CN102832625A - Mathematical model for optimal configuration of power distribution network filtering devices - Google Patents

Abstract

The invention discloses a mathematical model for optimal configuration of power distribution network filtering devices. Under the condition that harmonic voltage, condenser capacity and the like meet constraint conditions, in terms of system average voltage total distortion and investment cost, a comprehensive objective function is given in a linear weighting manner, a penalty function is added to the objective function so that a constrained optimization problem is converted into a unconstrained optimization problem, and an improved simulated annealing-particle swarm optimization (namely introduction of an adaptive inertia coefficient and a memorizer) is used for solving. The model can be adapted to the random variation of a harmonic source and network parameters, can ensure that the harmonicration and average voltage total harmonic distortion of each node of the network are in the specified limits, can optimize the installation type, installation location, installation quantity, capacity parameter and the like of active filters and passive filters in a centralized manner in the whole network; and by the mathematical model, the investment cost of filtering devices in the whole network is reduced to the minimum.

Description

The Mathematical Modeling that the power distribution network filter is distributed rationally

Technical field

The present invention relates to when distribution net work structure, when parameter harmonic electric current is known; Unified infield, Setup Type, installation group number and the capacity parameter of optimizing the active and passive filtration unit in the power distribution network scope; The filter that makes power distribution network in guaranteeing network each node harmonic voltage containing rate and the total percent harmonic distortion of average voltage in prescribed limits; And under the prerequisite of filter safe operation, make the investment cost of the whole network filter minimum.

Background technology

Development along with modern industrial technology; Various non-linear and time variation power electronic equipments (like inverter, rectifier and various Switching Power Supplies etc.) are large-scale to be used and makes that nonlinear-load rolls up in the electric power system; And a large amount of harmonic waves and order harmonic components have been injected in the switch motion of these power electronic equipments in power distribution network; Cause the serious distortion of voltage and current waveform in the power distribution network; From a large amount of on-the-spot test results, the harmonic current of these nonlinear-loads has significant randomness, is having a strong impact on the safety and economic operation of confession, power consumption equipment.The installing filter can reduce and control the harmonic current and the compensating reactive power loss of injection electrical network in user or electrical network, makes that each node harmonic voltage satisfies corresponding harmonic standard in the power distribution network.Active Power Filter-APF (APF) has high controllability and fast-response, can carry out the real-time tracking compensation to the harmonic wave that frequency and amplitude all change, but its capacity is big, cost is higher.Passive filter (PPF) is simple in structure, equipment investment is few, operational reliability is higher, but the influence that filter effect receives factors such as system impedance, harmonic source characteristic and system frequency variation when using separately greatly, is prone to and electric network impedance generation resonance.This is typical multiple target, contain the nonlinear optimal problem of a large amount of constraintss.Therefore, consider that distributing rationally of two kinds of filters is the inevitable choice of present practical applications.But because separately performance all can receive some effects when comprehensive the use; In order to adapt to the change at random of harmonic source and network parameter; Seek how in electrical network, to dispose filter and to make it to satisfy the inhibition harmonic requirement, so active and parameter passive filter are optimized design and seem extremely important with minimum cost.

Summary of the invention

The objective of the invention is (to satisfy at harmonic voltage and condenser capacity etc. under the prerequisite of constraints through setting up a kind of Mathematical Modeling; Angle from system's average voltage resultant distortion rate and cost of investment; Utilize the linear weighted function mode to provide integrated objective function; Penalty is added in the target function, makes constrained optimization problems be converted into unconstrained optimization problem, and adopt to improve simulated annealing-particle cluster algorithm and find the solution) to the infield of power distribution network filter; Installation group number and capacity parameter are optimized; Guaranteeing each node harmonic voltage containing rate of network and average voltage harmonic resultant distortion rate in prescribed limits, and under the prerequisite of filter safe and reliable operation, make Voltage Harmonic resultant distortion rate and cost of investment minimum.

