CN101944742A - Improved power flow feasible solution recovering method - Google Patents

Abstract

The invention provides a simple and convenient feasible improved power flow feasible solution recovering method, relating to the on-line safe monitoring of an electric system. The simple and convenient feasible improved power flow feasible solution recovering method comprises the following two substeps of: controlling quantity calculation and result checking which are iteratively carried out in the recovering process: the controlling quantity calculation step comprises the following procedures of: firstly solving a feasible region boundary point of a power injection space by dint of an optimal multiplier power flow, introducing a recovering parameter lambda, and calculating the sensitivity of the lambda on various control measures, and then estimating the adjusting quantity of various control measures on the basis of a sensitivity calculation result through mixed integer linear programming; and the result checking step comprises the following procedures of: verifying control effect through the optimal multiplier power flow, and returning to the controlling quantity calculation step to repeat the step till power flow calculation is convergent when the control measures can not ensure the power flow to be convergent because control strategies formed on the basis of the sensitivity which is the linear approximation of nonlinear relationship often have certain errors. The invention is mainly used for the electric system.

Description

Improved trend feasible solution restoration methods

Technical field

The present invention relates to the safety on line monitoring of electric power system, specifically relate to improved trend feasible solution restoration methods.

Background technology

The fast development of power industry, the dramatic growth of load and electric power enterprise all make the running environment of electric power system become increasingly complex to the pursuit of economy.How in economic pursuit, to guarantee its reliability, this to electric power system make rational planning for and safety on line monitoring is had higher requirement.

It is that system personnel is planned the important tool with on-line monitoring that trend is calculated.Whether exist trend to separate under the forecast accident condition is the problem that the operations staff very is concerned about.Cause trend to be calculated dispersing existing two kinds may: the one, this trend problem has itself separates and because the problem of numerical computation method causes computational process to be dispersed; The 2nd, there is not feasible point [1] in the system after the fault.Extend two subproblems thus: the one, when truly separating when existing, how to improve convergence, guarantee that its numerical convergence separates to trend; The 2nd, when truly separating when not existing, how by taking control measure to make trend separate recovery.

At first problem, optimum multiplier Newton method under the rectangular coordinate system has been proposed in the document [2], by introducing optimum multiplier the direction that Newton method produces is revised, that can effectively avoid calculating disperses, and [3] promote the use of optimum multiplier method under the polar coordinate system.[4] provided a kind of homotopy power flow algorithm of interior convergence on a large scale, [5] have provided a kind of serialization Newton method and have proved that itself and Homotopy Method are identical in essence.[6] by power flow equation being carried out again parametrization to improve its convergence, [7] have then proposed a kind of continuous tide method of fault parameterization.In the above method, the method for [2-3] only needs to increase a little amount of calculation on traditional Newton method basis, and the method amount of calculation of document [4-7] is relatively large.

At second problem, [8] the shortest Euclidean distance of injecting between infeasible operating point in space and the feasible zone border by power is measured trend intangibility degree, [9] having proposed linear programming model based on sensitivity on this basis recovers trend and separates, the control effect is better, but, in big system, calculate and have certain difficulty because this method need be calculated the left eigenvector of Jacobian matrix zero eigenvalue; [10-11] separates trend and recovers to be treated to nonlinear programming problem, and adopts interior point method to be found the solution, and its amount of calculation is bigger when big system applies; [11] trend is not had the reason of separating and be summed up as weak passway for transmitting electricity and surpass the transmission power limit, and recover trend and separate, but this method needs more artificial participation by alleviating weak passage loading level.

Summary of the invention

For overcoming the deficiencies in the prior art, a kind of simple and feasible improved trend feasible solution restoration methods is provided, solution is when truly separating when not existing, how by taking control measure to make trend separate recovery, the technical scheme that the present invention takes is, a kind of improved trend feasible solution restoration methods, recovery process is carried out controlled quentity controlled variable iteratively and calculated and two sub-steps of verification as a result: controlled quentity controlled variable is calculated the step and is at first tried to achieve the feasible zone boundary point that power injects the space by optimum multiplier trend, introduce and recover parameter lambda, calculate the sensitivity of λ, estimate the adjustment amount of each control device then based on the sensitivity calculations result by MILP each control measure; The verification step is then verified the control effect by optimum multiplier trend method, because sensitivity is the linear approximation to non-linear relation, the control strategy that forms based on it often has certain error, when control measure can not guarantee the trend convergence, then return controlled quentity controlled variable and calculate the step, repeat said process, calculate convergence until trend.

Obtain the trend feasible zone boundary point of power space based on optimum multiplier trend:

The electric power system tide equation can be written as:

F(x)＝f(x)-S＝0 (1)

Be constructed as follows the system power amount of not matching:

$g\left(x\right)=\frac{1}{2}{(f\left(x\right)-S)}^T(f\left(x\right)-S)---\left(2\right)$

X is a voltage vector in the formula, S is that node power injects and PV node voltage square vector, f (x) is node power equilibrium equation and PV node voltage constraint equation, adopt Newton method by constantly minimizing g (x) reaching the purpose of trend convergence, the iterative formula of Newton method be rewritten as by introducing optimum multiplier μ:

Δx ^k＝-J ^-1(x ^k)f(x ^k)

(3)

x ^k+1＝x ^k+μ ^kΔx ^k

Thereby in each iterative computation, the power amount of not matching is changed into:

