CN102545207B - Voltage source commutation-high voltage direct current (VSC-HVDC) alternating-direct current optimal power flow method based on predictor-corrector inner point method - Google Patents

Abstract

The invention discloses a voltage source commutation-high voltage direct current (VSC-HVDC) alternating-direct current optimal power flow method based on a predictor-corrector inner point method. A direct current network and an alternating current system are combined based on a stable model of VSC-HVDC, unite optimization solution is conducted on the alternating-direct current system, and simulation and analysis are conducted on a plurality of computing examples. The results of the computing examples show that the predictor-corrector inner point method is good in optimization effect and is few in iteration time and smaller in computing quantity compared with the prior antithesis inner point method on capability to resolve optimal power flow problems containing the VSC-HVDC.

Description

VSC-HVDC alternating current-direct current optimal load flow method based on prediction-correction interior point

Technical field

Invention relates to a kind of VSC-HVDC alternating current-direct current optimal load flow method based on prediction-correction interior point, belongs to electric power system optimization operation field.

Background technology

Electric power system is complexity in large scale not only, and it has extremely strong importance to the national economic development.This has just determined that the economy of power system operation is a problem meriting attention.Along with social development, the consumption of the energy is more and more large, and energy savings is subject to people's common concern.Electric power is again the most important aspect of current energy resource consumption, therefore, is meeting under the prerequisite of power system power supply reliability and the quality of power supply, improve as far as possible the economy of operation, reasonably utilizes the existing energy and equipment, with minimum fuel consumption.

Along with the development of power grid construction, direct current transportation is more and more extensive in electrical network, to there will be increasing Ac/dc Power Systems, with full-controlled switch device and voltage source converter (Voltage Source Converter, VSC) be basic high voltage direct current of new generation (High Voltage Direct Current, HVDC) transmission of electricity, than the direct current transportation based on thyristor, have directly to isolated distant loads power supply, more economical to advantages such as load center power transmission, operation control method are flexible and changeable.Therefore the research of VSC-HVDC becomes the focus of numerous scholar's research in recent years.

It is to approach to optimal solution in feasible zone inside that interior point method solves optimal load flow, rapid without the interior some algorithmic statement of difficult extensive use of estimating active constraint collection, and strong robustness is insensitive to the selection of initial value.Compared with former-dual interior point, prediction-correction interior point has only increased a former generation back substitution in each iteration to be calculated, but can obviously reduce convergence number of times, and optimal speed obviously improves.After adding VSC-HVDC, there is variation in system OPF model, need to, according to the steady-state load flow model of VSC-HVDC, ac and dc systems be combined, and carries out simultaneous Optimization Solution.

Summary of the invention

Technical problem to be solved by this invention is for the adding of new model, and a kind of prediction-correction interior point alternating current-direct current optimal load flow method containing VSC-HVDC is provided.

The present invention for achieving the above object, adopts following technical scheme:

The present invention is the VSC-HVDC alternating current-direct current optimal load flow method based on prediction-correction interior point, it is characterized in that described method realizes according to the following steps successively in computer:

(1) network parameter of acquisition electric power system, comprise: bus numbering, title, load are meritorious, reactive load, building-out capacitor, the branch road of transmission line number, headend node and endpoint node numbering, series resistance, series reactance, shunt conductance, shunt susceptance, transformer voltage ratio and impedance, generated power is exerted oneself, idle bound, economic parameters;

(2) program initialization, comprising: quantity of state is arranged initial value, Lagrange multiplier and penalty factor are arranged initial value, the optimization of node order, form node admittance matrix, recover iteration count k=1, maximum iteration time K is set _max, required precision ε is set;

(3) calculate duality gap C _gap, judge whether it meets C _gap< ε, if so, exports result of calculation, exits circulation, if not, continues;

(4) disturbance factor mu=0 is set, predicts according to following formula, obtain affine direction:

$\left[\begin{array}{cc}{H}^{\&prime;}& {\&dtri;}_{x}h\left(x\right)\\ {\&dtri;}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{x}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{y}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{L}_x^{\&prime;}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{z}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{l}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{-L}^{-1}{L}_l^{\mathrm{\&mu;}}\\ L_z+{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{w}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{u}_{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{-U}^{-1}{L}_u^{\mathrm{\&mu;}}\\ -L_w-{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

Wherein: Δ x ^aff, Δ y ^aff, Δ z ^aff, Δ l ^aff, Δ u ^aff, Δ w ^afffor the affine deflection correction of x, y, z, l, u, w.

