CN105337274B - Alternating current-direct current power grid decouples Power flow simulation model - Google Patents

Alternating current-direct current power grid decouples Power flow simulation model Download PDF

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CN105337274B
CN105337274B CN201510683135.7A CN201510683135A CN105337274B CN 105337274 B CN105337274 B CN 105337274B CN 201510683135 A CN201510683135 A CN 201510683135A CN 105337274 B CN105337274 B CN 105337274B
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CN105337274A (en
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李传栋
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State Grid Corp of China SGCC
Electric Power Research Institute of State Grid Fujian Electric Power Co Ltd
State Grid Fujian Electric Power Co Ltd
Economic and Technological Research Institute of State Grid Fujian Electric Power Co Ltd
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State Grid Corp of China SGCC
Electric Power Research Institute of State Grid Fujian Electric Power Co Ltd
State Grid Fujian Electric Power Co Ltd
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/36—Arrangements for transfer of electric power between ac networks via a high-tension dc link
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
    • Y02E60/60—Arrangements for transfer of electric power between AC networks or generators via a high voltage DC link [HVCD]

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  • Engineering & Computer Science (AREA)
  • Power Engineering (AREA)
  • Inverter Devices (AREA)
  • Other Investigation Or Analysis Of Materials By Electrical Means (AREA)

Abstract

The present invention relates to a kind of alternating current-direct current power grid to decouple Power flow simulation model, including AC network part Δ Sph, DC network part Δ Dph, inverter portion Δ VphAn and governing equation Δ Vkz；The AC network part Δ SphIncluding exchanging node active balance equation Δ PsAnd exchange node reactive balance equation Δ Qs, the DC network part Δ DphIncluding DC node power balance equation Δ Pd, the inverter portion Δ VphIncluding transverter exchange side active balance equationTransverter exchange side reactive balance equationAnd Converter DC-side power balance equationThe governing equation Δ VkzIncluding determining DC voltage control equationDetermine exchange side real power control equationDetermine the idle governing equation of exchange sideAnd determine exchange side voltage governing equation