$g(x,{\mathrm{\μ}}^k)=\frac{1}{2}{(f_1(x^k,{\mathrm{\μ}}^k)-S)}^T(f_1(x^k,{\mathrm{\μ}}^k)-S)---\left(4\right)$

Wherein: f ₁(x ^k, μ ^k)=f (x ^k+ μ ^kΔ x ^k), J (x ^k) be the Jacobian matrix of k step iteration, μ ^kBe the optimum multiplier of k step iteration, μ ^kEach the step iteration in by

Try to achieve, thereby guarantee that the amount of not matching progressively diminishes, if trend does not have feasible solution, then from initial infeasible solution, this method optimum multiplier μ after iteration repeatedly will be tending towards 0, function (4) value that do not match will be stabilized in certain on the occasion of last, promptly converge on power this moment and inject SNB point on the feasible zone boundary point of space.

Introduce and recover parameter lambda, use the sensitivity λ that recovers the parameter lambda control measure _uRecover trend and separate, determined a feasible boundary point x by optimum multiplier trend method ^*After being the SNB point, corresponding meritorious, idle injection vector are P ^*, Q ^*, power balance equation can be rewritten as:

$0=-e_i\underset{j\∈i}{\mathrm{\Σ}}(G_{\mathrm{ij}}{e}_j-B_{\mathrm{ij}}{f}_j)-f_i\underset{j\∈i}{\mathrm{\Σ}}(G_{\mathrm{ij}}{f}_j+B_{\mathrm{ij}}{e}_j)$

$-P_i^{*}+\mathrm{\λ}(P_i^{*}-P_{\mathrm{si}})$ $\left(5\right)$

$0=-f_i\underset{j\∈i}{\mathrm{\Σ}}(G_{\mathrm{ij}}{e}_j-B_{\mathrm{ij}}{f}_j)+e_i\underset{j\∈i}{\mathrm{\Σ}}(G_{\mathrm{ij}}{f}_j+B_{\mathrm{ij}}{e}_j)$

$-{Q_i}^{*}+\mathrm{\λ}({Q_i}^{*}-Q_{\mathrm{si}})$

E, f are the real part and the imaginary part of node voltage, and G, B are the real part and the imaginary part of line admittance, P _s, Q _sMeritorious, idle injection for initial point.At x ^*λ=0, place, and when λ=1, formula (5) is equivalent to original power flow equation, use sensitivity λ _u, can avoid asking for of Jacobian matrix zero eigenvalue left eigenvector by the following method:

Electric power system is at its steady stability operational limit (SNB) point (x ^*, λ ^*) locate, will satisfy following equation:

F(x，λ，u)＝0

F _x(x，λ，u)·v＝0 (6)

||v||≠0

F is a power flow equation, is x, u, the function of λ; U=(u ₁, u ₂..., u _m) be control measure, but control measure comprise switching reactive power compensation in parallel, OLTC no-load voltage ratio, series compensation and generator output adjustment; F _xBe the Jacobian matrix of system at this some place, v is its right characteristic vector; λ is a voltage stability margin, can try to achieve its sensitivity vector λ to u by following formula _u:

Λ·s＝-F _u (7a)

A _iBe the column vector of trend Jacobian matrix correspondence, the Λ matrix is that Jacobian matrix p is listed as by F _λReplace λ _uBe p element among the solution vector s, more than one when Control Parameter, when supposing to have m, (7a) formula can be rewritten as following form:

$\left[\begin{array}{cccc}{s}_{11}& s_{12}& ...& s_{1m}\\ s_{21}& s_{22}& ...& s_{2m}\\ M& M& & M\\ s_{(p-1)1}& s_{(p-1)2}& ...& s_{(p-1)m}\\ \frac{\mathrm{d\λ}}{{\mathrm{du}}_1}& \frac{\mathrm{d\λ}}{{\mathrm{du}}_2}& ...& \frac{\mathrm{d\λ}}{{\mathrm{du}}_m}\\ s_{(p+1)1}& s_{(p+1)2}& ...& s_{(p+1)m}\\ M& M& & M\\ s_{n1}& s_{n2}& ...& s_{\mathrm{nm}}\end{array}\right]={\mathrm{\Λ}}^{-1}\·\left[\begin{array}{cccc}\frac{{\∂F}_1}{{\∂u}_1}& \frac{{\∂F}_1}{{\∂u}_2}& \mathrm{\Λ}& \frac{{\∂F}_1}{{\∂u}_m}\\ \frac{\∂F_2}{{\∂u}_1}& \frac{{\∂F}_2}{{\∂u}_2}& \mathrm{\Λ}& \frac{{\∂F}_2}{{\∂u}_m}\\ & & M& \\ \frac{{\∂F}_n}{{\∂U}_1}& \frac{{\∂F}_n}{{\∂u}_2}& \mathrm{\Λ}& \frac{{\∂F}_n}{{\∂u}_m}\end{array}\right]---\left(8\right).$

Calculate the SNB point by optimum multiplier trend, the λ in the computing formula (8) _u, by the control measure screening, judge whether to exist control measure, do not reduce load if do not exist then by the node amount of not matching, then adopt MILP MILP model to determine its minimum control cost if exist:

$\min =\underset{i\∈n}{\mathrm{\Σ}}{c}_i*x_i+\underset{j\∈m}{\mathrm{\Σ}}{c}_j*y_j---\left(9a\right)$