$L_x^{\&prime;}=L_x+{\&dtri;}_{x}g\left(x\right)[L^{-1}(L_l^{\mathrm{\&mu;}}+Z{L}_z)+U^{-1}(L_u^{\mathrm{\&mu;}}+W{L}_w)]$

$H^{\&prime;}=H-{\&dtri;}_{x}g\left(x\right)[L^{-1}Z-U^{-1}W]{\&dtri;}_x^{T}g\left(x\right)$

$H=-[{\&dtri;}_x^{2}f\left(x\right)-{\&dtri;}_x^{2}h\left(x\right)y-{\&dtri;}_x^{2}g\left(x\right)(z+w)]$

Determine the iteration step length of affine direction:

${\mathrm{\&alpha;}}_p^{\mathrm{aff}}=0.9995\min \{\underset{i}{\min }(\frac{{-l}_i}{\mathrm{\&Delta;}{l}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{l}_i^{\mathrm{aff}}<0;\frac{{-u}_i}{\mathrm{\&Delta;}{u}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{u}_i^{\mathrm{aff}}<0),1\}$

${\mathrm{\&alpha;}}_d^{\mathrm{aff}}=0.9995\min \{\underset{i}{\min }(\frac{{-z}_i}{\mathrm{\&Delta;}{z}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{z}_i^{\mathrm{aff}}<0;\frac{{-w}_i}{\mathrm{\&Delta;}{w}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{w}_i^{\mathrm{aff}}>0),1\}$

Calculate the complementary gap of affine direction:

$C_{\mathrm{Gap}}^{\mathrm{aff}}=(l+{\mathrm{\&alpha;}}_p^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}})(z+{\mathrm{\&alpha;}}_d^{\mathrm{aff}}\mathrm{\&Delta;}{z}^{\mathrm{aff}})-(u+{\mathrm{\&alpha;}}_p^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}})(w+{\mathrm{\&alpha;}}_d^{\mathrm{aff}}\mathrm{\&Delta;}{w}^{\mathrm{aff}})$

Dynamic estimation Center Parameter:

$\mathrm{\&delta;}={(C_{\mathrm{Gap}}^{\mathrm{aff}}/C_{\mathrm{Gap}})}^3$

(5) calculation perturbation factor mu;

(6) complementary relaxation condition is revised:

$\mathrm{Z\&Delta;l}+\mathrm{L\&Delta;z}=-L_l^{\mathrm{\&mu;}}-\mathrm{\&Delta;}{Z}^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}}$

$\mathrm{W\&Delta;u}+\mathrm{U\&Delta;w}=-L_u^{\mathrm{\&mu;}}-\mathrm{\&Delta;}{W}^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}}$

Correspondingly, L' _xbe modified to:

$L_x^{\&prime;\&prime;}=L_x^{\&prime;}+{\&dtri;}_{x}g\left(x\right)(L^{-1}\mathrm{\&Delta;}{Z}^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}}-U^{-1}\mathrm{\&Delta;}{W}^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}})$

According to following equation solution Δ x, Δ y, Δ l, Δ u, Δ z, Δ w:

$\left[\begin{array}{cc}{H}^{\&prime;}& {\&dtri;}_{x}h\left(x\right)\\ {\&dtri;}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}x\\ \mathrm{\&Delta;}y\end{array}\right]=\left[\begin{array}{c}{L}_x^{\&prime;\&prime;}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}z\\ \mathrm{\&Delta;}l\end{array}\right]=\left[\begin{array}{c}{-L}^{-1}{L}_l^{\mathrm{\&mu;}}\\ L_z+{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}w\\ \mathrm{\&Delta;}u\end{array}\right]=\left[\begin{array}{c}{-U}^{-1}{L}_u^{\mathrm{\&mu;}}\\ -L_w-{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

Wherein: Δ x, Δ y, Δ z, Δ l, Δ u, Δ w are the correction of x, y, z, l, u, w.

(7) determine the iteration step length of original variable and dual variable:

${\mathrm{\&alpha;}}_p=0.9995\min \{\underset{i}{\min }(\frac{{-l}_i}{\mathrm{\&Delta;}{l}_i},\mathrm{\&Delta;}{l}_i<0;\frac{{-u}_i}{\mathrm{\&Delta;}{u}_i},\mathrm{\&Delta;}{u}_i<0),1\}$

${\mathrm{\&alpha;}}_d=0.9995\min \{\underset{i}{\min }(\frac{{-z}_i}{\mathrm{\&Delta;}{z}_i},\mathrm{\&Delta;}{z}_i<0;\frac{{-w}_i}{\mathrm{\&Delta;}{w}_i},\mathrm{\&Delta;}{w}_i>0),1\}$

(8) upgrade original variable and Lagrange multiplier;

(9) judge whether iterations is greater than K _max, if so, calculate and do not restrain, quit a program, if not, put iterations and add 1, return to (3).

Compared to traditional phased converter direct current transportation, VSC-HVDC have operation control method flexible and changeable, can be directly to isolated distant loads power supply, more economical to advantages such as load center power transmissions.

Some scholars have carried out many research to the alternating current-direct current optimal load flow containing the converter direct current transportation of conventional current source at present, but rarely have the optimal load flow achievement about VSC-HVDC.The present invention is based on the steady-state load flow model of VSC-HVDC, proposed a kind of interior point methods of prediction-correction of the VSC-HVDC of solution alternating current-direct current optimal load flow.