Description

Alternating current-direct current power grid decouples Power flow simulation model
Technical field
The present invention relates to a kind of alternating current-direct current power grid to decouple Power flow simulation model.
Background technology
It is to use newton La Fuxun method iterative solutions that the Power flow simulation of power grid, which calculates basic skills, and basic step is by electricity The power flow equation of net carries out partial differential, and solution is asked for by the matrix computations of the matrix of variable partial derivative and the deviation of power flow equation Direction is corrected, progressively corrects flow solution until deviation meets residual error requirement.
There is presently no the decoupling Power flow simulation computational methods for alternating current-direct current power grid.Conventional alternating current-direct current electric network swim is imitated Proper program mainly has simultaneous solution method and alternately solving method.
The basic principle of simultaneous solution method：By AC network, DC grid and the trend side for contacting their transverter Cheng Lianli gets up, and solves partial derivative matrix jointly, and Unified Solution is carried out using classical newton La Feixun methods.Simultaneous solution method The advantages of be that alternating current-direct current power grid equations simultaneousness calculates, better numerical value stability convergence is strong.The shortcomings that simultaneous solution method, is All equations need simultaneous, very difficult for the extension of model, it is difficult to realize being adjusted flexibly for control mode.At the same time Simultaneous solution method is for bulk power grid, it is difficult to realizes that block parallel calculates so that its Power flow simulation in ultra-large power grid It is difficult to meet higher and higher rate request in calculating speed.
Alternately solving method moral basic principle：Transverter is grouped into the side of alternating current-direct current power grid first, both sides power grid is handed over respectively For calculating.Think that the solution of offside power grid (direct current or exchange) has restrained when DC grid part is solved, this side power grid The Boundary Variables (voltage, electric current and power) of two power grids are changed with the last time of offside power grid iteration when (AC or DC) calculates The convergence solution in generation, tries to achieve this deuterzooid side (AC or DC) and the convergence result of this current iteration of side power grid is returned to offside electricity again Net is as Boundary Variables, and iteration is settled accounts the deviation of convergence solution and met the requirements twice before and after the power grid of both sides back and forth.Alternately The benefit of solving method be model easy to extension.The mould of the pattern of controller model and new equipment for DC grid Type continuously emerges, can be under conditions of AC network power flow algorithm and program is not influenced flexibly with alternately method for solving Increase new model.Alternately it is that introducing inside and outside two recirculates the shortcomings that solving method, the convergence for causing to calculate declines, on the one hand Number increase is restrained, the phenomenon for not receiving chain easily occurs in one side numerical stability variation, although also there are many correlative studys to adopt Overcome this shortcoming with various prediction initial values and the method for correction, but all cannot fundamentally solve, various compensation processes are all There is its applicable condition.Problem can be protruded more when this point is calculated for bulk power grid, and the situation of bulk power grid is sufficiently complex, with friendship It is difficult to obtain the guarantee of stabilization for the convergence of solving method.
Comprehensive existing various correlation models, have for various each class models of the Power flow simulation of extensive alternating current-direct current power grid Following two the problem of cannot taking into account at the same time:That is the flexibility body of the flexibility of model and the convergence calculated, model and program The block parallel that the supplement and increase of present new equipment and control mode also have power grid calculates, and the convergence of model is embodied in and adopts Convergent stability and convergence rate when being calculated with corresponding model.Specific equipment and mould among alternating current-direct current power grid simultaneous solution model The adjustment and addition of type are required for modifying to overall simultaneous equations, lack corresponding flexibility, while method is determined in itself Having determined can not be to the parallel computation of bulk power grid progress piecemeal.And the alternating solving model of alternating current-direct current power grid can not be solved fundamentally The stability of Algorithm Convergence, there are under initial value unfavorable condition the problem of poor astringency.
The content of the invention
In view of this, it is an object of the invention to provide a kind of alternating current-direct current power grid to decouple Power flow simulation model, has ensured mould Computational efficiency of the type in the parallel computation of the Power flow simulation of extensive AC-DC hybrid power grid.
To achieve the above object, the present invention adopts the following technical scheme that：A kind of alternating current-direct current power grid decouples Power flow simulation model, It is characterized in that：Including AC network part Δ Sph, DC network part Δ Dph, inverter portion Δ VphAn and governing equation ΔVkz；The AC network part Δ SphIncluding exchanging node active balance equation Δ PsAnd exchange node reactive balance equation Δ Qs, the DC network part Δ DphIncluding DC node power balance equation Δ Pd, the inverter portion Δ VphIncluding changing Flow device exchange side active balance equationTransverter exchange side reactive balance equationAnd Converter DC-side power-balance EquationThe governing equation Δ VkzIncluding determining DC voltage control equationDetermine exchange side real power control equationDetermine the idle governing equation of exchange sideAnd determine exchange side voltage governing equation
