✅作者简介热爱科研的Matlab仿真开发者擅长毕业设计辅导、数学建模、数据处理、建模仿真、程序设计、完整代码获取、论文复现及科研仿真。 往期回顾关注个人主页Matlab科研工作室 关注我领取海量matlab电子书和数学建模资料个人信条做科研博学之、审问之、慎思之、明辨之、笃行之是为博学慎思明辨笃行。 内容介绍在能源结构调整的大背景下天然气发电机组凭借其清洁高效的特点数量逐步增加使得电力网络与天然气网络的耦合程度不断加深。这种紧密耦合一方面提升了能源综合利用效率但另一方面不确定性新能源如风电的大规模接入给电 - 气互联系统的经济安全运行带来诸多挑战。风电功率的随机波动可能导致电力系统功率失衡进而影响天然气系统的稳定供气反之亦然。因此如何有效应对风电不确定性实现电 - 气互联系统的协同经济安全运行成为亟待解决的关键问题。应对风电不确定性的分布鲁棒机会约束方法一分布鲁棒性概念分布鲁棒性旨在处理模型中不确定参数的概率分布未知或难以精确估计的情况。在电 - 气互联系统中风电功率的不确定性使得传统基于精确概率分布的优化方法不再适用。分布鲁棒方法通过构建模糊集来描述不确定参数可能的取值范围在该模糊集内寻找使系统性能在最坏情况下仍能满足一定要求的最优解从而增强系统对不确定性的鲁棒性。二基于数据驱动的模糊集构建数据收集与分析收集少量的风电预测误差历史数据这些数据反映了风电实际功率与预测功率之间的偏差情况。通过对这些数据进行统计分析获取风电预测误差的一些矩信息如均值、方差等。模糊集形成利用这些矩信息构建与风电不确定性相关的模糊集。例如可以基于切比雪夫不等式或其他概率不等式以矩信息为基础确定模糊集的边界。该模糊集包含了所有可能的风电功率概率分布尽管我们不知道其确切形式但通过模糊集能够对不确定性进行有效界定。三机会约束转化机会约束定义机会约束是指在一定概率水平下满足某些约束条件。在电 - 气互联系统中例如要求在给定的置信水平下电力系统的功率平衡约束和天然气系统的流量平衡约束等仍能得到满足尽管存在风电功率的不确定性。转化为可求解形式将基于模糊集的机会约束问题通过数学变换转化为易于求解的形式。这通常涉及到利用对偶理论、凸优化等数学工具。例如对于一些具有特定结构的机会约束可以将其转化为线性或二次规划问题使得现有的优化求解器能够对其进行高效求解。通过这种转化在考虑风电不确定性的情况下仍能找到满足系统运行要求的最优调度方案。基于松弛交替乘子法的分布式协同运行一隐私保护需求电 - 气互联系统中电力系统和天然气系统各自拥有自身的运行数据和隐私信息如电力系统的电网拓扑、发电成本天然气系统的管道布局、气源成本等。在协同优化调度过程中双方都希望在共享必要信息以实现协同的同时保护自身的隐私不被泄露。二第三方可信任协调者假设为了实现电 - 气互联系统的分布式协同运行并保护双方隐私假设存在一个第三方可信任的协调者。该协调者不参与电 - 气系统的实际运行但负责收集和处理双方传递的信息并协调优化过程。电力系统和天然气系统将各自与优化相关的部分信息传递给协调者协调者根据这些信息进行统一的优化计算并将优化结果反馈给双方。三松弛交替乘子法原理与应用原理松弛交替乘子法是一种用于求解分布式优化问题的有效算法。它通过引入辅助变量和拉格朗日乘子将原问题分解为多个子问题然后在各个子问题之间交替迭代求解。在每次迭代中分别固定其他变量更新部分变量并通过乘子的调整来保证子问题之间的一致性。在电 - 气互联系统中的应用将电 - 气互联系统的优化问题按照电力系统和天然气系统进行分解。电力系统和天然气系统分别在本地根据自身的运行约束和部分信息进行优化计算并将结果传递给协调者。协调者利用松弛交替乘子法结合双方传递的信息对全局优化问题进行求解并将更新后的信息反馈给电力系统和天然气系统。通过多次迭代使得电 - 气互联系统逐步收敛到满足协同优化要求的分布式调度方案同时保护了双方的隐私信息。优化调度模型构建⛳️ 运行结果 部分代码function [F,h,failure] robustify(F,h,ops,w)%ROBUSTIFY Derives robust counterpart.%% [Frobust,objrobust,failure] ROBUSTIFY(F,h,options) is used to derive% the robust counterpart of an uncertain YALMIP model.%% min h(x,w)% subject to% F(x,w) () 0 for all w in W%% The constraints and objective have to satisfy a number of conditions for% the robustification to be possible. Please refer to the YALMIP Wiki for% the current assumptions.%% Some options for the robustification strategies can be altered via the% solver tag robust in sdpsettings%% robust.lplp : Controls how linear constraints with affine% parameterization in an uncertainty with polytopic% description is handled. Can be either duality or% enumeration%% robust.auxred: Controls how uncertainty dependent auxiliary variables% are handled% Can be either projection or enumeration (exact),% or none or affine (conservative)%% robust.reducedual Controls if the system equality constraints derived% when using the duality filter should be eliminated,% thus reducing the number of variables, possibly% destroying sparsity .