For realizing above-mentioned purpose; The technical scheme that the present invention adopted is: the complex optimum of whole power distribution network having been considered active filter and passive filter disposes, and from the angle of system voltage resultant distortion rate and cost of investment, adopts the mode of linear weighted function to provide integrated objective function; Make multi-objective optimization question be converted into the single goal optimization problem; Solved in actual conditions, two target functions can not arrive minimum value simultaneously, and can only be through coordinating the problem of the relation between each function.Adopt improved simulated annealing-particle cluster algorithm (promptly in algorithm, introducing a self adaptation inertia coeffeicent and memory) to find the solution.Penalty is added in the target function, come the adaptive value used in the computational algorithm, make constrained optimization problems be converted into unconstrained optimization problem.Fitness function is meant the needs based on practical problem, estimates the quality of particle by certain constraints.Stopping when algorithm is that the optimum particle of fitness function is the optimal solution that optimization searching searches out and must calculates.The memory record obtains globally optimal solution until the optimal solution that occurred in the current search process compares finally separating with it after the annealing process end again, has avoided particle cluster algorithm to be prone to be absorbed in local optimum.Unified infield and the relevant parameter of optimizing active and passive filtration unit reduces the loss of system in network-wide basis, and voltage, power factor etc. is remained in the prescribed limit, reaches the minimized purpose of cost of investment.Its basic skills is following

(1) sets up Mathematical Modeling

System's average voltage resultant distortion rate; In searching process; According to the GB GB/T14549-1993 of China, add the constraints of aberration rate, purpose is in order to guarantee that each node harmonic voltage containing rate HRU of network and average voltage harmonic resultant distortion rate THDU are in prescribed limits.Set up installing filter initial investment cost then; Draw investment cost and system the relation of the filter parameter that will install; And under the situation of voltage, electric current and capacity-constrained in trend constraint and passive filter branch road; Utilization improves simulated annealing-particle cluster algorithm whole network is carried out optimizing calculating; This mathematics model is through carrying out optimizing calculating to active with infield, Setup Type, installation group number and capacity parameter passive filtration unit, and the harmonic content that makes electrical network is on the basis of National standard, and is more little good more; Guaranteeing each node harmonic voltage containing rate of network and average voltage harmonic resultant distortion rate in prescribed limits, and under the prerequisite of filter safe and reliable operation, making that the cost of investment of power network current harmonic wave resultant distortion rate and the whole network filter is minimum.In reactive power compensation performance (compensation of the compensation harmonic reactive power of fundamental wave reactive power power) to filter; Because the power factor of nonlinear load mainly shows on the minimum rated installed capacity of filter the influence of system filter installation optimization configuration, therefore adopts the increasing filter capacity to satisfy the reactive power compensation requirement.Adopt the mode of linear weighted function to provide integrated objective function; Make multi-objective optimization question be converted into the single goal optimization problem; Again it is carried out the ISA-PSO search successively and consider, solve non-linear, the integer estimator problem that belongs to many discrete variables of distributing rationally from the angle of mathematics.

(2) algorithm basic principle of having improved:

Carry out probability according to the adaptive value after the population evolution and accept, both received optimization solution, also receive to worsen and separate, jump out local minimum.When the adaptive value of new particle increased, system received new particle; When the new particle fitness reduces, just receive by Probability p.This algorithm is jumped out from the local extremum zone, thereby finds globally optimal solution, has guaranteed Algorithm Convergence; It is accomplished in each relatively independent concurrent process, has guaranteed the diversity of each population, has improved convergence rate, and in each process, can introduce simulated annealing and jump out the population local extremum, has obtained globally optimal solution like this.In solution procedure, add a memory, be used for stored record until the optimal solution that occurred in the current search process is separated these again and compared, thereby obtains globally optimal solution after annealing finishes, this has just improved the accuracy of algorithm greatly.

Its beneficial effect is:

3) Mathematical Modeling that adopts the unification of passive filter and active filter to distribute rationally; When the power distribution network filter is optimized configuration; Because the groundwork of passive filter is filtering and compensating reactive power; Active filter then is responsible for the harmonic wave than high reps, and this has just reduced the capacity of required current transformer greatly, thereby has reached filter effect and optimistic economic benefit preferably.