It is specific as follows：
ΔPd=ucic+Pd-ud·Ydud=0
In above formula：PsActive vector, C are injected for exchange nodescConversion square for transverter sequence to exchange sequence node Battle array,For transverter exchange side active injection active power vector, U is exchange node voltage amplitude vector, and θ is exchange node electricity Press phase angle vector, Y be AC network bus admittance matrix, QsReactive power vector is injected for exchange node,Exchanged for transverter Survey injection reactive power vector, ucFor Converter DC-side node voltage vector, icDC grid is injected for Converter DC-side Current vector, PdFor DC node injecting power vector, YdFor the bus admittance matrix of DC network, GcFor the conductance of transverter Matrix, BcFor the susceptance matrix of transverter,It is vectorial for transverter AC system side gusset voltage magnitude,Changed for transverter The vector of device side equivalence alternating voltage amplitude is flowed, δ is equivalent for transverter AC system side gusset voltage and transverter transverter side The differential seat angle of alternating voltage, BfFor the susceptance matrix of transverter reactive-load compensator,DC grid is injected for Converter DC-side Vector power, CcsFor exchange sequence node to the transition matrix of transverter sequence, subscriptuc_setRepresent constant DC voltage control Transverter set, subscriptPs_setRepresent that fixed exchange has the set of transverter of public power, subscriptQs_setRepresent fixed exchange nothing The set of the transverter of work(power, subscriptUs_setExpression determines alternating voltage amplitude and obtains transverter set；Various released by above-listed：
Wherein：bu、bθ、kU2P,kU2Q、kθ2P、kθ2QIt is symbol, represents the result of calculation of levoform；
Wherein：PdIt is invariant；
Replace
Wherein：Replaced and eliminated with the transverter exchange node voltage subvector included in the dU in exchange；
Wherein：DU, dudLinear relation substitute into above formula, can obtain Exchange Station network-based control mode equation.
The present invention has the advantages that compared with prior art：The present invention both ensure that alternating current-direct current power grid Power flow simulation Calculate the numerical stability of simultaneous solution makes it possess and alternately interface flexibility and autgmentability similar in solving method again, can be with Computational efficiency of the assurance model in the parallel computation of the Power flow simulation of extensive AC-DC hybrid power grid.
Embodiment
With reference to embodiment, the present invention will be further described.
The present invention provides a kind of alternating current-direct current power grid decoupling Power flow simulation model, it is characterised in that：A kind of alternating current-direct current power grid solution Coupling Power flow simulation model, it is characterised in that：Including AC network part Δ Sph, DC network part Δ Dph, inverter portion Δ VphAn and governing equation Δ Vkz；The AC network part Δ SphIncluding exchanging node active balance equation Δ PsAnd exchange section Point reactive balance equation Δ Qs, the DC network part Δ DphIncluding DC node power balance equation Δ Pd, the change of current Device part Δ VphIncluding transverter exchange side active balance equationTransverter exchange side reactive balance equationAnd the change of current Device power balance of DC side equationThe governing equation Δ VkzIncluding determining DC voltage control equationIt is fixed to hand over Flow side real power control equationDetermine the idle governing equation of exchange sideAnd determine exchange side voltage governing equation
It is specific as follows：
ΔPd=ucic+Pd-ud·Ydud=0
In above formula：PsActive vector, C are injected for exchange nodescConversion square for transverter sequence to exchange sequence node Battle array,For transverter exchange side active injection active power vector, U is exchange node voltage amplitude vector, and θ is exchange node electricity Press phase angle vector, Y be AC network bus admittance matrix, QsReactive power vector is injected for exchange node,Exchanged for transverter Survey injection reactive power vector, ucFor Converter DC-side node voltage vector, icDC grid is injected for Converter DC-side Current vector, PdFor DC node injecting power vector, YdFor the bus admittance matrix of DC network, GcFor the conductance of transverter Matrix, BcFor the susceptance matrix of transverter,It is vectorial for transverter AC system side gusset voltage magnitude,Changed for transverter The vector of device side equivalence alternating voltage amplitude is flowed, δ is equivalent for transverter AC system side gusset voltage and transverter transverter side The differential seat angle of alternating voltage, BfFor the susceptance matrix of transverter reactive-load compensator,DC grid is injected for Converter DC-side Vector power, CcsFor exchange sequence node to the transition matrix of transverter sequence, subscriptuc_setRepresent constant DC voltage control Transverter set, subscriptPs_setRepresent that fixed exchange has the set of transverter of public power, subscriptQs_setRepresent fixed exchange nothing The set of the transverter of work(power, subscriptUs_setExpression determines alternating voltage amplitude and obtains transverter set；Various released by above-listed：
Wherein：bu、bθ、kU2P,kU2Q、kθ2P、kθ2QIt is symbol, represents the result of calculation of levoform；
Wherein：PdIt is invariant；
Replace
Wherein：Replaced and eliminated with the transverter exchange node voltage subvector included in the dU in exchange；
Wherein：DU, dudLinear relation substitute into above formula, can obtain Exchange Station network-based control mode equation.
It is above-listed it is various in：
DiagU=diag (U ∠ θ), diagIbus=diag (YU ∠ θ), diagUnorm=diag (1 ∠ θ)
For it is above-listed it is various in each parameter matrixing form.
The foregoing is merely presently preferred embodiments of the present invention, all equivalent changes done according to scope of the present invention patent with Modification, should all belong to the covering scope of the present invention.