Brief description of the drawings

Fig. 1: the inventive method flow chart.

Fig. 2: the Ac/dc Power Systems model that the present invention adopts.

Fig. 3: applied three the example systems of the VSC-HVDC alternating current-direct current optimal load flow based on prediction-correction interior point that the present invention proposes, wherein: figure (a) is IEEE-14 node system, figure (b) is IEEE-30 node system, and figure (c) is IEEE-57 node system.

Embodiment

Fig. 1 is the inventive method flow chart, and Fig. 2 is the Ac/dc Power Systems model that the present invention adopts.In Fig. 2, i represents to access i VSC of DC network.The fundamental voltage phasor of supposing i VSC output is with the voltage phasor of AC system junction be converter transformer impedance is jX _li, R _ibe the equivalent resistance of i inverter inside loss and converter transformer loss, active power and reactive power that AC system flows into converter transformer are respectively P _siand Q _si, the active power and the reactive power that flow into converter bridge are respectively P _ciand Q _ci, the electric current that wherein flows through converter transformer is suppose direction as shown in Figure 1,

${\stackrel{\&CenterDot;}{I}}_i=({\stackrel{\&CenterDot;}{U}}_{\mathrm{si}}-{\stackrel{\&CenterDot;}{U}}_{\mathrm{ci}})/(R_i+j{X}_{\mathrm{Li}})$

AC system flows into the complex power of converter transformer meet following relational expression:

${\stackrel{\&CenterDot;}{S}}_{\mathrm{si}}=P_{\mathrm{si}}+j{Q}_{\mathrm{si}}={\stackrel{\&CenterDot;}{U}}_{\mathrm{si}}{\left({\stackrel{\&CenterDot;}{I}}_i\right)}^{*}$

In order to discuss conveniently, make δ _i=θ _si-θ _ci, α _i=arctan (X _li/ R _i), further derivation can obtain

$P_{\mathrm{si}}=-\left|Y_i\right|U_{\mathrm{si}}{U}_{\mathrm{ci}}\cos ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)+\left|Y_i\right|U_{\mathrm{si}}^2\cos {\mathrm{\&alpha;}}_i$

$Q_{\mathrm{si}}=-\left|Y_i\right|U_{\mathrm{si}}{U}_{\mathrm{ci}}\sin ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)+\left|Y_i\right|U_{\mathrm{si}}^2\sin {\mathrm{\&alpha;}}_i$

In like manner can derive and obtain:

$P_{\mathrm{ci}}=-\left|Y_i\right|U_{\mathrm{si}}{U}_{\mathrm{ci}}\cos ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)+\left|Y_i\right|U_{\mathrm{ci}}^2\cos {\mathrm{\&alpha;}}_i$

$Q_{\mathrm{ci}}=-\left|Y_i\right|U_{\mathrm{si}}{U}_{\mathrm{ci}}\sin ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)+\left|Y_i\right|U_{\mathrm{ci}}^2\sin {\mathrm{\&alpha;}}_i$

Because the loss of the change of current brachium pontis of VSC is by R _iequivalence, so direct current power P _dishould with inject the P of converter bridge _ciequate, therefore can obtain

$P_{\mathrm{di}}=U_{\mathrm{di}}{I}_{\mathrm{di}}=\left|Y_i\right|U_{\mathrm{si}}{U}_{\mathrm{ci}}\cos ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)+\left|Y_i\right|U_{\mathrm{ci}}^2\cos {\mathrm{\&alpha;}}_i$

Wherein U _di, I _dibe respectively direct voltage and the electric current of direct current node.In addition, voltage equation is:

$U_{\mathrm{ci}}=\frac{\sqrt{6}}{4}{M}_i{U}_{\mathrm{di}}$

Wherein M _iit is the modulation degree of i VSC.

Above-mentioned 8 equations have formed the steady-state model of the lower VSC-HVDC of mark the one system.

In VSC-HVDC, the direct relation of whether stablizing of direct voltage that can system normally be moved and the stability of AC output voltage.Send to the active power of corresponding end AC system if the active power that the VSC of meritorious transmitting terminal absorbs from this end AC system is greater than receiving terminal VSC, direct voltage raises, otherwise direct voltage reduces.Therefore in order to realize this power-balance, wherein one end VSC must adopt constant DC voltage control.In addition, if direct voltage is constant, the variable quantity of direct current is proportional to the amount of unbalance of active power, and constant DC current control is equivalent with determining active power control.Comprehensive above analysis, the control mode that in VSC-HVDC, VSC can select has following several:

1.. determine direct voltage, determine Reactive Power Control;

2.. determine direct voltage, determine alternating voltage control;

3.. determine active power, determine Reactive Power Control;

4.. determine active power, determine alternating voltage control.