%% robust.polya : Controls the relaxation order of polynomials. If set to% NAN, the polynomials will be eliminated by forcing the% coefficients to zero%% See also UNCERTAIN% Author Johan L鰂berg% $Id: robustify.m,v 1.55 2010-03-10 15:19:05 joloef Exp $failure 0;if nargin 3ops sdpsettings;elseif isempty(ops)ops sdpsettings;endif nargin 4w [];endif nargin1if isa(h,double)h [];endelseh [];end% We keep track of auxilliary generated variablesnInitial yalmip(nvars);% Find the scenario, extract uncertainty model and classifiy variables[UncertainModel,Uncertainty,VariableType,ops] decomposeUncertain(F,h,w,ops);x VariableType.x;w VariableType.w;if isempty(x)error(There are no decision variables in the uncertain model.)endif isempty(UncertainModel.F_xw)error(The uncertainty does not enter the model anywhere.);end% Experimental code for conic-conic caseif ops.robust.coniclp.useconicconic || ((any(is(UncertainModel.F_xw,sdp)) || any(is(UncertainModel.F_xw,socp))) (any(is(Uncertainty.F_w,sdp)) || any(is(Uncertainty.F_w,socp))))SOSModel [];for i 1:length(UncertainModel.F_xw)if any(ismember(depends(UncertainModel.F_xw(i)),getvariables(VariableType.w)))SOSModel [SOSModel, dualtososrobustness(UncertainModel.F_xw(i),Uncertainty.F_w,VariableType.w,VariableType.x,ops.robust.conicconic.tau_degree,ops.robust.conicconic.gamma_degree,ops.robust.conicconic.Z_degree)];else% Misplaced?SOSModel [SOSModel, UncertainModel.F_xw(i)];endend%SOSModel expanded(SOSModel,1);F [SOSModel, UncertainModel.F_x];h UncertainModel.h;h expanded(h,1);F expanded(F,1); % This is actually done already in expandmodel% h expanded(h,1); % But this one has to be done manuallyreturnend% FIXME: SYNC with expandmodel?if ~isempty(UncertainModel.F_x)nv yalmip(nvars);yalmip(setbounds,1:nv,repmat(-inf,nv,1),repmat(inf,nv,1));LU getbounds(UncertainModel.F_x);yalmip(setbounds,1:nv,LU(:,1),LU(:,2));end% At this point, we have to decide on the algorithm we should use for% robustifying the constraints. There are a couple of alternatives,% depending on uncertainty and constraints% 1. Polya: Polynomial uncertainty dependence, simplex uncertainty,% can only be applied on LP constraints% 2. Elimination: Last resort, tries to cancel all nonlinear uncertainties% by setting coefficients to zero% 3. Explicit: Linear uncertainty dependence, box-model uncertainty, can% only be applied on LP constraints% 4. Enumeration: Linear uncertainty dependence, polytopic uncertainty,% arbitrary type of constraints (convex)% 5. Duality: Linear uncertainty dependence, conic uncertainty, can% only be applied on LP constraints% 6. S-procedure Special case, quadratic dependence in elementwise, one% quadratic constraint in W (obsolete)% 7. Conic conic Subsumes S-procedure% Robust modelF_robust ([]);% We begin by checking to see if the user wants to apply Polyas theorem.