4) adopt the mode of linear weighted function to provide integrated objective function, make multi-objective optimization question be converted into the single goal optimization problem, can solve non-linear, integer estimator problem that the network optimization configuration belongs to many discrete variables.

5). particle cluster algorithm adopts the inertial system numerical value that progressively reduces with iterations, can adjust the balance of particle between the overall situation and local search ability neatly, has guaranteed convergence rate and satisfied convergence precision of later stage that the initial stage is higher; And the memory in the simulated annealing, it has remedied to worsen in the simple analog algorithm separates the situation that overrides optimal solution, has improved the precision of algorithm; Has the advantage that bigger probability faster speed obtains globally optimal solution so improve later algorithm.

Below in conjunction with accompanying drawing the present invention is described further.

Description of drawings

Fig. 1 is an algorithm flow chart.

Embodiment

1. the foundation of target function

(1) analyzes electric network composition, calculate average voltage resultant distortion rate

If each node harmonic current of system is known, just can ask each node harmonic voltage according to the node admittance matrix of humorous wave network, that is:

U ^h＝[Y ^h] ^-1I ^h (h＝2，3，…….H) (1)

In the formula, h is a harmonic number, the higher harmonics number of times of H for considering, and this paper tests according to field data, gets H=19; U ^hBe h subharmonic voltage vector; I ^hBe the h subharmonic current vector of each harmonic source to the electrical network injection; Y ^hBe h subharmonic admittance matrix.

Behind the installing filter, make mains by harmonics content on the basis of National standard, more little good more.Therefore, the present invention is with the THDU of each node of power distribution network _iMean value is target function, and the offset current that obtains filter can make in the distribution network the total harmonic distortion of voltage for minimum.That is:

$\mathrm{<figure-callout\; id="1"\; label="min\; f"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">min</figure-callout>}{f}_{}{}_1=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left(U_{\mathrm{THD}{U}_i}\right)}^2=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left(\frac{\sqrt{\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}{U_{\mathrm{hi}}}^2}}{U_{1i}}\right)}^2---\left(2\right)$

In the formula, Voltage harmonic aberration rate for any node i; I is the grid nodes label, and N is the total node number of network, U _LiBe the fundamental voltage effective value that i is ordered, U _HiFor at i point h subharmonic voltage effective value.In searching process, according to the GB GB/T14549-1993 of China, the constraints that adds aberration rate is following:

U _hmin≤U _hi≤U _hmax (3)

${\mathrm{HRU}}_{\mathrm{hi}}=\left(\frac{U_{\mathrm{hi}}}{U_{1i}}\right)\&times;100\%=C_{\mathrm{HRU}}---\left(4\right)$

${\mathrm{THDU}}_i=\frac{\sqrt{\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}{U}_{\mathrm{hi}}^2}}{U_{1i}}\&times;100\%\&le;C_{\mathrm{THDU}}---\left(5\right)$

${\mathrm{THDU}}_{k,\mathrm{odd}}=\frac{\sqrt{\underset{h=1}{\overset{H/2}{\mathrm{\&Sigma;}}}{U}_{(2h+1)i}^2}}{U_{1i}}\&times;100\%\&le;C_{\mathrm{THDU},\mathrm{odd}}---\left(6\right)$

${\mathrm{THDU}}_{k,\mathrm{even}}=\frac{\sqrt{\underset{h=1}{\overset{H/2}{\mathrm{\&Sigma;}}}{U}_{\left(2h\right)i}^2}}{U_{1i}}\&times;100\%\&le;C_{\mathrm{THDU},\mathrm{even}}---\left(7\right)$

In the last formula, C _HRU, C _THDU, C _{THDU, odd}And C _{THDU, even}Be expressed as the limit value of i subharmonic voltage containing ratio, voltage resultant distortion rate, the idol time total percent harmonic distortion of voltage and the strange time voltage resultant distortion rate of regulation respectively.