The present invention adopts following four kinds of control modes combination to the VSC-HVDC at direct current branch two ends:

(1).①+③；(2).①+④；

(3).③+②；(4).④+②。

Whether be connected to converter transformer according to node, node is divided into direct current node and the pure node that exchanges.Direct current node refers to the node that the primary side of converter transformer connects, node as shown in Figure 1, owing to having connected converter exchanging on node, the control of its correspondence with state variable at the former exchange status variable U that exchanges node _i, θ _ion basis, increase DC Variable U _di, I _di, δ _i, M _i, P _si, Q _si, wherein δ _i, M _ifor phase angle and the modulation degree of converter; Pure interchange node refers to the node not being connected with converter transformer.The node of uniting of setting up departments adds up to n, and wherein the number of VSC is n _c, direct current nodes is n _c, pure interchange nodes is n _a=n-n _c.In order to compose a piece of writing conveniently, suppose that the node serial number order of Ac/dc Power Systems is: 1～n below _anode is pure interchange node; n _a+ 1～n node is direct current node.

For direct current node, its trend accounting equation is:

$\left\{\begin{array}{c}\mathrm{\&Delta;}{P}_{\mathrm{ti}}=P_{\mathrm{ti}}^s-U_{\mathrm{ti}}\underset{j\&Element;i}{\mathrm{\&Sigma;}}{U}_j(G_{\mathrm{ij}}\cos {\mathrm{\&theta;}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\&theta;}}_{\mathrm{ij}})-P_{\mathrm{si}}\\ \mathrm{\&Delta;}{Q}_{\mathrm{ti}}=Q_{\mathrm{ti}}^s-U_{\mathrm{ti}}\underset{j\&Element;i}{\mathrm{\&Sigma;}}{U}_j(G_{\mathrm{ij}}\sin {\mathrm{\&theta;}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\&theta;}}_{\mathrm{ij}})-Q_{\mathrm{si}}\end{array}\right.$

Wherein: Δ P _ti, Δ Q _tifor direct current node power amount of unbalance; for the node after deduction load sends power; U _tifor being connected to the interchange node voltage amplitude of VSC, subscript j is all nodes (representing with j ∈ i in formula) that are directly connected with node i; U _jfor exchanging node voltage amplitude, θ _ijfor node i, phase angle difference between j; G _ij, B _ijfor node i, the real part of admittance and imaginary part between j.P _si, Q _sifor AC system flows into gaining merit and reactive power of converter transformer.

According to the steady-state model of VSC-HVDC, the trend accounting equation that can obtain direct current system is:

$\mathrm{\&Delta;}{d}_{i1}=P_{\mathrm{si}}+(\sqrt{6}/4)M_i{U}_{\mathrm{si}}{U}_{\mathrm{di}}\left|Y_i\right|\cos ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)-U_{\mathrm{si}}^2\text{|}{Y}_i|\cos {\mathrm{\&alpha;}}_i$

$\mathrm{\&Delta;}{d}_{i2}=Q_{\mathrm{si}}+(\sqrt{6}/4)M_i{U}_{\mathrm{si}}{U}_{\mathrm{di}}\left|Y_i\right|\sin ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)-U_{\mathrm{si}}^2\left|Y_i\right|\sin {\mathrm{\&alpha;}}_i-U_{\mathrm{si}}^2/X_{\mathrm{fi}}$

$\mathrm{\&Delta;}{d}_{i3}=U_{\mathrm{di}}{I}_{\mathrm{di}}-(\sqrt{6}/4)M_i{U}_{\mathrm{si}}{U}_{\mathrm{di}}\left|Y_i\right|\cos ({\mathrm{\&delta;}}_i-{\mathrm{\&alpha;}}_i)+(3/8){\left(M_i{U}_{\mathrm{di}}\right)}^2\left|Y_i\right|\cos {\mathrm{\&alpha;}}_i$

In formula, subscript i represents i VSC.

Add the current deviation amount equation of DC network:

$\mathrm{\&Delta;}{d}_{i4}=I_{\mathrm{di}}-\underset{j=1}{\overset{n_c}{\mathrm{\&Sigma;}}}{g}_{\mathrm{dij}}{U}_{\mathrm{dj}}$

Wherein g _dijfor the electric conductivity value between direct current node i, j.

VSC-HVDC system access is after electrical network, and the steady-state load flow model of system changes, and need to adopt above-mentioned VSC steady-state model to carry out the calculating of OPF problem.

OPF model can be expressed as following Non-linear Optimal Model:

obj. min.f(x)

s.t. h(x)＝0

$\underset{\&OverBar;}{g}\&le;g\left(x\right)\stackrel{\&OverBar;}{g}$

Wherein: x=[P _g, Q _r, θ, V, U _d, I _d, δ, M, P _s, Q _s], compared to conventional OPF problem, in x, increase DC control amount and quantity of state U in VSC-HVDC system _d, I _d, δ, M, P _s, Q _s, f (x) is target function; H (x) is equality constraint, the power balance equation that comprises AC system, and the power of VSC-HVDC and current balance equation, This document assumes that equality constraint number is m; G (x) is inequality constraints condition, the voltage magnitude that comprises AC system, phase angle, and the constraint of circuit through-put power, voltage, the modulation degree etc. of direct current system, suppose that inequality constraints number is r.