% If that is the case, search for simplex structures, and apply Polyas.if ~isnan(ops.robust.polya) any(strcmp(Uncertainty.uncertaintyTypes,simplex)) ~ops.robust.forced_enumerationF_polya [];% Recursively apply Polya relaxation w.r.t each simplexfor i find(strcmp(Uncertainty.uncertaintyTypes,simplex))[UncertainModel.F_xw, F_polya] filter_polya(UncertainModel.F_xwF_polya,w(Uncertainty.uncertaintyGroups{i}),ops.robust.polya);end[UncertainModel.F_xw,F_robust] pruneCertain(F_polya,F_robust,UncertainModel.F_xw,w);end% LP constraints with quadratic dependence and quadratic uncertainty region% can be handled tightly using the S-procedureif (all(strcmp(Uncertainty.uncertaintyTypes,2-norm)) | all(strcmp(Uncertainty.uncertaintyTypes,quadratic))) length(Uncertainty.uncertaintyTypes)1 ~ops.robust.forced_enumeration[UncertainModel.F_xw,F_sprocedure] filter_sprocedure(UncertainModel.F_xw,w,Uncertainty.separatedZmodel,ops);F_robust F_robust F_sprocedure;end% There might still be nonlinearities left in the model. These have to be% removed. We remove all terms with w-degree larger than 1[UncertainModel.F_xw,F_elimination] filter_eliminatation(UncertainModel.F_xw,w,1,ops);F_robust F_robust F_elimination;% Equality constraints cannot be part of an uncertain problem. Any% dependence w.r.t w in equalities has to be removedF_eq extractConstraints(UncertainModel.F_xw,equality);UncertainModel.F_xw UncertainModel.F_xw - F_eq;[F_eq_left,F_eliminate_equality] filter_eliminatation(F_eq,w,0,ops);F_robust F_robust F_eliminate_equality F_eq_left;% The problem should now be linear in the uncertainty, with no uncertain% equality constraints. Hence, now we apply explicit maximization,% enumeration or duality-based robustification.% We start with the norm ballsif ~ops.robust.forced_enumerationfor i 1:length(Uncertainty.uncertaintyTypes)if ismember(Uncertainty.uncertaintyTypes{i},{1-norm,2-norm,inf-norm})F_lp extractConstraints(UncertainModel.F_xw,elementwise);UncertainModel.F_xw UncertainModel.F_xw - F_lp;F_flt filter_normball(F_lp,Uncertainty.separatedZmodel{i},x,w(Uncertainty.uncertaintyGroups{i}),w,Uncertainty.uncertaintyTypes{i},ops,VariableType);[UncertainModel.F_xw,F_robust] pruneCertain(F_flt,F_robust,UncertainModel.F_xw,w);endendend% Pick out the uncertain linear equalities and robustify using duality if% user has opted for this or the uncertainty is conic.conic ~isequal(Uncertainty.Zmodel.K.s,0) | ~isequal(Uncertainty.Zmodel.K.q,0);if (conic | isequal(ops.robust.lplp,duality)) ~ops.robust.forced_enumerationF_lp extractConstraints(UncertainModel.F_xw,elementwise);UncertainModel.F_xw UncertainModel.F_xw - F_lp;nv yalmip(nvars);F_filter filter_duality(F_lp,Uncertainty.Zmodel,x,w,ops);F_robust F_robust F_filter;if isa(F_filter,lmi) ops.verbosenewvars nnz(getvariables(F_filter)nv);disp([ - Duality introduced num2str(newvars) variables, num2str(nnz(is(F_filter,equality))) equalities, num2str(nnz(is(F_filter,elementwise))) LP inqualities and num2str(nnz(is(F_filter,sdp))nnz(is(F_filter,socp))) conic constraints]);endend% Robustify remaining uncertain LP/SOCP/SDP constraints and robustify by% enumeration.F_conic extractConstraints(UncertainModel.F_xw,{sdp,socc,elementwise});UncertainModel.F_xw UncertainModel.F_xw - F_conic;[F_temp,enumerationfailed] filter_enumeration(F_conic,Uncertainty.Zmodel,x,w,ops,Uncertainty.uncertaintyTypes,Uncertainty.separatedZmodel,VariableType);if enumerationfailed% Reset to previous stateUncertainModel.F_xw UncertainModel.F_xw F_conic;elseF_robust F_robust F_temp;endif enumerationfailed% Enumeration failed, probably due to lack of MPT. If problem is conic,% we are in trouble. If simple LP, we can resort to duality approachif conicif ops.verbosedisp( - Enumeration of uncertainty polytope failed. Missing Multiparametric Toolbox?)enderror(Enumeration failed (lacking MPT?), and due to conic constraints, duality cannot be used);elseF_lp extractConstraints(UncertainModel.F_xw,elementwise);UncertainModel.F_xw UncertainModel.F_xw - F_lp;nv yalmip(nvars);F_filter filter_duality(F_lp,Uncertainty.Zmodel,x,w,ops);if ops.verboseif isa(F_filter,lmi)disp([ - Duality introduced num2str(yalmip(nvars)-nv) variables, num2str(nnz(is(F_filter,equality))) equalities, num2str(nnz(is(F_filter,elementwise))) LP inqualities and num2str(nnz(is(F_filter,sdp))nnz(is(F_filter,socp))) conic constraints]);endendF_robust F_robust F_filter;endend% If there is anything left now, it means that we do not support it (such% as conic uncertainty in conic constraint)if length(UncertainModel.F_xw) 0if any(~islinear(UncertainModel.F_xw))error(There are some uncertain constraints which cannot be robustified by YALMIP)elseF_robust F_robust UncertainModel.F_xw;endend% Return the robustfied modelF F_robustUncertainModel.F_x;h UncertainModel.h;% The model has been expanded, so we have to remember this (trying to% expand an expanded model leads to nonconvexity error)F expanded(F,1); % This is actually done already in expandmodelh expanded(h,1); % But this one has to be done manuallynNow yalmip(nvars);if nNow nInitial% YALMIP has introduced auxilliary variables% We mark these as auxilliaryyalmip(addauxvariables,nInitial1:nNow);endif ops.verbosedisp(***** Derivation of robust counterpart done ***********************);endfunction [F_xw,F_robust] pruneCertain(F_new,F_robust,F_xw,w);for i 1:length(F_new)if ~isempty(intersect(depends(F_new(i)),depends(w)))F_xw F_xw F_new(i);elseF_robust F_robust F_new(i);endendfunction p indexIn(x,y)if ~isempty(x)for i 1:length(x)p(i) find(x(i)y);endelsep [];endfunction [F_x,F_w,F_xw,VariableType] partitionModel(F,F_original,VariableType);F_x [];F_w [];F_xw [];% x-var w_var aux_xw aux_wif ~(isempty(VariableType.aux_with_w_dependence) isempty(VariableType.aux_with_only_w_dependence))Dependency spalloc(length(F_original),4,length(F));for i 1:length(F_original)varF depends(F_original(i));Dependency(i,1) any(ismember(varF,VariableType.x_variables));Dependency(i,2) any(ismember(varF,VariableType.w_variables));Dependency(i,3) any(ismember(varF,VariableType.aux_with_w_dependence));Dependency(i,4) any(ismember(varF,VariableType.aux_with_only_w_dependence));endLiftedUncertaintiesDescription find(Dependency(:,1) 