to cope with system loads and wind power uncertainties.

Figure 5 provides the comparison results of the output boundaries of the thermal power units of the two methods in Scheme 2. As can be seen from the figure, the upper bound of the thermal power unit output of the piecewise affine method is larger than that of the thermal power unit output of the C&CG method, which means that the thermal power units of the piecewise affine method have more positive reserve capacity. As can be seen from the figure, the lower bound of the thermal power unit output of the piecewise affine method is very close to the lower bound of the thermal power unit output of the C&CG method. However, at 1:00, the piecewise affine method has a lower lower bound of the thermal power unit output, that is, the thermal power units of the piecewise affine method have more negative spinning reserves to cope with the uncertainty of system load and system wind power.

Table 2 shows the comparison results of the model solution time and system operation cost of the piecewise affine method and C&CG in Scheme 2. Scheme 2 was selected because there are more thermal power units in Scheme 2, which can better reflect the solution effect of the two-stage robust optimization unit combination problem of large-scale systems. As can be seen from Table 2, the proposed piecewise affine method has a faster solution efficiency, and its model solution time is about one-fourth of that of C&CG. In addition, the system operation cost of the piecewise affine method is higher than that of C&CG, which is consistent with the situation reflected in Figures 3, 4, and 5. Because having more spare capacity will mean higher system operation costs.

Table 2

It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims (7)

1. A piecewise affine method for solving a two-stage robust optimization unit commitment model, characterized by comprising the following steps: Based on the output regulation capability of thermal power units, a two-stage robust optimization unit combination model in the form of max-min-max is established; the output regulation capability of thermal power units is used to cope with the uncertainty of power system load and wind power, and the objective function of the first-stage unit model is to minimize the sum of the start-up and shutdown cost and fuel cost of the thermal power units in the power system; the objective function of the second-stage unit model is to minimize the sum of the positive spinning reserve cost and the negative spinning reserve cost under the worst case of power system load uncertainty and wind power uncertainty; the constraints include the total output regulation capability constraint of thermal power units; Based on the simplex piecewise affine method, the uncertain set in the two-stage robust optimization unit commitment model of max-min-max form is affine-mapped to the simplex space to obtain the two-stage robust optimization unit commitment model in the simplex space; Based on the polyhedral uncertain set, the scale factor in the simple space is defined as the minimum value of the scheduling duration and the unit duration, and the dominating vector is defined as the sum of unit vectors to obtain the robust optimization unit commitment model in the affine space. The robust optimization unit combination model in affine space is solved to obtain the combination mode of thermal power units. 2. According to the piecewise affine method described in claim 1, it is characterized in that the constraints also include: system power balance constraints, minimum start-up time and minimum shutdown time constraints of thermal power units, thermal power unit state transition constraints, thermal power unit output upper and lower limit constraints, thermal power unit climbing constraints, thermal power unit output adjustment range constraints and thermal power unit output adjustment capability constraints. 3. The piecewise affine method according to claim 1, characterized in that the objective function of the two-stage robust optimization unit commitment model in the form of max-min-max is: f＝f ₁ +f ₂

Among them, f is the objective function of the two-stage robust optimization unit commitment model; _f1 is the objective function of the first-stage unit model; t is the scheduling time; T is the scheduling duration; u _{i, t} = 1 means that the thermal power unit i is shut down at time t-1 and started at time t; vi _{, t} = 1 means that the thermal power unit i is started at time t-1 and shut down at time t; γ _{i, t} is the state variable of the thermal power unit i at time t;

is the output power of thermal power unit i at time t;

is the upward output regulation capability of thermal power unit i at time t;

is the downward output regulation capability of thermal power unit i at time t; Δ p _t is the output reduction of thermal power unit Δ p _t caused by the load and wind power forecast error at time t;

and Δ p _t are uncertain variables;

for

is the uncertain set of Δ p t; U is the uncertain set of Δ p _t ; C _start,i is the startup cost of unit i; C _shut,i is the shutdown cost of unit i; a _i , b _i , c _i are the secondary, primary and constant fuel cost coefficients of thermal power unit i; C _up,i is the positive standby cost coefficient of unit i; C _down,i is the negative standby cost coefficient of unit i; _NG is the number of thermal power units in the power system. 4. The piecewise affine method according to claim 3, characterized in that the objective function in the robust optimization unit commitment model in the affine space is:

5. The piecewise affine method according to claim 3 is characterized in that the total output regulation capability constraint of the thermal power units in the two-stage robust optimization unit commitment model in the max-min-max form is:

Among them, the load prediction error at time t is

The prediction error of wind power in the power system is

α _t ∈[-1,0],

β _t ∈[-1,0],

is the increase in thermal power unit output caused by the load forecast error and wind power forecast error at time t; Δ p _t is the decrease in thermal power unit output caused by the load and wind power forecast errors at time t. 6. The piecewise affine method according to claim 5, characterized in that the total output regulation capability constraint of the thermal power units in the robust optimization unit commitment model in the affine space is:

in,

is the unit vector with the tth dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit; e _t is the unit vector with the tth dimension being 1 corresponding to the fluctuation coefficient of the net residual value of the load forecast error and the wind power forecast error of the power system in the upward output regulation capability constraint of the thermal power unit. 7. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

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