CN104993525B - A kind of active distribution network coordinating and optimizing control method of meter and ZIP loads - Google Patents

A kind of active distribution network coordinating and optimizing control method of meter and ZIP loads Download PDF

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CN104993525B
CN104993525B CN201510460298.9A CN201510460298A CN104993525B CN 104993525 B CN104993525 B CN 104993525B CN 201510460298 A CN201510460298 A CN 201510460298A CN 104993525 B CN104993525 B CN 104993525B
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CN104993525A (en
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于汀
蒲天骄
赖祥生
李蕴
黄仁乐
杨占勇
徐绍军
李时光
王子安
韩巍
王伟
吴新景
杜佳桐
王建宇
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State Grid Corp of China SGCC
China Electric Power Research Institute Co Ltd CEPRI
North China Electric Power University
State Grid Beijing Electric Power Co Ltd
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State Grid Corp of China SGCC
China Electric Power Research Institute Co Ltd CEPRI
North China Electric Power University
State Grid Beijing Electric Power Co Ltd
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Abstract

The present invention relates to a kind of meter and the active distribution network coordinating and optimizing control method of ZIP loads, including：The control variable inputted in network topology, determine the active distribution network Coordination and Optimization Model of object function and constraints, structure meter and ZIP loads；According to the optimized algorithm based on functional transformation and generalized inverse, the active distribution network Optimized model optimal solution is obtained.This method is provided fundamental basis for optimal control, easy to operate, improves the efficiency of coordination optimization control, adds the running quality and stability of power network.

Description

A kind of active distribution network coordinating and optimizing control method of meter and ZIP loads
Technical field
The present invention relates to a kind of optimal control method, and in particular to a kind of active distribution network of meter and ZIP loads is coordinated excellent Change control method.
Background technology
For conventional electrical distribution net, because inside lacks active power supply, therefore when studying its running optimizatin, it is impossible to carry out active Optimization, and need to only carry out idle work optimization.With the continuous growth of electricity needs and the shortage of conventional energy resource, all kinds of distributed power sources (DG, Distributed Generation) has started to access power distribution network on a large scale, and the thing followed is a large amount of energy storage and new Extensive use of the controllable device in power distribution network.Traditional passive unidirectional supplying electricity and power distribution net is just progressively to more power supply bidirectional power supply distribution Net transformation, is increasingly highlighted in face of distribution network voltage level, capacity of short circuit, relay protection strategy, power supply reliability and the quality of power supply A series of problems, such as, the active distribution network (ADN, Active Distribution Network) as effective solution should Transport and give birth to.The power distribution network active optimization that exists for of the active power supplys of a large amount of DG and energy storage etc. provides hardware foundation in ADN.Consider It is active in power distribution network separately to carry out this traditional power transmission network with reactive power flow close-coupled, active optimization with idle work optimization and optimize The method of operation directly applies to power distribution network will necessarily be unreasonable, it is therefore necessary to by the adjustable active and idle amount in ADN simultaneously Control variable is included, carries out active reactive coordination optimization.
In currently being coordinated and optimized on ADN active reactives, load is handled according to constant power load model, does not consider its Static Load Voltage characteristic.But strictly speaking, load is constantly in optimization process among constantly variation, for this characteristic of power distribution network Particularly evident, the application is that this defect being directed in current research is improved.
The content of the invention
In order to make up the defects of prior art is present, the present invention provides a kind of active distribution network of meter and ZIP loads and coordinated Optimal control method, the optimization process suitable for considering Load static voltage characteristic.
The technical solution adopted in the present invention is：
A kind of active distribution network coordinating and optimizing control method of meter and ZIP loads, methods described include,
The control variable inputted in network topology, determine object function and constraints, the active of structure meter and ZIP loads Power distribution network Coordination and Optimization Model；According to the optimized algorithm based on functional transformation and generalized inverse, the active distribution network optimization is obtained Model optimal solution.
Preferably, the control variable includes：The active and reactive output of distributed power source, SVC is idle to be gone out Power and switched capacitors group switching group number.