$s.t.\underset{i\∈n}{\mathrm{\Σ}}{x}_i*{\mathrm{\λ}}_{x_i}+\underset{j\∈m}{\mathrm{\Σ}}{y}_i*{\mathrm{step}}_j*{\mathrm{\λ}}_{y_j}>=\mathrm{\Δ\λ}---\left(9b\right)$

$\underset{\‾}{x_i}\≤x_i\≤\stackrel{\‾}{x_i},i\∈n---\left(9c\right)$

$\underset{\‾}{y_j}\≤y_j\≤\stackrel{\‾}{y_j},j\∈m,y_j\∈Z---\left(9d\right)$

X wherein _iAnd y _iRepresent continuous and discrete control variables respectively, c is the expense of control.Step _jBe the step-length of discrete variable control, x _i , y _i Be the lower limit of corresponding control variables, Be the upper limit, Δ λ=1.0 o'clock are represented that trend is separated and are recovered.

Its characteristics of the present invention are:

1, utilizes the result of calculation of optimum multiplier Newton method, by introducing new control measure sensitivity, but avoided feasible zone border asking for of solution point and power flow equation critical point Jacobian matrix zero eigenvalue left eigenvector recently, improved computational efficiency.

2, further based on carrying sensitivity, the recovery that trend is separated is converted into a MILP problem and is found the solution, and the preferred dimension of finding the solution that has reduced problem by control measure.

Description of drawings

Fig. 1 trend is separated the recovery flow chart.

Schematic diagram is found the solution in two kinds of sensitivity of Fig. 2.

The improved controlled quentity controlled variable of Fig. 3 is calculated.

Fig. 4 IEEE118 node example subregion line chart.

Embodiment

When serious forecast accident occurred and causes system load flow to be dispersed, definite fast control strategy is separated with the recovery system trend, and was great for power system planning and safety on line meaning of monitoring.This method provides a kind of higher trend feasible solution recovery policy of efficient of finding the solution, utilize the result of calculation of optimum multiplier Newton method, by introducing new control measure sensitivity, but avoided asking for of nearest solution point in feasible zone border and power flow equation critical point Jacobian matrix zero eigenvalue left eigenvector, improved computational efficiency; Further based on carrying sensitivity, the recovery that trend is separated is converted into a MILP problem and is found the solution, and the preferred dimension of finding the solution that has reduced problem by control measure; Utilize New England-39 node and IEEE-118 node example to verify the validity of method.

Further describe the present invention below in conjunction with drawings and Examples.

1 linear programming trend recovery policy based on sensitivity

[8,9] but the intangibility degree of power flow equation is injected the infeasible operating point in space by power measures with the Euclidean distance (being designated as θ) between the nearest solution point (CBP) in feasible zone border, and trend to separate the controlled target of recovery be exactly that θ is decreased to zero and the cost minimum of control, its recovery policy flow chart is as follows: Fig. 1 trend is separated and is recovered flow chart (Fig1 Flow chart of load flowsolution restoration).

The trend recovery policy is divided into mainly that controlled quentity controlled variable is calculated and verification two parts as a result: the former at first asks for nearest border feasible point (CBP), further obtains the sensitivity θ of θ to control measure _u, then based on θ _uEstimate the adjustment amount of each control device, and then form corresponding trend recovery policy; Checking procedure is then verified the control effect by optimum multiplier trend method.Because sensitivity is the linear approximation to non-linear relation, the control strategy that forms based on its often has certain error, and when control measure can not guarantee the trend convergence, then optimum multiplier trend was tried to achieve one more near the convergence point on feasible zone border, repeat said process, restrain until trend.Two critical process of above-mentioned definite trend recovery policy are CBP and θ _uAsk for and the formation of trend recovery policy.

1. CBP and θ _uAsk for

Asking for of CBP need be by means of optimum multiplier Newton method, and [8] have proved that by optimum multiplier power flow algorithm, system necessarily can converge to the border of feasible zone, and this has guaranteed Fig. 1 convergence.

Optimum multiplier Newton method utilizes that power flow equation is these characteristics of quadratic function of variable under the rectangular coordinate, in iteration, the direction that Newton method produces is revised with optimum multiplier μ, avoided revising and owe correction to the mistake of voltage vector, the assurance iterative process is not dispersed.For the electric power system tide equation:

F(x)＝f(x)-S＝0 (1)

Can be constructed as follows the system power amount of not matching:

$g\left(x\right)=\frac{1}{2}{(f\left(x\right)-S)}^T(f\left(x\right)-S)---\left(2\right)$

X is a voltage vector in the formula, and S is that node power injects and PV node voltage square vector, and f (x) is node power equilibrium equation and PV node voltage constraint equation.Newton method is by constantly minimizing g (x) reaching the purpose of trend convergence, but owing to cross to revise or owe and revise, can cause Newton method to be dispersed ^[2]Be to improve the convergence that trend is calculated, optimum multiplier Newton method is rewritten as the iterative formula of Newton method by introducing optimum multiplier μ:

Δx ^k＝-J ^-1(x ^k)f(x ^k)

(3)

x ^k+1＝x ^k+μ ^kΔx ^k

Thereby in each iterative computation, the power amount of not matching is changed into:

$g(x,{\mathrm{\μ}}^k)=\frac{1}{2}{(f_1(x^k,{\mathrm{\μ}}^k)-S)}^T(f_1(x^k,{\mathrm{\μ}}^k)-S)---\left(4\right)$

