CN106026105A - Power system optimal power flow (OPF) control method based on punishment concave-convex optimization technology - Google Patents

Abstract

The invention discloses a power system optimal power flow (OPF) control method based on a punishment concave-convex optimization technology. The power system OPF control method comprises the steps of: firstly, acquiring network parameters of a power system, and determining a mathematical model of an OPF control problem; secondly, utilizing a punishment thought and Taylor expansion to obtain a current nearly convex problem of an OPF problem; thirdly, solving the OPF control nearly convex problem iteratively to obtain bus input power values; finally, completing the power system OPF control according to the bus input power values obtained through calculation. The power system OPF control method designs the power system OPF by utilizing the punishment concave-convex optimization technology, and can minimize system performance loss while ensuring feasibility of a power flow equation.

Description

A kind of based on the Optimal Power Flow Problems control method punishing concavo-convex optimisation technique

Technical field

The present invention relates to technical field of power systems, be specifically related to a kind of based on the power system punishing concavo-convex optimization thought Optimal load flow control method.

Background technology

Optimization Problems In Power Systems, including planning, dispatch, running on control, its target is security of system and economy Balance and compromise.As one of most important of which problem, optimal load flow (Optimal Power Flow, OPF) control is The structural parameters of finger power system and load condition have given timing the most, regulate available control variable (such as electromotor output work Rate, adjustable transformer tap etc.) find and can meet all operation constraints, and make system a certain performance indications (as Cost of electricity-generating or via net loss) power flowcontrol when minimizing value.In recent years, along with intelligent grid, distributed generation technology, The fast development of distribution type electric energy memory technology, on the premise of meeting power system security, improves economy as much as possible, Make rational use of resources configuration and existing equipment controls this classical problem with the optimal load flow reducing energy resource consumption and becomes again and grind Study carefully focus.

Since the sixties in 20th century, optimal load flow is as Operation of Electric Systems and the powerful tool of analysis, again always Concerned.Through the development of nearly 50 years, numerous optimization methods were introduced sequentially into this field, such as: linear programming, quadratic programming, Non-Linear Programming and Newton method and decoupling method etc..But optimal load flow is a typical nonlinear optimal problem, and due to about The complexity of bundle makes it calculate complexity, and difficulty is bigger.Currently, document [M.Farivar and S.H.Low, " Branch Flow model:Relaxations and convexification (parts I, II), " IEEE Trans.Power Syst., propose in vol.28, no.3, pp.2554-2572,2013] and utilize that convex relaxation method SOCP is lax to be solved Excellent Power Flow Problem, however its to demonstrate this under certain condition lax be tight.For the optimal power flow problems of non-convex, convex pine Relaxation method the most all cannot ensure to obtain the feasible solution of problem.Therefore, the present invention proposes based on the optimum tide punishing concavo-convex optimization Method of flow control.

Summary of the invention

Present invention aims to the deficiencies in the prior art, it is provided that a kind of power system based on concavo-convex optimization thought Optimal load flow control method, the inventive method considers the control problem of inverter in distributed power generation, it is ensured that in iteration During Power System Performance always monotone decreasing is lost, it is possible to realize while reaching each constraint requirements of power system reduce The purpose of Power System Performance loss, completes Optimal Power Flow Problems control.Specifically include following steps:

Step 1: obtain power system network parameter: bus set N and going and dig up the roots the set N after bus⁺=N { 0}；Electricity Net set of fingers ξ；The impedance z of branch road between bus_ij,Each bus nodes injecting power s_iConstraint set S_i；Female Square v of line voltage amplitude value_iLower limitv _iAnd the upper limit

Step 2: initialize iterations k=0, maximum iteration time K_max, convergence precision tol；Set initial pointDetermine penalty coefficient β；l_ijRepresent from bus i to the current amplitude of bus j square；S_ijRepresent bus i Trend to the line transmitting terminal between j；

Step 3: utilize Taylor expansion, obtains the convex problem of approximation of optimal power flow problems:

$OPF:\min \underset{i\∈N}{\Σ}{f}_i\left(\mathrm{Re}\right(s_i\left)\right)+\underset{(i,j)\∈\mathrm{\ξ}}{\Σ}\mathrm{\β}(l_{ij}-\frac{2\mathrm{Re}\left({\stackrel{\‾}{S}}_{ij}^k{S}_{ij}\right)}{v_i^k}+\frac{{\left|S_{ij}^k\right|}^2{v}_i}{{\left(v_i^k\right)}^2})$