(2) installing filter initial investment cost

Through adopting the Mathematical Modeling of passive filter and active filter, the target function that obtains the investment cost minimum is:

$\min {\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">f</figure-callout>}}_2=[\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{a}_{\mathrm{tj}}{f}_{\mathrm{Pij}}\left(Q_{\mathrm{CNij}}\right)+\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{b}_i{f}_{\mathrm{Ai}}\left(S_i\right)]---\left(8\right)$

M is the filter branch road number that each node can be installed in the formula; a _Ij, b _iWhether expression installs filter branches, works as a _Ij=1 o'clock, represent that the i node installs j bar passive filter branch road, and a _IjThe=0th, uneasiness is adorned corresponding branch road; Work as b _i, represent that the i node is equipped with the source filter branch road at=1 o'clock; f _Pij(Q _CNij) be the expense of capacitor and the functional relation between its rated capacity, f _Ai(S _i) be the functional relation between APF expense and its rated capacity.Have as follows:

f _Fij(Q _CNij)＝a _0ij+a _1ijQ _CNij (9)

f _Ai(Q _Ni)＝b _0i+b _1iS _Ni (10)

For avoiding the blindness of coefficient choosing value, make theoretical total investment expenses more near actual total investment of engineering expense, adopt market price decision method to confirm coefficient a _0ij, a _1ij, b _0ij, b _1ij

According to the relation of inductance L, resistance R and capacitor C in the filtering principle of PPF, can obtain L and R value, and then obtain the expense of PPF.It is following to add constraints:

$\sqrt{{\mathrm{<figure-callout\; id="1"\; label="Q\; C"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">Q</figure-callout>}}_C/\mathrm{\&omega;}}+\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}\sqrt{Q_{\mathrm{Chij}}\mathrm{h\&omega;}}\&le;K_U\sqrt{Q_{\mathrm{CNij}}/\mathrm{\&omega;}}---\left(11\right)$

$\sqrt{\mathrm{\&omega;}{\mathrm{<figure-callout\; id="1"\; label="Q\; C"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">Q</figure-callout>}}_C+\mathrm{\&omega;}\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}h{Q}_{\mathrm{Chij}}}\&le;K_i\sqrt{\mathrm{\&omega;}{Q}_{\mathrm{CNij}}}---\left(12\right)$

${\mathrm{<figure-callout\; id="1"\; label="Q\; C"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">Q</figure-callout>}}_C+\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}{Q}_{\mathrm{Chij}}\&le;K_Q{Q}_{\mathrm{CNij}}---\left(13\right)$

S _i≤K _SS _Ni (14)

Formula (13), (14), (15) are respectively voltage, electric current and the capacity-constrained in the passive filter branch road, K _U, K _IAnd K _QRepresent that respectively capacitor allows overcurrent, overvoltage and overcapacity coefficient, ω is the first-harmonic angular frequency, K _SIt is the overcapacity coefficient that APF allows.

The reactive power compensation performance of filter mainly comprises two aspects: the compensation of the compensation harmonic reactive power of fundamental wave reactive power power.And the power factor of nonlinear load mainly shows on the minimum rated installed capacity of filter the influence that the system filter installation optimization disposes; This paper gets nonlinear-load power factor 0.65～0.85, promptly adopts the increasing filter capacity to satisfy the reactive power compensation requirement.The specified installed capacity Q of the smallest capacitor of i node _CNijShould satisfy:

$Q_{\mathrm{CNij}}={\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+Q_{\mathrm{hi}}=\frac{I_{1i}^2}{\mathrm{\&omega;}{C}_i}+\frac{I_{\mathrm{hi}}^2}{\mathrm{h\&omega;}{C}_i}=\frac{h^2}{h^2-1}\left[\frac{U_{1i}^2{I}_{1i}^2}{h{Q}_1}\right]---\left(15\right)$

Wherein:

${\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000515973700011.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1=\mathrm{\&omega;}{C}_i\frac{h^2}{h^2-1}{U}_{1i}^2---\left(16\right)$