Preferably, the determination object function includes：Object function is minimised as with active distribution network active power loss, its table It is up to formula：
In formula (1), PlossFor active power loss, ViAnd VjFor node i and j voltage magnitude, GijAnd BijBetween node i and j The real and imaginary parts of transadmittance；θijFor node i and j phase difference of voltage；N is nodes.
Preferably, the constraints includes：Equality constraint and inequality constraints；Wherein,
The expression formula of the equality constraint is；
In formula (2), Δ PiFor the active power amount of unbalance of node i；ΔQiFor the reactive power amount of unbalance of node i； PGiAnd QGiThe active and reactive power that respectively power transmission network is injected by root node i to power distribution network, value is taken as 0 at non-root node； PDGiAnd QDGiDG active and idle output respectively at node i；QSVCiSVC idle output respectively at node i；PLiAnd QLi The respectively active and load or burden without work of node i；
The inequality constraints, including node voltage amplitude constraint, active/idle units limits, quiet of distributed power source Only the switching group number constraint of the idle units limits of reactive-load compensator, switched capacitors group, its expression formula are：
In formula (3),WithVi The respectively bound of node i voltage magnitude；WithPDGi Respectively DG has at node i Work(output bound；WithQDGi The idle output bounds of DG respectively at node i；WithQSVCi Respectively at node i Output bound that SVC is idle；KCjFor j-th of switching group number for being connected to switched capacitors group node,WithKCj Respectively Group number bound, nCTo be connected to the nodes of switched capacitors group.
Further, the active and load or burden without work P of ZIP loads is connected in node iLiAnd QLiExpression formula be：
In formula (4), PLiAnd QLiActual burden with power and the load or burden without work of ZIP loads are respectively connected in node i；PLNiAnd QLNi Respectively burden with power and load or burden without work of the node i under rated voltage；ViAnd VNiThe respectively virtual voltage amplitude and ZIP of node i The rated voltage of load；api、bpi、cpi、aqi、bqiAnd cqiIt is all the proportionality coefficient of ZIP loads, and meets
Preferably, the optimal solution for obtaining the active distribution network Optimized model comprises the steps：
A) according to the actual requirements, contraction factor ds initial values, shrinkage ratio ns, convergence threshold d are setminWith iterations N=0； It is determined that optimization opening flag position flag；
If b) flag ≠ 0, continue；If flag=0, go to step e) and carry out constrained load flow calculating；
C) judge whether contraction factor ds is more than convergence threshold dminIf being more than, each variable currency, iteration time are preserved Number N=0；Otherwise, step (l) is gone to；
If d) continuously convergence number reaches 5 times, first increase convergence factor ds=ds × ns, then shrink active power loss；It is no Then, the active power loss of current contraction factor ds values is directly shunk；
E) current voltage value is substituted into formula (4), exports the actual negative charge values of ZIP load access nodes, remaining node is born Lotus is constant；
F) each node power amount of unbalance is obtained, judges whether peak power amount of unbalance DPQ is more than convergence precision ite_ Jd, if being more than, continue；Otherwise, return to step c)；
G) iterations N=N+1, if N ＞ 30, step j) is gone to；Otherwise continue；
H) Expanded Jacobian matrix is generated, is connected to the active and load or burden without work P of ZIP loads to node i as the following formulaLiAnd QLi It is modified；
I) solve and obtain correction, each variable is modified, return to step e)；
If j) flag ≠ 0, each variable reverts to the convergence result of last time, and reduces convergence factor ds=ds/ns, returns Return step c)；If flag=0, continue；
K) constrained load flow is not restrained, and can not be optimized, optimization terminates；
L) optimal solution is exported, optimization terminates.
Compared with immediate prior art, beneficial effects of the present invention are：
(1) it is the optimized algorithm based on functional transformation and generalized inverse proposed in power transmission network is active/idle applied to ADN Coordination optimization, can easily handle inequality constraints；It is simple to operate and the algorithm is based on Newton-Raphson approach, easily In programming realization.
(2) ZIP load models are used to load in optimization process, so as to count and Load static voltage characteristic to load value and The influence of coherent element in Jacobian matrix is extended, makes optimum results more accurate reliable.