Optimized variable: s, S, v, l, s₀

$\begin{array}{cc}s.t& S_{ij}=s_i+\underset{h:h\→i}{\Σ}(S_{hi}-z_{hi}{l}_{hi}),\∀(i,j)\∈\mathrm{\ξ}\\ & 0=s_0+\underset{h:h\→0}{\Σ}(S_{h0}-z_{h0}{l}_{h0})\\ & v_i-v_j=2\mathrm{Re}\left({\stackrel{\‾}{z}}_{ij}{S}_{ij}\right)-{\left|z_{ij}\right|}^2{l}_{ij},\∀(i,j)\∈\mathrm{\ξ}\\ & l_{ij}\≥\frac{{\left|S_{ij}\right|}^2}{v_i},\∀(i,j)\∈\mathrm{\ξ}\\ & s_i\∈S_i,i\∈N^{+}\\ & {\underset{\‾}{v}}_i\≤v_i\≤{\stackrel{\‾}{v}}_i,i\∈N^{+}\end{array}---\left(P1\right)$

Wherein | a |,The plural number amplitude of a, conjugation and real part is represented respectively with Re (a)；Represent and note at bus i Enter the system performance loss caused by power；

Step 4: Solve problems (P1) obtains current result of calculationJudge whether to reach convergence Precision:Or whether reach maximum iteration time: k ＞ K_max；If so, the injecting power of bus is exported, meter Calculate system performance loss, perform step 5；Otherwise make iterations k=k+1, repeat step 3 and 4.

Step 5: complete optimal load flow control according to calculated bus injecting power.

Further, in described step 4, Solve problems (P1) obtains result of calculationSolution party Method is interior-point algohnhm.

The method have the benefit that first the inventive method constructs optimal load flow correspondence planning problem, think followed by punishment Think and Taylor expansion, obtain the optimal power flow problems convex problem of current approximation；Solve optimal load flow the most iteratively and control approximation Convex problem obtains each bus injecting power value；Power system optimum tide is completed finally according to calculated bus injecting power value Flow control.The present invention utilizes punishment concavo-convex optimisation technique design Optimal Power Flow Problems, it is possible to ensureing that power flow equation is feasible Make system performance loss minimum on the premise of property.

Accompanying drawing explanation

Fig. 1 is the system model figure that the embodiment of the present invention uses the method.

Fig. 2 is the particular flow sheet that the embodiment of the present invention uses the method.

Fig. 3 is the desired value graph of a relation with iterations of the embodiment of the present invention.

Fig. 4 is embodiment of the present invention system restriction feasibility criterion and iterations graph of a relation.

Detailed description of the invention

In order to make the purpose of the present invention and effect clearer, the specific embodiment party to the inventive method below in conjunction with the accompanying drawings Formula is described in detail.

As shown in Figure 1, it is considered to radial power distribution network, it is made up of the line of bus and connection bus.In this network Root node is substation bus bar (for convenience of describing, hereinafter referred to as root bus), and it is connected with power transmission network.Root bus uses fixing Voltage, the electric power received from transmission network is assigned to other buses simultaneously.The present invention defines this root bus for mother Line 0, other buses are 1 ..., n；It addition, make N:={0 ..., n} represents all buses in electrical network, defines N⁺:=N { 0}； (i, j) represents that bus i is connected with bus j, and direction is i → j, and bus j is on the bus i exclusive path with bus 0.Make ξ table Show the set of all branch roads in network, to arbitrarily (i, j) ∈ ξ represent vector branch i → j.

For any bus i ∈ N, make v_iRepresent voltage amplitude value at bus i square.As it has been described above, transformer station is female The voltage of line is fixed value v₀.Definition s_i=p_i+iq_iRepresent the injecting power at bus i, wherein p_i, qi represent injection respectively Active power and reactive power.It addition, definition P_iFor the exclusive path between bus i to bus 0, for Radial network, P_i It is unique.For any line, (i, j) ∈ ξ, make l_ijRepresent from bus i to the current amplitude of bus j square, z_ij= r_ij+ix_ijRepresent bus i, the impedance of line between j；Make S_ij=P_ij+iQ_ijThe tide of the line transmitting terminal between expression bus i to j Stream (or claiming power stream), wherein P_ijAnd Q_ijRepresent active power stream and reactive power flow respectively.It addition, for plural number a ∈ C, useRepresent the conjugation of a.