The basic ideas of interior point method are: inequality constraints is converted into equality constraint by slack variable, recycling Lagrange multiplier is incorporated into constraint in target function, and slack variable is retrained with barrier function method.For OPF problem, structure Lagrangian is as follows:

$L=f\left(x\right)-y^{T}h\left(x\right)-z^T[g\left(x\right)-l-\underset{\&OverBar;}{g}]-w^T[g\left(x\right)+u-\stackrel{\&OverBar;}{g}]-\mathrm{\&mu;}\underset{j=1}{\overset{r}{\mathrm{\&Sigma;}}}\ln \left(l_j\right)-\mathrm{\&mu;}\underset{j=1}{\overset{r}{\mathrm{\&Sigma;}}}\ln \left(u_j\right)$

Wherein y=[y ₁..., y _m] ^tfor the Lagrange multiplier of equality constraint, z=[z ₁..., z _r] ^t, w=[w ₁..., w _r] ^tfor the Lagrange multiplier of inequality constraints, l=[l ₁..., l _r] ^t, u=[u ₁..., u _r] ^tfor the slack variable of inequality constraints, μ is the penalty factor of barrier function.

KKT (Karush-Kuhn-Tucker) condition of this problem is:

$\left\{\begin{array}{c}{L}_x={\&dtri;}_{x}f\left(x\right)-{\&dtri;}_{x}h\left(x\right)y-{\&dtri;}_{x}g\left(x\right)(z+w)\\ L_y=h\left(x\right)=0\\ L_z=g\left(x\right)-l-\underset{\&OverBar;}{g}=0\\ L_w=g\left(x\right)+u-\stackrel{\&OverBar;}{g}=0\\ L_l=z-\mathrm{\&mu;}{L}^{-1}e=0\\ L_u=-w-\mathrm{\&mu;}{U}^{-1}e=0\end{array}\right.$

In formula:

for 1 order derivative of f (x) to x, be respectively the Jacobian matrix of h (x), g (x).L＝diag(l ₁,…,l _r)U＝diag(u ₁,…,u _r)，Z＝diag(z ₁,…,z _r)W＝diag(w ₁,…,w _r)，

L ^-1＝diag(1/l ₁,…,1/l _r)，U ^-1＝diag(1/u ₁,…,1/u _r)，e＝[1,…,1] ^T。

Can be in the hope of by latter two equation in formula KKT condition

μ＝(l ^Tz-u ^Tw)/2r

Definition C _gap=l ^tz-u ^tw.

But facts have proved, when the parameter in target function during according to above formula value convergence poor, generally adopt

μ＝σC _Gap/2r，

Center Parameter σ is the important parameter that affects algorithm performance, and the core concept of predict-correction interior point is the dynamic estimation to Center Parameter σ.Compared with former dual interior point, this algorithm only increases a former generation back substitution computing in each iteration, but can obviously reduce iterations, and convergence rate obviously improves.

Prediction-correction interior point, in each iteration, is walked and is obtained affine direction by prediction, then utilizes its second order term of estimating complementary equation Taylor expansion, obtains and proofreaies and correct step.

Prediction step:

1, set Center Parameter σ=0;

2, solve following equation, obtain affine direction Δ x ^aff, Δ l ^aff, Δ u ^aff, Δ y ^aff, Δ z ^aff, Δ w ^aff:

$\left[\begin{array}{cc}{H}^{\&prime;}& {\&dtri;}_{x}h\left(x\right)\\ {\&dtri;}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{x}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{y}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{L}_x^{\&prime;}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{z}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{l}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{-L}^{-1}{L}_l^{\mathrm{\&mu;}}\\ L_z+{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{w}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{u}_{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{-U}^{-1}{L}_u^{\mathrm{\&mu;}}\\ -L_w-{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

Wherein: Δ x ^aff, Δ y ^aff, Δ z ^aff, Δ l ^aff, Δ u ^aff, Δ w ^afffor the affine deflection correction of x, y, z, l, u, w.

$L_x^{\&prime;}=L_x+{\&dtri;}_{x}g\left(x\right)[L^{-1}(L_l^{\mathrm{\&mu;}}+Z{L}_z)+U^{-1}(L_u^{\mathrm{\&mu;}}+W{L}_w)]$

$H^{\&prime;}=H-{\&dtri;}_{x}g\left(x\right)[L^{-1}Z-U^{-1}W]{\&dtri;}_x^{T}g\left(x\right)$

$H=-[{\&dtri;}_x^{2}f\left(x\right)-{\&dtri;}_x^{2}h\left(x\right)y-{\&dtri;}_x^{2}g\left(x\right)(z+w)]$

3, determine the iteration step length of affine direction:

${\mathrm{\&alpha;}}_p^{\mathrm{aff}}=0.9995\min \{\underset{i}{\min }(\frac{{-l}_i}{\mathrm{\&Delta;}{l}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{l}_i^{\mathrm{aff}}<0;\frac{{-u}_i}{\mathrm{\&Delta;}{u}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{u}_i^{\mathrm{aff}}<0),1\}$