Brief description of the drawings
Fig. 1 is the active distribution network coordinating and optimizing control method flow chart of a kind of meter of the present invention and ZIP loads.
Embodiment
1 the present invention is further described below in conjunction with the accompanying drawings.
As shown in figure 1, the active distribution network coordinating and optimizing control method of a kind of meter and ZIP loads, methods described includes,
The control variable inputted in network topology, determine object function and constraints, the active of structure meter and ZIP loads Power distribution network Coordination and Optimization Model；According to the optimized algorithm based on functional transformation and generalized inverse, the active distribution network optimization is obtained Model optimal solution.
The control variable includes：The active and reactive output of distributed power source, SVC is idle to contribute and can throw Cut capacitor group switching group number.
The determination object function includes：Object function is minimised as with active distribution network active power loss, its expression formula is：
In formula (1), PlossFor active power loss, ViAnd VjFor two node is and j voltage magnitude arbitrarily chosen, GijAnd Bij The real and imaginary parts of transadmittance between node i and j；θijFor node i and j phase difference of voltage；N is nodes.
The constraints includes：Equality constraint and inequality constraints；Wherein,
The expression formula of the equality constraint is；
In formula (2), Δ PiFor the active power amount of unbalance of node i；ΔQiFor the reactive power amount of unbalance of node i； PGiAnd QGiThe active and reactive power that respectively power transmission network is injected by root node i to power distribution network, value is taken as 0 at non-root node； PDGiAnd QDGiDG active and idle output respectively at node i；QSVCiSVC idle output respectively at node i；PLiAnd QLi The respectively active and load or burden without work of node i；
The inequality constraints, including node voltage amplitude constraint, active/idle units limits, quiet of distributed power source Only the switching group number constraint of the idle units limits of reactive-load compensator, switched capacitors group, its expression formula are：
In formula (3),WithVi The respectively bound of node i voltage magnitude；WithPDGi Respectively DG has at node i Work(output bound；WithQDGi The idle output bounds of DG respectively at node i；WithQSVCi Respectively at node i Output bound that SVC is idle；KCjFor j-th of switching group number for being connected to switched capacitors group node,WithKCj Respectively Group number bound, nCTo be connected to the nodes of switched capacitors group.
The active and load or burden without work P of ZIP loads is connected in node iLiAnd QLiExpression formula be：
In formula (4), PLiAnd QLiActual burden with power and the load or burden without work of ZIP loads are respectively connected in node i；PLNiWith QLNiRespectively burden with power and load or burden without work of the node i under rated voltage；ViAnd VNiThe respectively virtual voltage width of node i The rated voltage of value and ZIP loads；api、bpi、cpi、aqi、bqiAnd cqiIt is all the proportionality coefficient of ZIP loads, and meets
The optimal solution for obtaining the active distribution network Optimized model comprises the steps：
A) according to the actual requirements, contraction factor ds initial values, shrinkage ratio ns, convergence threshold d are setminWith iterations N=0； It is determined that optimization opening flag position flag；Active distribution network is operated in optimum state according to sets target, ensure the economy of system Operation；
If b) flag ≠ 0, continue；If flag=0, go to step e) and carry out constrained load flow calculating；
C) judge whether contraction factor ds is more than convergence threshold dminIf being more than, each variable currency, iteration time are preserved Number N=0；Otherwise, step (l) is gone to；
If d) continuously convergence number reaches 5 times, first increase convergence factor ds=ds × ns, then shrink active power loss；It is no Then, the active power loss of current contraction factor ds values is directly shunk；
E) current voltage value is substituted into formula (4), exports the actual negative charge values of ZIP load access nodes, remaining node is born Lotus is constant；
F) each node power amount of unbalance is obtained, judges whether peak power amount of unbalance DPQ is more than convergence precision ite_ Jd, if being more than, continue；Otherwise, return to step c)；
G) iterations N=N+1, if N ＞ 30, step j) is gone to；Otherwise continue；
H) Expanded Jacobian matrix is generated, is connected to the active and load or burden without work P of ZIP loads to node i as the following formulaLiAnd QLi It is modified；
I) solve and obtain correction, each variable is modified, return to step e)；
If j) flag ≠ 0, each variable reverts to the convergence result of last time, and reduces convergence factor ds=ds/ns, returns Return step c)；If flag=0, continue；
K) constrained load flow is not restrained, and can not be optimized, optimization terminates；
L) optimal solution is exported, optimization terminates.