Given network topology (N, ξ), impedance z and substation bus bar voltage v₀Time, then other electrical network parameters (s, S, v, L, s₀) can be expressed as follows by a flow model (branch flow model) for radial network:

$S_{ij}=s_i+\underset{h:h\→i}{\Σ}(S_{hi}-z_{hi}{l}_{hi}),\∀(i,j)\∈\mathrm{\ξ}---\left(1a\right)$

$0=s_0+\underset{h:h\→0}{\Σ}(S_{h0}-z_{h0}{l}_{h0})---\left(1b\right)$

$v_i-v_j=2\mathrm{Re}\left({\stackrel{\‾}{z}}_{ij}{S}_{ij}\right)-{\left|z_{ij}\right|}^2{l}_{ij},\∀(i,j)\∈\mathrm{\ξ}---\left(1c\right)$

$l_{ij}=\frac{{\left|S_{ij}\right|}^2}{v_i},\∀(i,j)\∈\mathrm{\ξ}---\left(1d\right)$

Formula (1a) and (1b) are power balance equations, and formula (1c) and (1d) are the identical transformations of ohm formula.

The present invention considers following several power distribution network controllable device: distributed generator, inverter, controllable load, such as electricity Motor-car, intelligent appliance, shnt capacitor.In actual applications, electrical network is injected by control shnt capacitor and inverter Reactive power regulates voltage.After setting injecting power s, may determine that other electrical quantity (S, v, l, s by formula (1)₀)。

Dissimilar according to controllable device, electrical network median generatrix i ∈ N⁺Injecting power s_iThere is different constraint set S_i, it may be assumed that

s_i∈S_i, i ∈ N⁺ (2)

According to device type definition set S_iFor:

If 1. s_iRepresenting a rated capacity isShnt capacitor, then If s_iRepresenting a maximum generating watt isSolar energy electroplax, it by a capacity isInverter be connected with electrical network, that ?

If 2. s_iRepresent a power factor be η, active power consumption is in intervalContinually varying adjustable negative Carry, then

Note, s_iThe injecting power that multiple the said equipment is total can be represented.

In addition, it is necessary to by square v of the voltage amplitude value of bus i_iControl at voltage lower limit value set in advancev _iAnd voltage Higher limitBetween, i.e. need to meet

${\underset{\‾}{v}}_i\≤v_i\≤{\stackrel{\‾}{v}}_i,i\∈N^{+}$

Under conditions of the constraint of power stream, voltage constraint, injecting power constraint, optimal power flow problems can be described as follows:

$OPF:min\underset{i\∈N}{\Σ}{f}_i\left(\mathrm{Re}\right(s_i\left)\right)$

Optimized variable: s, S, v, l, s₀

$\begin{array}{cc}s.t& s_{ij}=s_i+\underset{h:h\→i}{\Σ}(S_{hi}-z_{hi}{l}_{hi}),\∀(i,j)\∈\mathrm{\ξ}\end{array}---\left(3a\right)$

$0=s_0+\underset{h:h\→0}{\Σ}(S_{h0}-z_{h0}{l}_{h0})---\left(3b\right)$

$v_i-v_j=2\mathrm{Re}\left({\stackrel{\‾}{z}}_{ij}{S}_{ij}\right)-{\left|z_{ij}\right|}^2{l}_{ij},\∀(i,j)\∈\mathrm{\ξ}---\left(3c\right)$

$l_{ij}=\frac{{\left|S_{ij}\right|}^2}{v_i},\∀(i,j)\∈\mathrm{\ξ}---\left(3d\right)$

s_i∈S_i, i ∈ N⁺(3e)

${\underset{\‾}{v}}_i\≤v_i\≤{\stackrel{\‾}{v}}_i,i\∈N^{+}---\left(3f\right)$

Wherein in object functionRepresent the system performance loss that bus i injecting power is caused.If for appointing Meaning i ∈ N, has f_i(x)=x, thenI.e. represent the total-power loss in electrical network.