Wherein: f ₁(x ^k, μ ^k)=f (x ^k+ μ ^kΔ x ^k), J (x ^k) be the Jacobian matrix of k step iteration, μ ^kBe the optimum multiplier of k step iteration, μ ^kEach the step iteration in by

Try to achieve, thereby guarantee that the amount of not matching progressively diminishes.From initial infeasible solution, this method optimum multiplier μ after iteration repeatedly will be tending towards 0, and the function that do not match (4) value will be stabilized in certain on the occasion of last, promptly converge on power this moment and inject on the feasible zone boundary point of space (SNB point) ^[8]

The convergence point of one suboptimum multiplier trend can not guarantee it is the CBP of system, and CBP generally need obtain by two-layer iterative process: external iteration is constantly adjusted power and is injected vector with convergence CBP, and number of iterations depends on the evenness on feasible zone border; The internal layer iteration is carried out the calculating of optimum multiplier trend, and iterations depends on the convergence of optimum multiplier algorithm at this point.After CBP tried to achieve, θ can be expressed as formula (5):

θ＝||S-S _m|| (5)

S and S _mThe power that is respectively system's initial point and CBP place injects vector, and θ can be obtained by following formula the sensitivity of control measure u ^[9]:

${\mathrm{\θ}}_u=-{\mathrm{\ω}}^m*\∂(f\left(x_m\right)-S)/\∂u---\left(6\right)$

In the formula, ω ^mLeft eigenvector for CBP point place Jacobian matrix zero eigenvalue.Work as S _mWhen injecting for the power of CBP correspondence, vectorial S-S _mWill with ω ^mParallel.Further, need the power adjusted to inject in the direction and satisfy following formula:

θ＝ω ^m*(S-S _m)＝-ω _m*[f(x _m)-S] (7)

Formula (7) both sides then can get formula (6) to the u differentiate.

2. determine based on the trend recovery policy of sensitivity

Power injects the sensitivity θ of the shortest Euclidean distance in space to control measure _uAfter trying to achieve, each control measure adjustment amount that the recovery trend is separated can obtain by finding the solution following linear programming problem:

$\min =\underset{i\∈m}{\mathrm{\Σ}}{c}_i*u_i---\left(8a\right)$

$s.t.\underset{i\∈m}{\mathrm{\Σ}}{u}_i*{\mathrm{\θ}}_{u_i}=\mathrm{\θ}---\left(8b\right)$

$\underset{\&OverBar;}{u_i}<u_i\&le;\stackrel{\&OverBar;}{u_i},i\&Element;m---\left(8c\right)$

u _iRepresent i control device, as generator output adjustment, reduction load, input reactive power compensation in parallel etc.,

And u _iThe bound of expression adjusting range; c _iThe adjustment expense of expression control device;

For θ to control variables u _iSensitivity, satisfy

Formula (8a) is the optimization aim of problem, i.e. master control expense minimum; Formula (8b) is separated for trend and is recovered constraint, promptly gives regularly at θ, by optimizing to determine best control measure, guarantees the regulate expenditure minimum; The restriction range that formula (8c) is adjusted for controlled quentity controlled variable.

2 improved control strategy model and methods

1 method that saves has convergence and control effect preferably through a large amount of example checkings ^[8,9], but find that by analyzing us there are following 2 deficiencies in it: 1) in asking for the CBP of system process, when power injected near the feasible zone border, the iterations of optimum multiplier algorithm increased greatly, caused amount of calculation to increase; 2) need the left eigenvector of accurate Calculation CBP place Jacobian matrix zero eigenvalue, when large scale system was used, its difficulty in computation was bigger.At above-mentioned deficiency, this paper avoids the calculating of Jacobian matrix zero eigenvalue left eigenvector by introducing new parametric sensitivity, and the efficient of finding the solution that preferably improves whole algorithm by control measure.

1. improve Control Parameter sensitivity

At first this paper uses for reference [13] method, proposes a kind of new sensitivity λ _uThe θ that replaces former method _u, to avoid asking for of CBP and Jacobian matrix zero eigenvalue left eigenvector.

Electric power system is at its steady stability operational limit (SNB) point (x ^*, λ ^*) locate, will satisfy following equation:

F(x，λ，u)＝0

F _x(x，λ，u)·v＝0 (9)

||v||≠0

F is a power flow equation, is x, u, the function of λ; U=(u ₁, u ₂..., u _m) be control measure (but as switching reactive power compensation in parallel, OLTC no-load voltage ratio, series compensation and generator output adjustment etc.); F _xBe the Jacobian matrix of system at this some place, v is its right characteristic vector; λ is a voltage stability margin, can try to achieve its sensitivity vector λ to u by following formula _u ^[12]:

Λ·s＝-F _u (10a)

A _iBe the column vector of trend Jacobian matrix correspondence, the Λ matrix is that Jacobian matrix p is listed as by F _λReplace.λ _uBe p element among the solution vector s.When more than one of Control Parameter (suppose to have m), (10a) formula can be rewritten as following form:

$\left[\begin{array}{cccc}{s}_{11}& s_{12}& ...& s_{1m}\\ s_{21}& s_{22}& ...& s_{2m}\\ M& M& & M\\ s_{(p-1)1}& s_{(p-1)2}& ...& s_{(p-1)m}\\ \frac{\mathrm{d\&lambda;}}{{\mathrm{du}}_1}& \frac{\mathrm{d\&lambda;}}{{\mathrm{du}}_2}& ...& \frac{\mathrm{d\&lambda;}}{{\mathrm{du}}_m}\\ s_{(p+1)1}& s_{(p+1)2}& ...& s_{(p+1)m}\\ M& M& & M\\ s_{n1}& s_{n2}& ...& s_{\mathrm{nm}}\end{array}\right]={\mathrm{\&Lambda;}}^{-1}\&CenterDot;\left[\begin{array}{cccc}\frac{{\&PartialD;F}_1}{{\&PartialD;u}_1}& \frac{{\&PartialD;F}_1}{{\&PartialD;u}_2}& \mathrm{\&Lambda;}& \frac{{\&PartialD;F}_1}{{\&PartialD;u}_m}\\ \frac{\&PartialD;F_2}{{\&PartialD;u}_1}& \frac{{\&PartialD;F}_2}{{\&PartialD;u}_2}& \mathrm{\&Lambda;}& \frac{{\&PartialD;F}_2}{{\&PartialD;u}_m}\\ & & M& \\ \frac{{\&PartialD;F}_n}{{\&PartialD;U}_1}& \frac{{\&PartialD;F}_n}{{\&PartialD;u}_2}& \mathrm{\&Lambda;}& \frac{{\&PartialD;F}_n}{{\&PartialD;u}_m}\end{array}\right]---\left(11\right)$

The capable sensitivity that is relative each Control Parameter of λ of formula (11) left side p.Be not difficult to find out, when calculating the sensitivity information of a plurality of Control Parameter, only need carry out finding the solution of linear function (11).In finding the solution, F derives simple relatively to the partial derivative of static reactive, generator output and line admittance, repeat no more, to the derivation of the partial derivative of transformer voltage ratio referring to appendix.

This paper is with λ _uBe used for the recovery that trend is separated, because in this article, λ no longer has the meaning of load margin, so be referred to as to recover parameter.It realizes that principle is as follows, and under rectangular coordinate system, the original power equation of system is:

$0=-e_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{e}_j-B_{\mathrm{ij}}{f}_j)$

$-f_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{f}_j+B_{\mathrm{ij}}{e}_j)-P_{\mathrm{si}}$ $\left(12\right)$

$0=-f_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{e}_j+B_{\mathrm{ij}}{f}_j)$

$+e_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{f}_j+B_{\mathrm{ij}}{e}_j)-Q_{\mathrm{si}}$

E, f are the real part and the imaginary part of node voltage, and G, B are the real part and the imaginary part of line admittance, P _s, Q _sMeritorious, idle injection for initial point.After having determined a feasible boundary point x* by optimum multiplier trend method, corresponding meritorious, idle injection vector are P ^*, Q ^*, formula (12) can be rewritten as:

$0=-e_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{e}_j-B_{\mathrm{ij}}{f}_j)-f_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{f}_j+B_{\mathrm{ij}}{e}_j)$

$-P_i^{*}+\mathrm{\&lambda;}(P_i^{*}-P_{\mathrm{si}})$ $\left(13\right)$

$0=-f_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{e}_j-B_{\mathrm{ij}}{f}_j)+e_i\underset{j\&Element;i}{\mathrm{\&Sigma;}}(G_{\mathrm{ij}}{f}_j+B_{\mathrm{ij}}{e}_j)$

$-{Q_i}^{*}+\mathrm{\&lambda;}({Q_i}^{*}-Q_{\mathrm{si}})$

At x ^*λ=0, place, and when λ=1, formula (13) is equivalent to original power flow equation (12).By introducing parameter lambda with θ _uAsk for and be converted into λ _uAsk for, the benefit of doing has like this avoided iteration to ask for a large amount of calculating that CBP brings on the one hand, only needs a suboptimum multiplier trend to calculate, and tries to achieve the sensitivity λ that can obtain this some place behind the SNB point fast _uAvoid asking for SNB place Jacobian matrix zero eigenvalue left eigenvector on the other hand, significantly reduced amount of calculation.

θ _uAnd λ _uThe schematic diagram (Fig. 2) that can be in conjunction with following one the two-dimentional power of the concrete implication of two kinds of sensitivity injects the space is illustrated: schematic diagram (Fig2 relationship between θ is found the solution in two kinds of sensitivity of Fig. 2 _uAnd λ _u)

1. θ _uAsk at first and need track S _m, draw according to formula (6) then || S-S _m|| to the sensitivity of u, can be with the feasible zone border extension to S; But owing to reasons such as Algorithm Error, optimum multiplier trend method does not track S accurately _mIn time, (supposed at S ¹Point), then be equivalent to along ω according to the result of calculation of formula (6) ¹(ω ¹Be S ¹Normal vector corresponding to section, feasible zone border) with the feasible zone border extension || S-S ¹|| distance to the A point.θ _uPrerequisite is S accurately ¹With S _mFully approaching, ω ¹With ω _mApproximate parallel, this also is a major reason that increases former method amount of calculation.

2. λ _uonly ask for and to need be calculated to S by a suboptimum multiplier trend ¹(demand is not separated CBP), its implication are along vector S-S ¹Direction expansion feasible zone border, recover the trend solvability.Owing to do not use S _mPlace's Jacobian matrix zero eigenvalue left eigenvector, it does not need accurately to ask for S _m, the process of therefore asking for is simple relatively and implication is clear and definite, has simplified the Sensitivity calculation process greatly.