Finally it should be noted that：Above example is only illustrating the technical scheme of the application rather than to its protection domain Limitation, although the application is described in detail with reference to above-described embodiment, those of ordinary skill in the art should Understand：Those skilled in the art read the embodiment of application can be still carried out after the application a variety of changes, modification or Person's equivalent substitution, these changes, modification or equivalent substitution, it applies within pending right at it.

Claims (3)

  1. A kind of 1. active distribution network coordinating and optimizing control method of meter and ZIP loads, it is characterised in that methods described includes,
    The control variable inputted in network topology, determine the active distribution of object function and constraints, structure meter and ZIP loads Net Coordination and Optimization Model；According to the optimized algorithm based on functional transformation and generalized inverse, the active distribution network Optimized model is obtained Optimal solution；
    The determination object function includes：Object function is minimised as with active distribution network active power loss, its expression formula is：
    <mrow> <mi>min</mi> <mi> </mi> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>V</mi> <mi>i</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>V</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>cos&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>sin&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    In formula, PlossFor active power loss, ViAnd VjFor node i and j voltage magnitude, GijAnd BijThe transadmittance between node i and j Real and imaginary parts；θijFor node i and j phase difference of voltage；N is nodes；
    The optimal solution for obtaining the active distribution network Optimized model comprises the steps：
    A) according to the actual requirements, contraction factor ds initial values, shrinkage ratio ns, convergence threshold d are setminWith iterations N=0；It is determined that Optimize opening flag position flag；
    If b) flag ≠ 0, continue；If flag=0, go to step e) and carry out constrained load flow calculating；
    C) judge whether contraction factor ds is more than convergence threshold dminIf being more than, each variable currency, iterations N=are preserved 0；Otherwise, step (l) is gone to；
    If d) continuously convergence number reaches 5 times, first increase convergence factor ds=ds × ns, then shrink active power loss；Otherwise, directly Receive the active power loss of contracting current contraction factor ds values；
    E) by the virtual voltage amplitude V of node iiSubstitute into following formula in, export node i be connected to ZIP loads actual burden with power and Load or burden without work, remaining node load are constant；
    <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mrow> <mi>L</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>L</mi> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>a</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>V</mi> <mi>i</mi> </msub> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mfrac> <msub> <mi>V</mi> <mi>i</mi> </msub> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <msub> <mi>c</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>Q</mi> <mrow> <mi>L</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mi>L</mi> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>a</mi> <mrow> <mi>q</mi> <mi>i</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>V</mi> <mi>i</mi> </msub> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow> <mi>q</mi> <mi>i</mi> </mrow> </msub> <mfrac> <msub> <mi>V</mi> <mi>i</mi> </msub> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>+</mo> <msub> <mi>c</mi> <mrow> <mi>q</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
    In formula, PLiAnd QLiActual burden with power and the load or burden without work of ZIP loads are respectively connected in node i；PLNiAnd QLNiRespectively The burden with power for being node i under rated voltage and load or burden without work；ViAnd VNiThe respectively virtual voltage amplitude and ZIP of node i The rated voltage of load；api、bpi、cpi、aqi、bqiAnd cqiIt is all the proportionality coefficient of ZIP loads, and meets
    F) each node power amount of unbalance is obtained, judges whether peak power amount of unbalance DPQ is more than convergence precision ite_jd, if It is more than, then continues；Otherwise, return to step c)；
    G) iterations N=N+1, if N ＞ 30, step j) is gone to；Otherwise continue；
    H) Expanded Jacobian matrix is generated, is connected to the active and load or burden without work P of ZIP loads to node i as the following formulaLiAnd QLiRepaiied Just；
    <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>dP</mi> <mrow> <mi>L</mi> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>dV</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>L</mi> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mfrac> <msub> <mi>V</mi> <mi>i</mi> </msub> <msubsup> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>b</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>dQ</mi> <mrow> <mi>L</mi> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>dV</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mi>L</mi> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow> <mi>q</mi> <mi>i</mi> </mrow> </msub> <mfrac> <msub> <mi>V</mi> <mi>i</mi> </msub> <msubsup> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>b</mi> <mrow> <mi>q</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    I) solve and obtain correction, each variable is modified, return to step e)；
    If j) flag ≠ 0, each variable reverts to the convergence result of last time, and reduces convergence factor ds=ds/ns, returns to step It is rapid c)；If flag=0, continue；
    K) constrained load flow is not restrained, and can not be optimized, optimization terminates；
    L) optimal solution is exported, optimization terminates.