${\mathrm{\&alpha;}}_d^{\mathrm{aff}}=0.9995\min \{\underset{i}{\min }(\frac{{-z}_i}{\mathrm{\&Delta;}{z}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{z}_i^{\mathrm{aff}}<0;\frac{{-w}_i}{\mathrm{\&Delta;}{w}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{w}_i^{\mathrm{aff}}>0),1\}$

4, calculate the complementary gap of affine direction:

$C_{\mathrm{Gap}}^{\mathrm{aff}}=(l+{\mathrm{\&alpha;}}_p^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}})(z+{\mathrm{\&alpha;}}_d^{\mathrm{aff}}\mathrm{\&Delta;}{z}^{\mathrm{aff}})-(u+{\mathrm{\&alpha;}}_p^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}})(w+{\mathrm{\&alpha;}}_d^{\mathrm{aff}}\mathrm{\&Delta;}{w}^{\mathrm{aff}})$

5, dynamic estimation Center Parameter:

$\mathrm{\&delta;}={(C_{\mathrm{Gap}}^{\mathrm{aff}}/C_{\mathrm{Gap}})}^3$

Proofread and correct step:

1, complementary relaxation condition is revised:

$\mathrm{Z\&Delta;l}+\mathrm{L\&Delta;z}=-L_l^{\mathrm{\&mu;}}-\mathrm{\&Delta;}{Z}^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}}$

$\mathrm{W\&Delta;u}+\mathrm{U\&Delta;w}=-L_u^{\mathrm{\&mu;}}-\mathrm{\&Delta;}{W}^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}}$

Correspondingly, L' _xbe modified to:

$L_x^{\&prime;\&prime;}=L_x^{\&prime;}+{\&dtri;}_{x}g\left(x\right)(L^{-1}\mathrm{\&Delta;}{Z}^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}}-U^{-1}\mathrm{\&Delta;}{W}^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}})$

2, according to following equation solution Δ x, Δ y, Δ l, Δ u, Δ z, Δ w:

$\left[\begin{array}{cc}{H}^{\&prime;}& {\&dtri;}_{x}h\left(x\right)\\ {\&dtri;}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}x\\ \mathrm{\&Delta;}y\end{array}\right]=\left[\begin{array}{c}{L}_x^{\&prime;\&prime;}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}z\\ \mathrm{\&Delta;}l\end{array}\right]=\left[\begin{array}{c}{-L}^{-1}{L}_l^{\mathrm{\&mu;}}\\ L_z+{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}w\\ \mathrm{\&Delta;}u\end{array}\right]=\left[\begin{array}{c}{-U}^{-1}{L}_u^{\mathrm{\&mu;}}\\ -L_w-{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

Wherein: Δ x, Δ y, Δ z, Δ l, Δ u, Δ w are the correction of x, y, z, l, u, w.

Fig. 3 is applied three the example systems of the VSC-HVDC alternating current-direct current optimal load flow based on prediction-correction interior point that the present invention proposes.Figure (a) is IEEE-14 node system, wherein VSC ₁, VSC ₂be connected on node 13,14; Figure (b) is IEEE-30 node system, and wherein VSC1, VSC2 are connected on node 29,30; Figure (c) is IEEE-57 node system, wherein VSC1, VSC ₂be connected on node 54,55 VSC ₃, VSC ₄be connected on node 56,57.

The present invention adopts prediction-correction interior point to carry out optimal load flow calculating to VSC-HVDC Ac/dc Power Systems, prediction-correction interior point effect of optimization of having verified the present invention's proposition is remarkable, and solving in the ability of VSC-HVDC AC/DC Optimal Power, still less, amount of calculation is less for the more former dual interior point iterations of the method.

Introduce three embodiment of the present invention below:

Example one:

The present invention adopts the modified IEEE-14 node standard example shown in Fig. 3 (a), and direct current branch parameter is in table 1.

The each variable initial value of table 1VSC

N	R	X L	P l	Q l	U	θ	P s	Q s	U d
										2	0.006	0.150	2.000	1.000	1.078	0.000	0.919	0.122	2.000
3	0.006	0.150	3.700	1.300	1.036	0.000	-0.899	0.173	2.000

Wherein N represents female wire size at VSC place, P _l, Q _lmeritorious, idle for the load that VSC place ac bus connects, U, θ represent alternating voltage amplitude, the phase angle of VSC place bus, P _s, Q _sfor the injection of direct current system meritorious, idle, U _dfor direct voltage.R is the equivalent resistance of inverter inside loss and converter transformer loss, R, X _l, P _l, Q _lgiven by system; P _s, Q _sinitial value is made as and revises front system branch power and equate, is calculated by the trend of original system.I _d, δ, M be by obtaining below:

I _di＝(P _gi-P _li)/U _di

${\mathrm{\&delta;}}_i=\arctan (P_{\mathrm{ti}}/(U_{\mathrm{ti}}^2/X_{\mathrm{Li}}-Q_{\mathrm{ti}}))$

$M_i=\sqrt{6}{P}_{\mathrm{ti}}{X}_{\mathrm{Li}}\left(4U_{\mathrm{ti}}{U}_{\mathrm{di}}\sin {\mathrm{\&delta;}}_i\right)$

Wherein P _gimeritorious for node generating, if this node is not generator node, be 0.