C in the formula _iIt is the capacitance of the filter of i node installation.And the capacity S of active filter _iBy the decision of the each harmonic current value that compensated, irrelevant with fundamental current, its capacity is decided by total harmonic current effective value of being compensated, that is:

$S_i=\sqrt{(U_{1i}^2+\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}{U}_{\mathrm{hi}}^2)\left(\underset{h=2}{\overset{H}{\mathrm{\&Sigma;}}}{I}_{\mathrm{Ahi}}^2\right)}---\left(17\right)$

Consider from the angle of mathematics, more than distribute non-linear, the integer estimator problem that belong to many discrete variables rationally.In actual conditions, make two target functions arrive minimum value simultaneously is impossible exist, and can only make them reach more excellent separating simultaneously through coordinating the relation between each function as far as possible.Therefore; This paper adopts the mode of linear weighted function to provide integrated objective function; Make multi-objective optimization question be converted into the single goal optimization problem, again it is carried out the ISA-PSO search successively, just infield, Setup Type, installation group number and the capacity parameter to filter carries out optimizing.

2. the calculating of fitness function

Fitness function is meant the needs based on practical problem, estimates the quality of particle by certain constraints.When algorithm stop be, the optimum particle of fitness function is the optimal solution that optimization searching searches out, this paper introduces penalty function and comes the adaptive value used in the computational algorithm in target function.The basic thought of penalty function method is certain penalty of characteristics structure according to constraint, and penalty is added in the target function, makes finding the solution of constrained optimization problems be converted into finding the solution of unconstrained optimization problem.That is:

F＝V-f ₁-f ₂-[∑r _iG _i+∑c _jH _j] (18)

In the formula, V is a suitable big positive integer, r _iAnd c _jBe penalty factor, but their value be difficult to be held in Practical Calculation, is not had a punishment effect too for a short time, too big then since the influence of error can lead to errors.This paper gets less positive number with it earlier in computational process, obtain the optimal solution of F (x); If when this separates the constraints that does not satisfy the bundle optimization problem of having an appointment, amplify penalty factor and repeat, till satisfying condition.G _i, H _jBe respectively constraints g _iAnd h _jFunction, as follows:

$g_i=\left\{\begin{array}{cc}{U}_{\mathrm{hi}}-U_{h\max },& U_{\mathrm{hi}}>U_{h\max }\\ 0,& U_{\mathrm{hi}}\&le;U_{h\max }\end{array}\right.---\left(19\right)$

$h_i=\left\{\begin{array}{cc}{\mathrm{THDU}}_i-C_{\mathrm{THDU}},& {\mathrm{THDU}}_i>C_{\mathrm{THDU}}\\ 0,& {\mathrm{THDU}}_i\&le;C_{\mathrm{THDU}}\end{array}\right.---\left(20\right)$

G _i＝max[0，g _i] ²，H _i＝|h _i| ² (21)

Formula (18) is carried out the ISA-PSO search successively will carry out optimizing to the parameters such as infield, Setup Type, installation group number and capacity of filter exactly.Can impact adaptive value for fear of different constraint condition, considered constraints is dispersed in multilevel optimization's process in the process of optimal design, can be bundled into the reliability of separating like this, can accelerate convergence of algorithm speed again.That is to say that the filtering parameter that just can make gained can adapt to the situation that node load changes, thereby be applicable to the various operation conditionss in the power distribution network.

3. carry out optimizing with improved simulation-annealing particle cluster algorithm

As shown in Figure 1 in particle swarm optimization algorithm, in the population particle add up to N, each particle has a position x in the space _iThis particle is from x _iWith speed v _iFlight forward, the optimal location that each particle searches in the space is p _i, the optimal location that whole population searches in the space is p _g, x _iThe correction of the k time iteration be v ^k _i=[v ^k _I1, v ^k _I2..., v ^k _In], its computing formula is as follows:

v ^k _i＝wv ^k-1 _i+c ₁rand ₁(p _i-x ^k-1 _i)+c ₂rand ₂(p _g-x ^k-1 _i)

x ^k _i＝x ^k-1 _i+v ^k-1 _i i＝1，2，...，N (22)

In the formula (22), k is an iterations; c ₁And c ₂Be accelerated factor, rand ₁And rand ₂Be two independently random numbers between [0,1]; W is an inertia coeffeicent, adjusts its size and can change search capability.The fitness that the stopping criterion for iteration of algorithm is elected the optimal location that maximum iteration time or population search up to now as satisfies predetermined minimum fitness threshold value.

