CN102545207A - 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 complicacy not only in large scale, and it has extremely strong importance to the national economic development.This is a problem that merits attention with regard to the economy that has determined power system operation.Along with the development of society, the consumption of the energy is more and more big, and energy savings receives people's common concern.Electric power is again the most important aspect of current energy resource consumption, therefore, under the prerequisite that satisfies the power system power supply reliability and the quality of power supply, improve the economy of operation as far as possible, reasonably utilizes the existing energy and equipment, with minimum fuel consumption.

Development along with power grid construction; Direct current transportation in electrical network more and more widely; Increasing alternating current-direct current hybrid system will appear; (Voltage Source Converter VSC) is HVDC of new generation (High Voltage Direct Current, HVDC) transmission of electricity on basis with full-controlled switch device and voltage source converter; Than direct current transportation, have directly to isolated distant loads power supply, more economical send advantages such as electricity, operation control method be flexible and changeable to load center based on thyristor.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 is found the solution optimal load flow, need not to estimate the active constraint collection difficult extensive use interior some algorithm convergence rapidly, strong robustness, insensitive to the selection of initial value.Compare with former-antithesis interior point method, prediction-correction interior point has only increased a former generation back substitution in each iteration calculates, but can obviously reduce the convergence number of times, and optimal speed obviously improves.Add after the VSC-HVDC, variation has taken place in system OPF model, need ac and dc systems be combined according to the stable state tide model of VSC-HVDC, carries out simultaneous optimization and finds the solution.

Summary of the invention

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

The present invention adopts following technical scheme for realizing above-mentioned purpose:

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 said method realizes in computer successively 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; Generator is meritorious exerts oneself, idle bound, economic parameters;

(2) program initialization comprises: quantity of state is provided with initial value, Lagrange multiplier and penalty factor are provided with 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 satisfies C _Gap＜ε, if, then export result of calculation, withdraw from circulation, if not, then continue;

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

$\left[\begin{array}{cc}{H}^{\′}& {\▿}_{x}h\left(x\right)\\ {\▿}_x^{T}h\left(x\right)& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{\Δx}}^{\mathrm{aff}}\\ {\mathrm{\Δy}}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}{L}_x^{\′}\\ -L_y\end{array}\right]$

$\left[\begin{array}{cc}I& L^{-1}Z\\ 0& I\end{array}\right]\left[\begin{array}{c}{\mathrm{\Δz}}^{\mathrm{aff}}\\ {\mathrm{\Δl}}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}-L^{-1}{L}_l^{\mathrm{\μ}}\\ L_z+{\▿}_x^{T}g\left(x\right)\mathrm{\Δx}\end{array}\right]$

$\left[\begin{array}{cc}I& U^{-1}W\\ 0& I\end{array}\right]\left[\begin{array}{c}{\mathrm{\Δw}}^{\mathrm{aff}}\\ {\mathrm{\Δu}}^{\mathrm{aff}}\end{array}\right]=\left[\begin{array}{c}-U^{-1}{L}_u^{\mathrm{\μ}}\\ -L_w-{\▿}_x^{T}g\left(x\right)\mathrm{\Δx}\end{array}\right]$

Wherein: Δ x ^Aff, Δ y ^Aff, Δ z ^Aff, Δ l ^Aff, Δ u ^Aff, Δ w ^AffAffine deflection correction for x, y, z, l, u, w.

$L_x^{\′}=L_x+{\▿}_{x}g\left(x\right)[L^{-1}(L_l^{\mathrm{\μ}}+{\mathrm{ZL}}_z)+U^{-1}(L_u^{\mathrm{\μ}}+{\mathrm{WL}}_w)]$

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

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

Confirm 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:

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

The dynamic estimation Center Parameter:

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

(5) calculate the disturbance factor mu;

(6) complementary relaxation condition is revised:

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

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

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) confirm 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 that whether iterations is greater than K _MaxIf,, then calculate and do not restrain, quit a program, if not, then put iterations and add 1, return (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 the advantage such as electricity of sending to load center.

Some scholars have carried out many researchs to the alternating current-direct current optimal load flow that contains 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 stable state tide 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.