  2. 2. the method as described in claim 1, it is characterised in that the control variable includes：Distributed power source it is active and reactive go out Power, SVC is idle to contribute and switched capacitors group switching group number.
  3. 3. according to the method for claim 1, it is characterised in that the constraints includes：Equality constraint and inequality are about Beam；Wherein,
    The expression formula of the equality constraint is；
    <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;P</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>L</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>V</mi> <mi>i</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>V</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>cos&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>sin&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;Q</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mi>S</mi> <mi>V</mi> <mi>C</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mrow> <mi>L</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>V</mi> <mi>i</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>V</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>sin&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>cos&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    In formula, Δ PiFor the active power amount of unbalance of node i；ΔQiFor the reactive power amount of unbalance of node i；PGiAnd QGiPoint Not Wei active and reactive power from root node i to power distribution network that injected by of power transmission network, value is taken as 0 at non-root node；PDGiAnd QDGi DG active and idle output respectively at node i；QSVCiSVC idle output respectively at node i；PLiAnd QLiRespectively save Point i active and load or burden without work；
    The inequality constraints, including node voltage amplitude constraint, active/idle units limits of distributed power source, static nothing The idle units limits of work(compensator, the switching group number constraint of switched capacitors group, its expression formula are：
    <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <munder> <msub> <mi>V</mi> <mi>i</mi> </msub> <mo>&amp;OverBar;</mo> </munder> <mo>&amp;le;</mo> <msub> <mi>V</mi> <mi>i</mi> </msub> <mo>&amp;le;</mo> <mover> <msub> <mi>V</mi> <mi>i</mi> </msub> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munder> <msub> <mi>P</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;OverBar;</mo> </munder> <mo>&amp;le;</mo> <msub> <mi>P</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;le;</mo> <mover> <msub> <mi>P</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munder> <msub> <mi>Q</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;OverBar;</mo> </munder> <mo>&amp;le;</mo> <msub> <mi>Q</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;le;</mo> <mover> <msub> <mi>Q</mi> <mrow> <mi>D</mi> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munder> <msub> <mi>Q</mi> <mrow> <mi>S</mi> <mi>V</mi> <mi>C</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;OverBar;</mo> </munder> <mo>&amp;le;</mo> <msub> <mi>Q</mi> <mrow> <mi>S</mi> <mi>V</mi> <mi>C</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;le;</mo> <mover> <msub> <mi>Q</mi> <mrow> <mi>S</mi> <mi>V</mi> <mi>C</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munder> <mrow> <msub> <mi>KC</mi> <mi>j</mi> </msub> </mrow> <mo>&amp;OverBar;</mo> </munder> <mo>&amp;le;</mo> <msub> <mi>KC</mi> <mi>j</mi> </msub> <mo>&amp;le;</mo> <mover> <mrow> <msub> <mi>KC</mi> <mi>j</mi> </msub> </mrow> <mo>&amp;OverBar;</mo> </mover> <mo>,</mo> <mrow> <mo>(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>n</mi> <mi>C</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    In formula,And ViThe respectively bound of node i voltage magnitude；WithPDGi Respectively at node i in DG active power outputs Lower limit；WithQDGi The idle output bounds of DG respectively at node i；WithQSVCi Respectively at node i SVC it is idle go out Power bound；KCjFor j-th of switching group number for being connected to switched capacitors group node,WithKCj Respectively organize above and below number Limit, nCTo be connected to the nodes of switched capacitors group.
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