2. improved controlled quentity controlled variable is calculated

Introduce improved sensitivity λ _uAfter, the controlled quentity controlled variable computational process among Fig. 1 is amended as follows shown in the figure: the improved controlled quentity controlled variable of Fig. 3 is calculated (Fig3modified control variables computation).

For the improvement of Fig. 3, what time have needs explanation:

1. as previously mentioned, utilize sensitivity λ _uReplace θ _uAfter, solution procedure can be simplified greatly, thereby reduces the amount of calculation of algorithm.

2. control measure is preferred: find by studying us, although all can separating to improve to trend, various control devices play a role, but have only some crucial control devices to play a crucial role, control measure are divided into crucial and non-key two classes: the sensitivity of key measure is along with S-S for this reason ¹Euclidean distance reduce rapid increase, show that the recovery that it is separated trend plays more and more important function; But not the sensitivity meeting of key measure is along with S-S ¹The reducing of Euclidean distance changes not obvious even reduces.In this paper recovery policy iterative process, by the relatively variation of front and back control measure sensitivity, only select crucial control measure to participate in the improvement that trend is separated, guaranteed the effect of control measure on the one hand, significantly reduce the search volume of optimizing process on the other hand, further improved computational speed.

3. in the electric power system control means, but discrete variables such as switching reactive-load compensation equipment group number, transformer voltage ratio had both been comprised, also comprise continuous variables such as generator output adjustment, so determine its minimum control cost by following MILP (MILP) model:

$\min =\underset{i\&Element;n}{\mathrm{\&Sigma;}}{c}_i*x_i+\underset{j\&Element;m}{\mathrm{\&Sigma;}}{c}_j*y_j---\left(14a\right)$

$s.t.\underset{i\&Element;n}{\mathrm{\&Sigma;}}{x}_i*{\mathrm{\&lambda;}}_{x_i}+\underset{j\&Element;m}{\mathrm{\&Sigma;}}{y}_i*{\mathrm{step}}_j*{\mathrm{\&lambda;}}_{y_j}>=\mathrm{\&Delta;\&lambda;}---\left(14b\right)$

$\underset{\&OverBar;}{x_i}\&le;x_i\&le;\stackrel{\&OverBar;}{x_i},i\&Element;n---\left(14c\right)$

$\underset{\&OverBar;}{y_j}\&le;y_j\&le;\stackrel{\&OverBar;}{y_j},j\&Element;m,y_j\&Element;Z---\left(14d\right)$

X wherein _iAnd y _iRepresent continuous and discrete control variables respectively, stepj is the step-length of discrete variable control, x _i, y _iBe the lower limit of corresponding control variables,

Be the upper limit.Δ λ=1.0 o'clock are represented that trend is separated and are recovered.

4. the setting of regulate expenditure: according to the actual conditions of power system operation, the adjustment expense of each control measure has following rule: but switching reactive power compensation＜OLTC adjustment＜generator output adjustment＜controlled series compensation, concrete expense can be set according to the actual conditions of system, and the measure that regulate expenditure is low will be able in optimizing process preferably.In this paper example, as example, but the regulate expenditure of switching reactive power compensation is made as 0.01K, and OLTC is 0.1K, and the regulate expenditure of generator output adjustment and controlled series compensation is respectively 1K and 10K, and K is certain monetary unit.

5. we find by research, and in computational process, suitably increase the recovery dynamics, promptly the Δ λ value in (14b) is slightly larger than 1.0 and can obtains better to restrain effect.

6. when being selected in, crucial control mode represents that having had no idea to recover trend by above control measure separates when no longer including, can only go to reduce load that this method wouldn't be given consideration according to the amount of not matching of each node.

3 embodiments

Example adopts New England-39 node and two examples of IEEE-118 node to verify the validity of this method, and wherein the realization of lingo software is partly called in mixed linear programming.The parameter of test computer: Intel Pentium Dual 2.00GHz, internal memory 2.00GB.

1. example one

Example one adopts New England-39 node system ^[14], as follows to the original system parameter modification:

1. with load bus 3,4,15,16 load increases to original 2.7 times, and the increment of load is by generator 1,3, and 4,8 bear.

2. circuit 2-3 fault withdraws from.

Available control measure comprise:

Node 3,4,15, but 16} is equipped with switching reactive power compensation (electric capacity and reactance), the group number be respectively 2,1,3,2}, every pool-size 30Mvar.

Transformer place branch road be 6-31,10-32,20-34,2-30}, its no-load voltage ratio is adjustable, establishes ± 5 grades, every grade of step-length is 0.02.

Circuit 3-18,7-8,16-17, the 22-23} admittance is adjustable.

4. { 33,34,35, the 36} generator output is adjustable for node.