Description of drawings

Fig. 1: the inventive method flow chart.

Fig. 2: the alternating current-direct current hybrid system 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 the IEEE-14 node system, and figure (b) is the IEEE-30 node system, and figure (c) is the IEEE-57 node system.

Embodiment

Fig. 1 is the inventive method flow chart, the alternating current-direct current hybrid system model that Fig. 2 adopts for the present invention.I representes to insert i VSC of DC network among Fig. 2.The fundamental voltage phasor of supposing i VSC output does

With the voltage phasor of AC system junction do

The 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 does

Suppose that direction is as shown in Figure 1, then

The complex power

that AC system flows into converter transformer satisfies following relational expression:

In order to discuss conveniently, make δ _i=θ _Si-θ _Ci,

α _i=arctan (X _Li/ R _i), further derivation can get

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

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

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 _iSo equivalence is direct current power P _DiShould with the P that injects converter bridge _CiEquate, therefore can get

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

U wherein _Di, I _DiBe respectively the direct voltage and the electric current of direct current node.In addition, voltage equation is:

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

M wherein _iIt is the modulation degree of i VSC.

Above-mentioned 8 equations have constituted mark one system steady-state model of VSC-HVDC down.

Among the VSC-HVDC, the direct relation of whether stablizing of direct voltage that can system normally move and the stability of AC side output voltage.If the active power that the VSC of meritorious transmitting terminal absorbs from this end AC system sends to the active power of corresponding end AC system greater than receiving terminal VSC, direct voltage raises, otherwise direct voltage reduces.Therefore in order to realize this power-balance, wherein an end VSC must adopt and decide direct voltage control.In addition, if direct voltage is constant, then the variable quantity of direct current is proportional to the amount of unbalance of active power, and it is equivalent then deciding direct current control and deciding active power control.Comprehensive above the analysis, the control mode that VSC can select among the VSC-HVDC has following several kinds:

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

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

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

4.. decide active power, decide 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.The direct current node is meant the node that primary side connected of converter transformer, node as shown in Figure 1, owing to connected converter exchanging on the node, its corresponding control and state variable are at the former exchange status variable U that exchanges node _i, θ _iIncreased direct current variable U on the basis _Di, I _Di, M _i, P _Si, Q _Si, δ wherein _i, M _iPhase angle and modulation degree for converter; Pure interchange node is meant the node that does not link to each other with converter transformer.The node of uniting of setting up departments adds up to n, and wherein the number of VSC is n _c, then direct current node number is n _c, pure interchange node number is n _a=n-n _cIn order to compose a piece of writing conveniently, suppose that the node serial number of alternating current-direct current hybrid system is in proper order: 1～n below _aNode is pure interchange node; n _a+ 1～n node is the direct current node.

For the 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 _TiBe direct current node power amount of unbalance; For the node behind the 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 the formula) that directly link to each other with node i; U _jFor exchanging node voltage amplitude, θ _IjBe node i, phase angle difference between j; G _Ij, B _IjBe node i, the real part of admittance and imaginary part between j.P _Si, Q _SiFlow into the meritorious and reactive power of converter transformer for AC system.

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

${\mathrm{\&Delta;<figure-callout\; id="1"\; label="d\; i"\; filenames="HDA0000123417890000023.png,HDA0000123417890000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_i=P_{\mathrm{si}}+(\sqrt{6}/4)M_i{U}_{\mathrm{si}}{U}_{\mathrm{di}}\left|Y_i\right|\cos ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)-{\mathrm{<figure-callout\; id="2"\; label="U\; si"\; filenames="HDA0000123417890000023.png"\; state="\{\{state\}\}">U</figure-callout>}}_{\mathrm{si}}^{}\left|Y_i\right|\cos {\mathrm{\&alpha;}}_i$