Owing to existing such as Non-convex constraint, above-mentioned optimal power flow problems is non-convex optimization problem, It is difficult to solve.Document [M.Farivar and S.H.Low, " Branch flow model:Relaxations and Convexification (parts I, II), " IEEE Trans.Power Syst., vol.28, no.3, pp.2554-2572, 2013] propose in and utilize SOCP relaxation method to solve optimal power flow problems, but demonstrate this most under certain condition and relax It is tight.For one optimal power flow problems, convex relaxation method the most all cannot ensure to obtain the feasible solution of problem.Therefore, The present invention proposes based on the optimal load flow control method punishing concavo-convex optimization.

As a example by non-convex constraint (3d), first (3d) is converted into two inequality constraints:

$l_y\≥\frac{{\left|S_{ij}\right|}^2}{v_i},l_{ij}\≤\frac{{\left|S_{ij}\right|}^2}{v_i}$

Wherein the former is convex constraint, and the latter retrains for non-convex.For the latter, by introducing penalty term, move it to target In function, obtain punishment problem:

$\begin{array}{c}\min \underset{i\∈N}{\Σ}{f}_i\left(\mathrm{Re}\right(s_i\left)\right)+\mathrm{\β}\underset{(i,j)\∈\mathrm{\ξ}}{\Σ}(l_{ij}-\frac{{\left|S_{ij}\right|}^2}{v_i})\\ overs,S,v,l,s_0\\ \begin{array}{cc}s.t& \left(3a\right)-\left(3c\right),\left(3e\right)-\left(3f\right)\\ & l_{ij}\≥\frac{{\left|S_{ij}\right|}^2}{v_i},\∀(i,j)\∈\mathrm{\ϵ}\end{array}\end{array}---\left(4\right)$

May certify that, when punishing parameter beta more than certain threshold value.Problem (4) can be solved by concavo-convex optimization. Specifically, given S_ijAnd v_iCurrency S_pre_ijAnd v_pre_i, the penalty term in problem (4) is carried out line by Taylor expansion Property is approached, it may be assumed that

$\underset{i\∈N}{\Σ}\mathrm{\β}\×(l_{ij}-\frac{2\×\mathrm{Re}(S\_{\mathrm{pre}}_{ij}^{\′}\·S_{ij})}{v\_{\mathrm{pre}}_i}+\frac{abs{(S\_{\mathrm{pre}}_{ij})}^2\·v_i}{{(v\_{\mathrm{pre}}_i)}^2})---\left(5\right)$

Following convex problem can be obtained,

$\begin{array}{c}\min \underset{i\∈N}{\Σ}{f}_i\left(\mathrm{Re}\right(s_i\left)\right)+\underset{(i,j)\∈\mathrm{\ξ}}{\Σ}\mathrm{\β}(l_{ij}-\frac{2\×\mathrm{Re}(S\_{\mathrm{pre}}_{ij}^{\′}\·S_{ij})}{v\_{\mathrm{pre}}_i}+\frac{abs{(S\_{\mathrm{pre}}_{ij})}^2\·v_i}{{(v\_{\mathrm{pre}}_i)}^2})\\ overs,S,v,l,s_0\\ \begin{array}{cc}s.t& \left(3a\right)-\left(3c\right),\left(3e\right)-\left(3f\right)\\ & l_{ij}\≥\frac{{\left|S_{ij}\right|}^2}{v_i},\∀(i,j)\∈\mathrm{\ϵ}\end{array}\end{array}---\left(6\right)$

Solve above-mentioned convex problem iteratively until algorithmic statement, optimal load flow can be obtained and control result.

Fig. 2 gives the flow chart of above-mentioned Optimal Power Flow Problems control method based on concavo-convex optimisation technique.Specifically Ground, can be described as follows:

A kind of Optimal Power Flow Problems control method based on the concavo-convex optimisation technique of punishment, the method includes walking as follows Rapid:

Step 1: obtain power system network parameter: bus set N and going and dig up the roots the set N after bus⁺=N { 0}；Electricity Net set of fingers ξ；The impedance z of branch road between bus_ij,Each bus nodes injecting power s_iConstraint set S_i；Female Square v of line voltage amplitude value_iLower limitvI and the upper limit