Adopt prediction-correction interior point to carry out optimal load flow calculating (direct current branch two ends VSC control mode be combined as 1.+3.) to VSC-HVDC Ac/dc Power Systems, simulation result is as shown in following table 2 and 3.In table, except generating expense unit is $, other data unit is perunit value.

Table 2 AC system result

Table 3 direct current system result

From table 2, total generating of system is meritorious, idle all to be reduced to some extent, and each generator distributes by economic index, and expense has reduced nearly 10%.Table 3 is each VSC-HVDC control and state variable value before and after having listed and having optimized, and the modulation degree M of VSC all increases to some extent, has improved the utilance of direct voltage, has reduced AC percent harmonic distortion.Example two:

In service in practical power systems, in order to realize stable operation, to reduce the functions such as loss, load disturbance, VSC-HVDC usually need to move under various control pattern.Table 4 has been listed different examples under different control modes, the economic parameters before and after optimizing.In table, the economic parameters unit of each system is $.

Each system cost result under table 4 different control modes

Slightly difference of the effect of optimization that each example obtains under different control modes, main cause is to be provided with different control modes, is in fact equivalent to strengthen the operation constraint of VSC-HVDC.Visible this algorithm applicability of effect of optimization by each example is stronger, can both successfully be optimized, and effect of optimization is remarkable to the system under various control modes, and taking IEEE-57 node as example, expense reduction reaches 35%.

Table 5 is under different control modes, and the iterations while utilizing former dual interior point and prediction-correction interior point to optimize each example compares, and in table, A represents former dual interior point, and B represents prediction-correction interior point.

The former dual interior point of table 5 and the comparison of prediction-correction interior point iterations

Hence one can see that, and under equal conditions, compared to former dual interior point, prediction-correction interior point only need increase prediction step and proofread and correct a little amount of calculation of step in the time of each iteration, and it is 30% even more that iterations but can reduce.

Claims (1)

1. the VSC-HVDC alternating current-direct current optimal load flow method based on prediction-correction interior point, is characterized in that described method realizes according to the following steps:

(1) network parameter of acquisition electric power system, comprise: bus numbering, title, load are meritorious, reactive load, building-out capacitor, the branch road of transmission line number, headend node and endpoint node numbering, series resistance, series reactance, shunt conductance, shunt susceptance, transformer voltage ratio and impedance, generated power is exerted oneself, idle bound, economic parameters;

(2) program initialization, comprising: quantity of state is arranged initial value, Lagrange multiplier and penalty factor are arranged initial value, the optimization of node order, form node admittance matrix, recover iteration count k=1, maximum iteration time K is set _max, required precision ε is set;

(3) calculate duality gap C _gap, judge whether it meets C _gap< ε, if so, exports result of calculation, exits circulation, if not, continues;

(4) disturbance factor mu=0 is set, predicts according to following formula, obtain affine deflection correction:

$\left[\begin{array}{cc}{H}^{\&prime;}& {\&dtri;}_{x}h\left(x\right)\\ {\&dtri;}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{x}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{y}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{L}_x^{\&prime;}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{z}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{l}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{-L}^{-1}{L}_l^{\mathrm{\&mu;}}\\ L_z+{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}{w}^{\mathrm{aff}}\\ \mathrm{\&Delta;}{u}_{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{-U}^{-1}{L}_u^{\mathrm{\&mu;}}\\ -L_w-{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

Wherein: f (x) is target function, h (x) is equality constraint, and g (x) is inequality constraints condition; Y=[y ₁..., y _m] ^tfor the Lagrange multiplier of equality constraint, z=[z ₁..., z _r] ^t, w=[w ₁..., w _r] ^tfor the Lagrange multiplier of inequality constraints, l=[l ₁..., l _r] ^t, u=[u ₁..., u _r] ^tfor the slack variable of inequality constraints, for 1 order derivative of f (x) to x, be respectively the Jacobian matrix of h (x), g (x); L=diag (l ₁..., l _r), U=diag (u ₁..., u _r), Z=diag (z ₁..., z _r), W=diag (w ₁..., w _r); Δ x ^aff, Δ y ^aff, Δ z ^aff, Δ l ^aff, Δ u ^aff, Δ w ^afffor the affine deflection correction of x, y, z, l, u, w;

$L_x^{\&prime;}=L_x+{\&dtri;}_{x}g\left(x\right)[L^{-1}(L_l^{\mathrm{\&mu;}}+Z{L}_z)+U^{-1}(L_u^{\mathrm{\&mu;}}+W{L}_w)]$