${\mathrm{\&Delta;<figure-callout\; id="2"\; label="d\; i"\; filenames="HDA0000123417890000023.png"\; state="\{\{state\}\}">d</figure-callout>}}_i=Q_{\mathrm{si}}+(\sqrt{6}/4)M_i{U}_{\mathrm{si}}{U}_{\mathrm{di}}\left|Y_i\right|\sin ({\mathrm{\&delta;}}_i+{\mathrm{\&alpha;}}_i)-{\mathrm{<figure-callout\; id="2"\; label="U\; si"\; filenames="HDA0000123417890000023.png"\; state="\{\{state\}\}">U</figure-callout>}}_{\mathrm{si}}^{}\left|Y_i\right|\sin {\mathrm{\&alpha;}}_i-{\mathrm{<figure-callout\; id="2"\; label="U\; si"\; filenames="HDA0000123417890000023.png"\; state="\{\{state\}\}">U</figure-callout>}}_{\mathrm{si}}^{}/X_{\mathrm{fi}}$

${\mathrm{\&Delta;<figure-callout\; id="3"\; label="d\; i"\; filenames="HDA0000123417890000023.png,HDA0000123417890000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_i=U_{\mathrm{di}}{I}_{\mathrm{di}}-(\sqrt{6}/4)M_i{U}_{\mathrm{si}}{U}_{\mathrm{di}}\left|Y_i\right|\cos ({\mathrm{\&delta;}}_i-{\mathrm{\&alpha;}}_i)+(3/8){\left(M_i{U}_{\mathrm{di}}\right)}^2\left|Y_i\right|\cos {\mathrm{\&alpha;}}_i$

In the formula, subscript i representes i VSC.

Add the current deviation amount equation of DC network:

${\mathrm{\&Delta;<figure-callout\; id="4"\; label="d\; i"\; filenames="HDA0000123417890000023.png,HDA0000123417890000031.png"\; state="\{\{state\}\}">d</figure-callout>}}_i=I_{\mathrm{di}}-\underset{j=1}{\overset{n_c}{\mathrm{\&Sigma;}}}{g}_{\mathrm{dij}}{U}_{\mathrm{dj}}$

G wherein _DijBe the electric conductivity value between direct current node i, the j.

The VSC-HVDC system is linked into after the electrical network, and the stable state tide model of system changes, and needs to adopt above-mentioned VSC steady-state model to carry out the calculating of OPF problem.

The OPF model can be expressed as following nonlinear optimization model:

obj.min.f(x)

s.t.h(x)＝0

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

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

The basic ideas of interior point method are: with slack variable inequality constraints is converted into equality constraint, utilizes Lagrange multiplier that constraint is incorporated in the target function again, and slack variable is retrained with barrier function method.For the OPF problem, the structure Lagrangian is following:

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

Y=[y wherein ₁, L, y _m] ^TBe the Lagrange multiplier of equality constraint, z=[z ₁, L, z _r] ^T, w=[w ₁, L, w _r] ^TBe the Lagrange multiplier of inequality constraints, l=[l ₁, L, l _r] ^T, u=[u ₁, L, u _r] ^TBe the slack variable of inequality constraints, μ is the penalty factor of barrier function.

The KKT of this problem (Karush-Kuhn-Tucker) condition 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 the formula:

is 1 order derivative of f (x) to x, and

is respectively the Jacobian matrix of h (x), g (x).

L＝diag(l ₁，L，l _r)U＝diag(u ₁，L，u _r)，Z＝diag(z ₁，L，z _r)W＝diag(w ₁，L，w _r)，

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

By latter two equation in the formula KKT condition can in the hope of

μ＝(l ^Tz-u ^Tw)/2r

Definition C _Gap=l ^TZ-u ^TW.

But facts have proved, when the parameter in the target function during according to the following formula value convergence poor, the general employing

μ＝σC _Gap/2r，

Center Parameter σ is the important parameter that influences algorithm performance, and predict-core concept of correction interior point is the dynamic estimation to Center Parameter σ.Compare with former antithesis interior point method, 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 is obtained affine direction through predicting the step in each iteration, utilize its second order term of estimating complementary equation Taylor expansion then, obtains and proofreaies and correct the step.

The prediction step:

1, sets Center Parameter σ=0;

2, find the solution 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 ^AffImitative for x, y, z, l, u, w

Penetrate deflection correction.