In the PSO algorithm, when particle under the effect of big inertia coeffeicent, might lack and cause search precision not high the fine search of optimal solution.Adopt adaptive inertia coeffeicent, w is carried out the self adaptation adjustment,, gradually reduce the w value promptly along with the increase of iterations by formula (23).Bigger w value helps improving algorithm the convergence speed, and less w then can improve arithmetic accuracy.

$w\left(k\right)=[2/(1+e^{\mathrm{\&lambda;k}/k_{\max }})]w^0---\left(23\right)$

In the formula, λ is a positive coefficient, is used for regulating the pace of change of w; K is an iterations; k _MaxBe the iterations upper limit; w ⁰Be w (k) upper limit.

According to annealing temperature, designed simulated annealing probability acceptance criterion, promptly as f (x _Ij)＜f (x _{I (j+1)}) time, p=1; As f (x _Ij)>=f (x _{I (j+1)}),

When the approaching convergence of algorithm, the ratio of local maximum adaptation value and individual average maximum adaptation value reduces gradually and trends towards 1, and at this moment t also approaches 0 thereupon.Like this, near the speed that temperature descends globally optimal solution is enough slow, accepts to worsen and separates also minimizing gradually of probability, so population can form the ground state of minimum energy surely.When the adaptive value of new particle increased, system necessarily received new particle; When the new particle fitness reduced, the Probability p of just pressing in the following formula received.Algorithm is jumped out from the local extremum zone, finds globally optimal solution, and has guaranteed convergence.

In solution procedure, add a memory, be used for stored record until the optimal solution that occurred in the current search process is separated these again and compared, thereby obtains globally optimal solution after annealing finishes, this has just improved the accuracy of algorithm greatly.

Claims (2)

1. the power distribution network filter that proposes of the present invention Mathematical Modeling of distributing rationally; It is characterized in that: in whole power distribution network, considered the complex optimum configuration of active filter and passive filter; From system voltage resultant distortion rate and two angles of cost of investment; Adopt the mode of linear weighted function to provide integrated objective function, make multi-objective optimization question be converted into the single goal optimization problem, solved in actual conditions; Two target functions can not arrive minimum value simultaneously, and can only be through coordinating the problem of the relation between each function.Penalty is added in the target function, come the adaptive value used in the computational algorithm, make constrained optimization problems be converted into unconstrained optimization problem.Adopt improvement simulated annealing-particle cluster algorithm (promptly introducing a self adaptation inertia coeffeicent and memory) to find the solution, this self adaptation inertia coeffeicent can carry out the self adaptation adjustment, improves arithmetic accuracy; The memory record obtains globally optimal solution until the optimal solution that occurred in the current search process compares finally separating with it after the annealing process end.

2. according to right 1 described improvement simulated annealing-particle cluster algorithm; Introduce a self adaptation inertia coeffeicent and memory; It is characterized in that: particle cluster algorithm adopts the inertial system numerical value that progressively reduces with iterations; The balance of particle between the overall situation and local search ability be can adjust neatly, convergence rate and satisfied convergence precision of later stage that the initial stage is higher guaranteed; And the memory in the simulated annealing, it has remedied to worsen in the simple analog algorithm separates the situation that overrides optimal solution, has improved the precision of algorithm.Improved algorithm has the ability of the Local Extremum jumped out, and can search out global optimum or approximate optimal solution, and irrelevant with the selection of initial point, can bigger probability faster speed obtain globally optimal solution.

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