Step 2: initialize iterations k=0, maximum iteration time K_max, convergence precision tol；Set initial pointDetermine penalty coefficient β；l_ijRepresent from bus i to the current amplitude of bus j square；S_ijRepresent bus i Trend to the line transmitting terminal between j；

Step 3: utilize Taylor expansion, obtains the convex problem of approximation of optimal power flow problems:

$OPF:\min \underset{i\∈N}{\Σ}{f}_i\left(\mathrm{Re}\right(s_i\left)\right)+\underset{(i,j)\∈\mathrm{\ξ}}{\Σ}\mathrm{\β}(l_{ij}-\frac{2\mathrm{Re}\left({\stackrel{\‾}{S}}_{ij}^k{S}_{ij}\right)}{v_i^k}+\frac{{\left|S_{ij}^k\right|}^2{v}_i}{{\left(v_i^k\right)}^2})$

Optimized variable: s, S, v, l, s₀

$\begin{array}{cc}s.t& S_{ij}=s_i+\underset{h:h\→i}{\Σ}(S_{hi}-z_{hi}{l}_{hi}),\∀(i,j)\∈\mathrm{\ξ}\\ & 0=s_0+\underset{h:h\→0}{\Σ}(S_{h0}-z_{h0}{l}_{h0})\\ & v_i-v_j=2\mathrm{Re}\left({\stackrel{\‾}{z}}_{ij}{S}_{ij}\right)-{\left|z_{ij}\right|}^2{l}_{ij},\∀(i,j)\∈\mathrm{\ξ}\\ & l_{ij}\≥\frac{{\left|S_{ij}\right|}^2}{v_i},\∀(i,j)\∈\mathrm{\ξ}\\ & s_i\∈S_i,i\∈N^{+}\\ & {\underset{\‾}{v}}_i\≤v_i\≤{\stackrel{\‾}{v}}_i,i\∈N^{+}\end{array}---\left(P1\right)$

Wherein | a |,The plural number amplitude of a, conjugation and real part is represented respectively with Re (a)；Represent at bus i System performance loss caused by injecting power；

Step 4: Solve problems (P1) obtains current result of calculationJudge whether to reach convergence Precision:Or whether reach maximum iteration time: k ＞ K_max；If so, the injecting power of bus is exported, meter Calculate system performance loss, perform step 5；Otherwise make iterations k=k+1, repeat step 3 and 4.

Step 5: complete optimal load flow control according to calculated bus injecting power.

Further, in described step 4, Solve problems (P1) obtains result of calculationSolution party Method is interior-point algohnhm.

Below by instantiation, technical scheme is further elaborated.In experiment, use SCE-47 and SCE-56 network system carries out proof of algorithm.Specifically, following experiment parameter is used:

1. setting power loss is minimised as target, and the voltage V of substation bus bar₀It it is the reference voltage of a unit Value；

2. for the setting of injecting power restrained boundary, any bus i ∈ N in electrical network⁺Place there may be multiple equipment, Such as shunt capacitance, tunable load, solar energy electroplax etc.；Assume electrical network always co-exists in D_iIndividual equipment and by its numbered 1, 2 ..., D_i；For d=1,2 ..., D_i, s_idThe injecting power of expression equipment d.

If equipment d is a load, and known active power consumes p and reactive power consumption q, then now s_id=-p- i·q；If the apparent energy peak value S of known load d_peak, then s_id=-S_peakexp(jθ).Wherein, θ=arccos (0.9), Now, the injecting power s of load_idNamely a constant；

If equipment d is a capacityElectric capacity, then have

If equipment d is a capacityPhotovoltaic electroplax, then

Setting according to above, now total for bus i injecting power is

Other parameters set as follows: determine each reference capacity value according to electrical network practical situation, initialize bus sum N, electricity Resistance r_ij, reactance x_ijAnd at bus the capacity of relevant device type or active power consumption figures, concurrently set transformer station's node The reference power value that power is a unit, make iteration total degree K_max=20, v_maxEqual to 1.1 cell voltage base values, v_minDeng In 0.9 cell voltage base value, convergence precision tol=0.001, punish parameter beta=0.001, initialize bus voltage upper limitLower voltage limitIt addition, in the present embodiment, definitionFor constraint feasibility criterion, its value Arrive feasible close to explanation when 0.

Fig. 3,4 it is by the Matlab simulation results figure to designed method.