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

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

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

3, confirm 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:

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

5, dynamic estimation Center Parameter:

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

Proofread and correct the step:

1, complementary relaxation condition is revised:

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

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

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.

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

The present invention adopts prediction-correction interior point that VSC-HVDC alternating current-direct current hybrid system is carried out optimal load flow and calculates; Prediction-correction interior point optimization effect of having verified the present invention's proposition is remarkable; And on the ability that solves VSC-HVDC alternating current-direct current optimal power flow problems; The more former antithesis interior point method of this method iterations still less, amount of calculation is littler.

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 the direct current branch parameter is seen table 1.

Each variable initial value of table 1VSC

N	R	X L	P 1	Q 1	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 representes female wire size at VSC place, P ₁, Q ₁Meritorious, 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 _dBe direct voltage.R is the equivalent resistance of inverter inside loss and converter transformer loss, R, X _L, Q ₁, Q ₁Given by system; P _s, Q _sInitial value be made as with revise before system branch power equate, calculate by the trend of original system.I _d, δ, M obtain by following:

I _di＝(P _gi-P _li)/U _di

${\mathrm{\&delta;}}_i=\arctan (P_{\mathrm{ti}}/({\mathrm{<figure-callout\; id="2"\; label="U\; ti"\; filenames="HDA0000123417890000023.png"\; state="\{\{state\}\}">U</figure-callout>}}_{\mathrm{ti}}^{}/X_{\mathrm{Li}}-Q_{\mathrm{ti}}))$

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

P wherein _GiMeritorious for the node generating, if this node is not the generator node, then be 0.

Adopt prediction-correction interior point that VSC-HVDC alternating current-direct current hybrid system is carried out optimal load flow calculating (direct current branch two ends VSC control mode be combined as 1.+3.), simulation result is shown in following table 2 and 3.Except that generation cost unit was $, other data unit was perunit value in the table.

Table 2 AC system result

Table 3 direct current system result

Visible by 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%.Each VSC-HVDC control and state variable value before and after table 3 has been listed and optimized, the modulation degree M of VSC all increases to some extent, has improved the utilance of direct voltage, has reduced the AC side percent harmonic distortion.

Example two:

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

Each system cost result under table 4 different control modes

The optimization effect that each example obtains under different control modes is difference slightly, and main cause is to be provided with the Different control mode, in fact is equivalent to strengthen the operation constraint of VSC-HVDC.Stronger by visible this algorithm applicability of the optimization effect of each example, can both successfully be optimized the system under the various control modes, and it is remarkable to optimize effect, is example with the IEEE-57 node, the expense reduction reaches 35%.

Table 5 is pair under different control modes, and the iterations when utilizing former antithesis interior point method and prediction-each example of correction interior point optimization compares, and A representes former antithesis interior point method in the table, and B representes prediction-correction interior point.

Former antithesis interior point method of table 5 and prediction-correction interior point iterations relatively

Hence one can see that, and under equal conditions, compared to former antithesis interior point method, prediction-correction interior point only needs when each iteration, to increase a little amount of calculation in the prediction step and the step of correction, and iterations but can reduce 30% even more.

Claims (1)

1. VSC-HVDC alternating current-direct current optimal load flow method based on prediction-correction interior point is characterized in that said 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; Generator is meritorious exerts oneself, idle bound, economic parameters;

(2) program initialization comprises: quantity of state is provided with initial value, Lagrange multiplier and penalty factor are provided with 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 satisfies C _Gap＜ε, if, then export result of calculation, withdraw from circulation, if not, then continue;

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

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

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

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

Wherein: Δ x ^Aff, Δ y ^Aff, Δ z ^Aff, Δ l ^Aff, Δ u ^Aff, Δ w ^AffAffine deflection correction for x, y, z, l, u, w.

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

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

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

Confirm 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:

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

The dynamic estimation Center Parameter:

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

(5) calculate the disturbance factor mu;

(6) complementary relaxation condition is revised:

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

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

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) confirm 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 that whether iterations is greater than K _MaxIf,, then calculate and do not restrain, quit a program, if not, then put iterations and add 1, return (3).

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