Models And Algorithms Of Reactive Power Optimization Considering Uncertainties Based On Interval Theory

There are many kinds of uncertainties existing in power system data,such as intermittence of renewable power generation,uncertainty of load demand,and measurement errors of network parameters.These uncertainties cause the fluctuation and flicker,even the overrun more seriously,of the voltage,and it will threaten the operation security of the power grid.To ensure the voltage security under uncertain circumanstances,two kinds of solution methods mainly have been proposed at present,i.e.,1)stochastic programming-based reactive power optimization;2)robust reactive power optimizatio.However,the stochastic programmingbased method should collect amounts of data for constructing the probability distribution functions of the uncertainties,and its Monte Carlo simulation greatly increases the computating burden,not to say the obtained voltage control stategy cannot ensure the voltage reside in the security limits in theory.The power flow equations should be convexified before the robust programming method can be applied to solve the reactive power optimization models,therefore the obtained voltage control strategy also cannot ensure the voltage satisfying the security constraints of original models.To overcome the demerits of the aforementioned two methods,this paper proposes the solution algorithms and models of reactive power optimization considering uncertainties based on interval theory.The interval reactive power optimization model regards the uncertain input data(e.g.,renewable power generation and load demand)as intervals,the state variables(e.g.,load voltage,angle,and reactive power generation)as intervals,control variables(e.g.,ratios of transformers,VAR compensator output,and voltage of generators)as real numbers.The meaning of this model is to find a group of optimal control variables for ensuring the intervals of the state variables residing in the set security constraints,as well as minimizing the level of the operation cost(e.g.,real power losses).The interval reactive power optimization model is a discrete non-convex nonlinear programming including interval parameters,and no effective solution algorithms are proposed to solve this kind of programming at present.To solve the interval reactive power optimization moedel,this paper first proposed the interval power flow algorithms to solve the interval power flow equations,thus dealing with the interval nonlinear equaitons.Consequently,we proposed four different solution algorithms for solving interval reactive power optimization model.The main work of this paper mainly including:1)Constructing the interval(dynamic)reactive power optimization model.The model expresses the uncertain power data as intervals and regards the real power losses as objective function,and it takes the operating and security constraints of the power grid into accunt.Its target is to obtain an optimal(dynamic)voltage control strategy for ensuring the state variables satisfying the security constraints under uncertain circumanstances.2)Proposing the interval power flow algorithms with better accuracy.The previously proposed most effective interval power flow algorithms are based on the affine arithmetic theory,and they use the linear combination of the noise elements to approach the power flow intervals essentially,thus producing the Chebyshev approximation errors.To improve the accuracy of interval power flow results,on one hand,we proposed an affine arithmetic-based interval power flow algorithm under the mixed coordination axis,which can reduce the errors caused by the Chebyshev approximation;on the other hand,we proposed an interval power flow algorithm based on an optimizing-scenarios method,which can obtain the accurate interval power flow results,due to no usage of approximation computations.3)Putting forward interval reactive power optimization solution algorithms based on the improved genetic algorithms.On one hand,the non-dominated sorting is introduced to obtain the Pareto front of the interval reactive power optimization model.On the other hand,to enhance the computation efficiency and obtain better results of genetic algorithm,the interval power flow algorithm with more accurate results is adopted,and the constraints are expressed as penalty functions.Meanwhile,the adaptive genetic algorithm is proposed to solve the interval reactive power optimization model.4)Raising a linear approximation method for solving the interval reactive power optimization model.The Taylor expansion of interval functions are applied to linearize the interval reactive power optimization model,thus constructing the linear approximation iteration algorithm.In addition,the positive-curve penalty function is used to deal with the discrete variables,and the interval power flow algorithm is utilized to improve the precision and convergence of the proposed algorithm.5)Presenting the interval reactive power optimization algorithm based on an interval sequential quadratic programming.Based on the linear approximation method,we keep the quadratic items of interval function Taylor expansion,thus establishing the interval sequential quadratic programming iteration algorithm,which is used to solve the interval dynamic reactive power optimization model.To improve the performance of the proposed algorithm,the optimizing-scenarios method is used to obtain more accurate interval results of state variables,thus expanding the feasible area of iteration models of the proposed algorithm.6)Proposing the interval reactive power optimization algorithm based on the definition of the security limits.We defined the upper and lower security limits for the interval reactive power optimization model,by which the interval reactive power optimization model is converted to a deterministic reactive power optimization model,where the upper and lower limits of the state variables are the formerly defined upper and lower security limits,respectively.By solving the deterministic reactive power optimization model,the voltage control strategy satisfying the security limits is obtained,and it is also validated that the obtained strategy can ensure the interval state variables of the interval reactive power optimization model within their upper and lower limits.By using the aforementioned four different interval reactive power optimization algorithms,we finally solve the voltage overrun problem under various uncertainties,as well as maintaining a relatively lower level of real power losses.The merits of using models and algorithms for interval reactive power optimization mainly include: 1)Its obtained voltage control strategy can ensure the load voltage within the security state under the uncertain circumstances all the time;2)The information of uncertainty incorporated in the model is easy to acquire;3)Its solution efficiency is much higher than that of stochastic programming models,which validates its engineering application value.