$H^{\&prime;}=H-{\&dtri;}_{x}g\left(x\right)[L^{-1}Z-U^{-1}W]{\&dtri;}_x^{T}g\left(x\right)$

$H=-[{\&dtri;}_x^{2}f\left(x\right)-{\&dtri;}_x^{2}h\left(x\right)y-{\&dtri;}_x^{2}g\left(x\right)(z+w)]$

Determine the iteration step length of affine direction:

${\mathrm{\&alpha;}}_p^{\mathrm{aff}}=0.9995\min \{\underset{i}{\min }(\frac{{-l}_i}{\mathrm{\&Delta;}{l}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{l}_i^{\mathrm{aff}}<0;\frac{{-u}_i}{\mathrm{\&Delta;}{u}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{u}_i^{\mathrm{aff}}<0),1\}$

${\mathrm{\&alpha;}}_d^{\mathrm{aff}}=0.9995\min \{\underset{i}{\min }(\frac{{-z}_i}{\mathrm{\&Delta;}{z}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{z}_i^{\mathrm{aff}}<0;\frac{{-w}_i}{\mathrm{\&Delta;}{w}_i^{\mathrm{aff}}},\mathrm{\&Delta;}{w}_i^{\mathrm{aff}}>0),1\}$

Calculate the complementary gap of affine direction:

$C_{\mathrm{Gap}}^{\mathrm{aff}}=(l+{\mathrm{\&alpha;}}_p^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}})(z+{\mathrm{\&alpha;}}_d^{\mathrm{aff}}\mathrm{\&Delta;}{z}^{\mathrm{aff}})-(u+{\mathrm{\&alpha;}}_p^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}})(w+{\mathrm{\&alpha;}}_d^{\mathrm{aff}}\mathrm{\&Delta;}{w}^{\mathrm{aff}})$

Dynamic estimation Center Parameter:

$\mathrm{\&delta;}={(C_{\mathrm{Gap}}^{\mathrm{aff}}/C_{\mathrm{Gap}})}^3$

(5) calculation perturbation factor mu;

(6) complementary relaxation condition is revised:

$\mathrm{Z\&Delta;l}+\mathrm{L\&Delta;z}=-L_l^{\mathrm{\&mu;}}-\mathrm{\&Delta;}{Z}^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}}$

$\mathrm{W\&Delta;u}+\mathrm{U\&Delta;w}=-L_u^{\mathrm{\&mu;}}-\mathrm{\&Delta;}{W}^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}}$

Correspondingly, L' _xbe modified to:

$L_x^{\&prime;\&prime;}=L_x^{\&prime;}+{\&dtri;}_{x}g\left(x\right)(L^{-1}\mathrm{\&Delta;}{Z}^{\mathrm{aff}}\mathrm{\&Delta;}{l}^{\mathrm{aff}}-U^{-1}\mathrm{\&Delta;}{W}^{\mathrm{aff}}\mathrm{\&Delta;}{u}^{\mathrm{aff}})$

According to following equation solution Δ x, Δ y, Δ l, Δ u, Δ z, Δ w:

$\left[\begin{array}{cc}{H}^{\&prime;}& {\&dtri;}_{x}h\left(x\right)\\ {\&dtri;}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}x\\ \mathrm{\&Delta;}y\end{array}\right]=\left[\begin{array}{c}{L}_x^{\&prime;\&prime;}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}z\\ \mathrm{\&Delta;}l\end{array}\right]=\left[\begin{array}{c}{-L}^{-1}{L}_l^{\mathrm{\&mu;}}\\ L_z+{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}\mathrm{\&Delta;}w\\ \mathrm{\&Delta;}u\end{array}\right]=\left[\begin{array}{c}{-U}^{-1}{L}_u^{\mathrm{\&mu;}}\\ -L_w-{\&dtri;}_x^{T}g\left(x\right)\mathrm{\&Delta;x}\end{array}\right]$

Wherein: Δ x, Δ y, Δ z, Δ l, Δ u, Δ w are the correction of x, y, z, l, u, w;

(7) determine the iteration step length of original variable and dual variable:

${\mathrm{\&alpha;}}_p=0.9995\min \{\underset{i}{\min }(\frac{{-l}_i}{\mathrm{\&Delta;}{l}_i},\mathrm{\&Delta;}{l}_i<0;\frac{{-u}_i}{\mathrm{\&Delta;}{u}_i},\mathrm{\&Delta;}{u}_i<0),1\}$

${\mathrm{\&alpha;}}_d=0.9995\min \{\underset{i}{\min }(\frac{{-z}_i}{\mathrm{\&Delta;}{z}_i},\mathrm{\&Delta;}{z}_i<0;\frac{{-w}_i}{\mathrm{\&Delta;}{w}_i},\mathrm{\&Delta;}{w}_i>0),1\}$

(8) upgrade original variable and Lagrange multiplier;

(9) judge whether iterations is greater than K _max, if so, calculate and do not restrain, quit a program, if not, put iterations and add 1, return to (3).