Fig. 3 sets forth and applies the calculating of the inventive method to tie in SCE-47 bus-bar system and SCE-56 bus-bar system Really.In order to compare, figure give also the result of SOCP relaxation method.It can be seen that the inventive method can be quick Convergence, makes system total power consumption along with iterations while meeting electric power system tide equation, power and voltage constraint Constantly reduce until restraining, and optimal load flow control method based on concavo-convex optimisation technique has reached and optimum tide based on SOCP The desired value that method of flow control is almost identical, illustrates that technical solution of the present invention is capable of optimal load flow control.

Fig. 4 sets forth after applying the inventive method in SCE-47 bus-bar system and SCE-56 bus-bar system, model Middle constraint feasibility criterion and iterations graph of a relation.It can be seen that along with the increase of iterations, constraints by Gradually it is met, and after iteration the 2nd time, the inventive method can meet constraint feasibility criterion.

The present invention is not only limited to above-mentioned detailed description of the invention, and one technical staff of this area is according to disclosed by the invention interior Hold, other multiple specific embodiments can be used to implement the present invention.Therefore, the design structure of every employing present invention and think of Road, does some simply change or designs of change, both falls within scope.

Claims (2)

1. an Optimal Power Flow Problems control method based on the concavo-convex optimisation technique of punishment, it is characterised in that the method bag Include following steps:

Step 1: obtain power system network parameter: bus set N and going and dig up the roots the set N after bus⁺=N { 0}；Grid branch Set ξ；The impedance z of branch road between bus_ij,Each bus nodes injecting power s_iConstraint set S_i；Busbar voltage Square v of range value_iLower limitv _iAnd the upper limit

Step 2: initialize iterations k=0, maximum iteration time K_max, convergence precision tol；Set initial pointDetermine penalty coefficient β；l_ijRepresent from bus i to the current amplitude of bus j square；S_ijRepresent bus i Trend to the line transmitting terminal between j；

Step 3: utilize Taylor expansion, obtains the convex problem of approximation of optimal power flow problems:

$OPF:\min \underset{i\∈N}{\mathrm{\Σ}}{f}_i\left(\mathrm{Re}\left(s_i\right)\right)+\underset{\left(i,j\right)\∈\mathrm{\ξ}}{\mathrm{\Σ}}\mathrm{\β}\left(l_{ij}-\frac{2\mathrm{Re}\left({\stackrel{\‾}{S}}_{ij}^k{S}_{ij}\right)}{v_i^k}+\frac{|S_{ij}^k{|}^2{v}_i}{{\left(v_i^k\right)}^2}\right)$

Optimized variable: s, S, v, l, s₀

$\begin{array}{cc}s.t& S_{ij}=s_i+\underset{h:h\→i}{\Σ}(S_{hi}-z_{hi}{l}_{hi}),\∀(i,j)\∈\mathrm{\ξ}\end{array}$

$0=s_0+\underset{h:h\→0}{\Σ}(S_{h0}-z_{h0}{l}_{h0})$

$v_i-v_j=2\mathrm{Re}\left({\stackrel{\‾}{z}}_{ij}{S}_{ij}\right)-|z_{ij}{|}^2{l}_{ij},\∀(i,j)\∈\mathrm{\ξ}---\left(P1\right)$

$l_{ij}\≥\frac{|S_{ij}{|}^2}{v_i},\∀(i,j)\∈\mathrm{\ξ}$

s_i∈S_i, i ∈ N⁺

${\underset{\‾}{v}}_i\≤v_i\≤{\stackrel{\‾}{v}}_i,i\∈N^{+}$

Wherein | a |,The plural number amplitude of a, conjugation and real part is represented respectively with Re (a)；Represent and at bus i, inject merit System performance loss caused by rate；

Step 4: Solve problems (P1) obtains current result of calculationJudge whether to reach convergence precision:Or whether reach maximum iteration time: k ＞ K_max；If so, export the injecting power of bus, calculate system System performance loss, performs step 5；Otherwise make iterations k=k+1, repeat step 3 and 4.

Step 5: complete optimal load flow control according to calculated bus injecting power.

A kind of Optimal Power Flow Problems control method based on the concavo-convex optimisation technique of punishment the most according to claim 1, It is characterized in that, in described step 4, Solve problems (P1) obtains result of calculationSolution be interior point Algorithm.