Amended example calculates by Newton method to be dispersed.Adopt optimum multiplier method, through 22 iteration, optimum multiplier goes to zero, and θ is fixed on 7.226e-001, Jacobian matrix unusual (SNB point).This paper uses literary composition [9] respectively and improves one's methods and carries out trend and separate recovery, and all the trend that is restorable system by a control measure adjustment is separated.But the switching reactive power compensation is preferentially selected in the adjustment process, uses up behind the available capacitance #37 transformer voltage ratio and adjusts 1 grade downwards, and control back system satisfies the trend constraint.The sensitivity error comparison of two kinds of methods and computing time efficient see Table 1 and table 2 respectively:

Two kinds of sensitivity errors of table 1 relatively

Table1?Difference?between?two?sensitivities

Two kinds of methods of table 2 compare computing time

Table2?Time?consumption?comparison?between?original?method?and?modified?method

Two kinds of Sensitivity calculation results have been listed in the table 1 respectively, because two kinds of sensitivity are respectively θ and the λ sensitivity to control measure, so in table, be that benchmark converts with θ.The result shows that the sensitivity error of two kinds of methods is less, all below 2%, improves sensitivity λ so use _uReplace θ _uBe feasible.Though error is little on the result, there is very big difference the time of asking for, based on θ _uMethod when asking for CBP, need three external iteration, the trend iterations of internal layer is respectively 21,32,184 times, be not difficult to find out, when injecting power when the feasible zone border, the iterations of the optimum multiplier Newton method of internal layer increases to 184 times, and be 0.281s computing time; Sensitivity θ _uCalculating need calculate the Jacobian matrix left eigenvector, but because this example is less, computing time and not obvious.Can see that have only 19% of former method the computing time of this paper method from table, computational efficiency is greatly improved.

2. example two

Example two adopts the IEEE-118 node system ^[15], accompanying drawing 4 is a part of line charts of 118 node examples, wherein node 100 links to each other with the outside.Example is carried out following modification:

1. the load in should the zone increases to original 1.5 times, keeps power factor constant.

2. circuit 100-104,100-103 is out of service.

Available control measure comprise:

1. but all load buses are equipped with the switching reactive power compensation in the zone, wherein, node 106,107} be separately installed with 2,1} group, all the other nodes all have 3 groups, single pool-size is 30Mvar;

2. { the 100-106} admittance is adjustable for 103-104,103-105 for circuit.

3. { 103,111} place generator output is adjustable for node.

Amended example adopts the Newton method trend to calculate and disperses, and stops by after the optimum multiplier Newton method iteration 30 times, and optimum multiplier levels off to 0, θ stuck-at-.842.This example is still used based on θ _uAnd λ _uTwo kinds of methods are carried out trend respectively and are separated recovery, and the control measure adjustment process sees the following form 3:

Table 3 trend is separated the iterative process of recovery

Table3?Iteratively?restore?load?flow?solution

As can be seen from Table 3, two kinds of methods have all been recovered the trend feasible solution by twice control measure adjustment.And control measure are adjusted basically identical, and only slightly different on the adjustment amount of line admittance, the adjustment amount of this paper method is smaller.Two kinds of methods relatively see Table 4 computing time, are not difficult to find out, this paper method is calculated required time, less than 1/4th of former method.Our find by research simultaneously, and when big and accident was even more serious when system scale, the advantage on this paper method computational efficiency can be more obvious.

Two kinds of methods of table 4 compare computing time

Table4?Time?consumption?comparison?between?original?method?and?modified?method

4 conclusions

This method is recovered on the control strategy basis in original trend, by introducing new Control Parameter sensitivity, but avoided finding the solution of nearest solution point (CBP) in feasible zone border and Jacobian matrix zero eigenvalue left eigenvector, reduced finding the solution the time of fail-over policy, improve computational efficiency, verified the correctness and the validity of method by examples such as New England-39 node and IEEE-118 node systems.

5 lists of references

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GUO?Li，ZHANG?Yao，HU?Jinlei，et?al.Optimal?control?strategy?to?restore?load?flowsolution.Automation?ofElectric?Power?Systems，2007，31(16)：24-28.

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[13] Jiang Wei, Wang Chengshan, Yu Yixin. the new method that voltage stability margin is found the solution parametric sensitivity. Proceedings of the CSEE, 2006,26 (2): 13-18.

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6 appendix

Because transformer voltage ratio is positioned at denominator when forming admittance matrix,, use variable β to replace the inverse of no-load voltage ratio in order to obtain better linearization condition.Load margin is derived as follows to the sensitivity of transformer voltage ratio β reciprocal:

The node power equation is suc as formula (F-1):

F(x)＝V.×conj(Y _bus×V)-S (F-1)

V is the node voltage column vector, Y _BusBe the node admittance matrix of system, S is that node injects column vector.Both sides are to β _iAsk local derviation, can get formula (F2):

$\frac{\&PartialD;F}{\&PartialD;{\mathrm{\&beta;}}_i}=V\&times;\mathrm{conj}(\frac{\&PartialD;V_{\mathrm{bus}}}{\&PartialD;{\mathrm{\&beta;}}_i}\&times;V)---(F-2)$

Y _BusExpression formula see formula (F-3):

Y _bus＝C _f*diag(Y _ff)*C _f+C _f*diag(Y _ft)*C _t

(F-3a)

+C _t*diag(Y _tf)*C _f+C _t*diag(Y _tt)*C _t

Y _tt＝Y _s+j*B _s/2 (F-3b)

Y _ff＝Y _tt.*(β.*β) (F-3c)

Y _ft＝Y _tf＝-Y _s.*β (F-3d)

Y in the formula _sBe line admittance column vector, B _sBe line charging electric capacity column vector, C _f, C _tBe respectively the incidence matrices of circuit and top node and endpoint node.Formula (F-3a) both sides are to β _iAsk partial derivative to obtain formula (F-4):