0 Dependency(:,3)0);if ~isempty(LiftedUncertaintiesDescription)% for i LiftedUncertaintiesDescription(:)% vars depends(F_original(i));% vars intersect(vars,VariableType.aux_with_only_w_dependence);%% endreclassifyAsUncertain depends(F_original(LiftedUncertaintiesDescription));[notused,reclassifyAsUncertain] find(VariableType.Graph(reclassifyAsUncertain,:));reclassifyAsUncertain unique(reclassifyAsUncertain);reclassifyAsUncertain intersect(unique(reclassifyAsUncertain),getvariables(F));VariableType.aux_with_only_w_dependence setdiff(VariableType.aux_with_only_w_dependence,reclassifyAsUncertain);VariableType.w_variables union(VariableType.w_variables,reclassifyAsUncertain);VariableType.aux_with_w_dependence union(VariableType.aux_with_w_dependence,VariableType.aux_with_only_w_dependence);VariableType.aux_with_only_w_dependence [];endend% x-var w_var aux_xw aux_wDependency spalloc(length(F),4,length(F));for i 1:length(F)varF depends(F(i));Dependency(i,1) any(ismember(varF,VariableType.x_variables));Dependency(i,2) any(ismember(varF,VariableType.w_variables));Dependency(i,3) any(ismember(varF,VariableType.aux_with_w_dependence));Dependency(i,4) any(ismember(varF,VariableType.aux_with_only_w_dependence));endpureX find(Dependency*[1;2;4;8] 1);pureW find(Dependency(:,1) 0 Dependency(:,3)0);mixedXW find(Dependency(:,1) | Dependency(:,3));%mixedXW find((Dependency(:,1) | Dependency(:,3));mixedXW setdiff(1:size(Dependency,1),union(pureW,pureX));F_x F(pureX);F_w F(pureW);F_xw F(mixedXW);reclassifyAsUncertain depends(F_w);VariableType.aux_with_only_w_dependence setdiff(VariableType.aux_with_only_w_dependence,reclassifyAsUncertain);VariableType.w_variables union(VariableType.w_variables,reclassifyAsUncertain);function [VariableType,h_fixed,F_xw] reformatObjective(h,F_xw,VariableType)% Some pre-calcx recover(VariableType.x_variables);w recover(VariableType.w_variables);xw [x;w];xind find(ismembcYALMIP(getvariables(xw),getvariables(x)));wind find(ismembcYALMIP(getvariables(xw),getvariables(w)));% Analyze the objective and try to rewrite any uncertainty into the format% assumed by YALMIPif ~isempty(h)[Q,c,f,dummy,nonquadratic] vecquaddecomp(h,xw);Q Q{1};c c{1};f f{1};if nonquadraticerror(Objective can be at most quadratic, with the linear term uncertain);endQ_ww Q(wind,wind);Q_xw Q(xind,wind);Q_xx Q(xind,xind);c_x c(xind);c_w c(wind);if nnz(Q_ww) 0error(Objective can be at most quadratic, with the linear term uncertain);end% Separate certain and uncertain terms, place uncertain terms in the% constraints insteadif is(h,linear)if isempty(intersect(getvariables(w),getvariables(h)))h_fixed h;elsesdpvar tF_xw F_xw (h t);h_fixed t;x [x;t];endelseh_fixed x*Q_xx*x c_x*x f;h_uncertain 2*w*Q_xw*x c_w*w;if ~isa(h_uncertain,double)sdpvar tF_xw F_xw (h_uncertain t);h_fixed h_fixed t;x [x;t];endendelseh_fixed [];endVariableType.x_variables getvariables(x); 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