$\frac{\&PartialD;Y_{\mathrm{bus}}}{\&PartialD;{\mathrm{\&beta;}}_i}=C_f*\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{ff}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}*C_f+C_f*\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{ft}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}*C_t$ $(F-4)$

$+C_t*\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{tf}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}*C_f+C_t*\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{tt}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}*C_t$

Wherein:

$\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{ff}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}=\mathrm{diag}(Y_{\mathrm{tt}}.*2*\mathrm{\&beta;}*e_i),\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{tt}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}=0$

$\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{ft}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}=\frac{\&PartialD;\mathrm{diag}\left(Y_{\mathrm{tf}}\right)}{\&PartialD;{\mathrm{\&beta;}}_i}=\mathrm{diag}(-Y_s*e_i)$

e _iFor i element only is 1 unit column vector.

With formula (F-4) substitution formula (F-2), can get

Concrete outcome.

Claims (3)

1. improved trend feasible solution restoration methods, it is characterized in that, recovery process is carried out controlled quentity controlled variable iteratively and calculated and two sub-steps of verification as a result: controlled quentity controlled variable is calculated the step and is at first tried to achieve the feasible zone boundary point that power injects the space by optimum multiplier trend, introduce and recover parameter lambda, calculate the sensitivity of λ, estimate the adjustment amount of each control device then based on the sensitivity calculations result by MILP each control measure; The verification step is then verified the control effect by optimum multiplier trend method, because sensitivity is the linear approximation to non-linear relation, the control strategy that forms based on it often has certain error, when control measure can not guarantee the trend convergence, then return controlled quentity controlled variable and calculate the step, repeat said process, calculate convergence until trend.

2. a kind of improved trend feasible solution restoration methods according to claim 1 is characterized in that, obtains the trend feasible zone boundary point of power space based on optimum multiplier trend:

The electric power system tide equation can be written as:

F(x)＝f(x)-S＝0 (1)

Be constructed as follows the system power amount of not matching:

X is a voltage vector in the formula, S is that node power injects and PV node voltage square vector, f (x) is node power equilibrium equation and PV node voltage constraint equation, adopt Newton method by constantly minimizing g (x) reaching the purpose of trend convergence, the iterative formula of Newton method be rewritten as by introducing optimum multiplier μ:

Δx ^k＝-J ^-1(x ^k)f(x ^k)

(3)

x ^k+1＝x ^k+μ ^kΔx ^k

Thereby in each iterative computation, the power amount of not matching is changed into:

Wherein: f ₁(x ^k, μ ^k)=f ( ^xK+ μ ^kΔ x ^k), J (x ^k) be the Jacobian matrix of k step iteration, μ ^kBe the optimum multiplier of k step iteration, μ ^kEach the step iteration in by

Try to achieve, thereby guarantee that the amount of not matching progressively diminishes, if trend does not have feasible solution, then from initial infeasible solution, this method optimum multiplier μ after iteration repeatedly will be tending towards 0, function (4) value that do not match will be stabilized in certain on the occasion of last, promptly converge on power this moment and inject SNB point on the feasible zone boundary point of space.

3, a kind of improved trend feasible solution restoration methods according to claim 1 is characterized in that, introduces and recovers parameter lambda, uses the sensitivity λ that recovers the parameter lambda control measure _uRecover trend and separate, determined a feasible boundary point x by optimum multiplier trend method ^*After being the SNB point, corresponding meritorious, idle injection vector are P ^*, Q ^*, power balance equation can be rewritten as:

E, f are the real part and the imaginary part of node voltage, and G, B are the real part and the imaginary part of line admittance, at x ^*λ=0, place, and when λ=1, formula (5) is equivalent to original power flow equation, use sensitivity λ _u, can avoid asking for of Jacobian matrix zero eigenvalue left eigenvector by the following method: electric power system is at its steady stability operational limit (SNB) point (x ^*, λ ^*) locate, will satisfy following equation:

F(x，λ，u)＝0

F _x(x，λ，u)·v＝0 (6)

||v||≠0

F is a power flow equation, is x, u, the function of λ; U=(u ₁, u ₂..., u _m) be control measure, but control measure comprise switching reactive power compensation in parallel, 0LTC no-load voltage ratio, series compensation and generator output adjustment; F _xBe the Jacobian matrix of system at this some place, v is its right characteristic vector; λ is a voltage stability margin, can try to achieve its sensitivity vector λ to u by following formula _u:

Λ·s＝-F _u (7a)

(7b)

A _iBe the column vector of trend Jacobian matrix correspondence, the Λ matrix is that Jacobian matrix p is listed as by F _λReplace λ _uBe p element among the solution vector s, more than one when Control Parameter, when supposing to have m, (7a) formula can be rewritten as following form:

4. a kind of improved trend feasible solution restoration methods according to claim 3 is characterized in that, calculates the SNB point by optimum multiplier trend, the λ in the computing formula (8) _u, by the control measure screening, judge whether to exist control measure, do not reduce load if do not exist then by the node amount of not matching, then adopt MILP MILP model to determine its minimum control cost if exist:

X wherein _iAnd y _iRepresent continuous and discrete control variables respectively, step _jBe the step-length of discrete variable control, x _i , y _i Be the lower limit of corresponding control variables,

Be the upper limit, Δ λ=1.0 o'clock are represented that trend is separated and are recovered.

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