Claims (1)

1. a kind of alternating current-direct current power grid decouples Power flow simulation model, it is characterised in that：Including AC network part Δ Sph, DC network Part Δ Dph, inverter portion Δ VphAn and governing equation Δ Vkz；The AC network part Δ SphIt is active including exchange node Equilibrium equation Δ PsAnd exchange node reactive balance equation Δ Qs, the DC network part Δ DphPut down including DC node power Weigh equation Δ Pd, the inverter portion Δ VphIncluding transverter exchange side active balance equationTransverter exchange side without Work(equilibrium equationAnd Converter DC-side power balance equationThe governing equation Δ VkzIncluding determining DC voltage Governing equationDetermine exchange side real power control equationDetermine the idle governing equation of exchange sideAnd fixed exchange Side voltage governing equation
It is specific as follows：
<mrow> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> <msubsup> <mi>P</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>-</mo> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>&amp;angle;</mo> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>j</mi> <mo>(</mo> <mrow> <mi>Y</mi> <mi>U</mi> <mo>&amp;angle;</mo> <mi>&amp;theta;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> <msubsup> <mi>Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>-</mo> <mi>i</mi> <mi>m</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>&amp;angle;</mo> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>j</mi> <mo>(</mo> <mrow> <mi>Y</mi> <mi>U</mi> <mo>&amp;angle;</mo> <mi>&amp;theta;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow>
ΔPd=ucic+Pd-ud·Ydud=0
<mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>=</mo> <msubsup> <mi>P</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>-</mo> <msub> <mi>G</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mi>c</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;delta;</mi> <mo>+</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;delta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>=</mo> <msubsup> <mi>Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mi>c</mi> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;delta;</mi> <mo>-</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;delta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>B</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>d</mi> </msubsup> <mo>=</mo> <mn>2</mn> <msub> <mi>u</mi> <mi>c</mi> </msub> <msub> <mi>i</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>c</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mi>c</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;delta;</mi> <mo>-</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;delta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msubsup> <mi>&amp;Delta;i</mi> <mi>c</mi> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>i</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>Y</mi> <mi>d</mi> </msub> <msub> <mi>u</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mrow> <mi>P</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>P</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>-</mo> <msubsup> <mi>P</mi> <mi>c</mi> <mrow> <mi>P</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mrow> <mi>Q</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>-</mo> <msubsup> <mi>Q</mi> <mi>c</mi> <mrow> <mi>Q</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mrow>
<mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow>
In above formula：PsActive vector, C are injected for exchange nodescTransition matrix for transverter sequence to exchange sequence node, For transverter exchange side active injection active power vector, U is exchange node voltage amplitude vector, and θ is exchange node voltage phase Angular amount, Y are the bus admittance matrix of AC network, QsReactive power vector is injected for exchange node,Noted for transverter exchange side Enter reactive power vector, ucFor Converter DC-side node voltage vector, icThe electric current of DC grid is injected for Converter DC-side Vector, PdFor DC node injecting power vector, udFor DC network node voltage vector, YdFor the node admittance of DC network Matrix, GcFor the conductance matrix of transverter, BcFor the susceptance matrix of transverter,For transverter AC system side gusset voltage amplitude Value vector,For the vector of transverter transverter side equivalence alternating voltage amplitude, δ for transverter AC system side gusset voltage and The differential seat angle of transverter transverter side equivalence alternating voltage, BfFor the susceptance matrix of transverter reactive-load compensator,For transverter DC side injects the vector power of DC grid, CcsFor exchange sequence node to the transition matrix of transverter sequence, subscriptuc_set Represent the set of the transverter of constant DC voltage control, subscriptPs_setRepresent the set of the transverter of exchange active power surely, on MarkQs_setRepresent the set of the transverter of exchange reactive power surely, subscriptUs_setThe transverter collection of alternating voltage amplitude is determined in expression Close,For the calibration value of transverter exchange side active injection active power vector,Nothing is injected for transverter exchange side The calibration value of work(vector power；
By above-listed various release：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;S</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>P</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>d</mi> <mi>U</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;DoubleRightArrow;</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>d</mi> <mi>U</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>b</mi> <mi>U</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mi>&amp;theta;</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>k</mi> <mrow> <mi>U</mi> <mn>2</mn> <mi>P</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow> <mi>U</mi> <mn>2</mn> <mi>Q</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow> <mi>&amp;theta;</mi> <mn>2</mn> <mi>P</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow> <mi>&amp;theta;</mi> <mn>2</mn> <mi>Q</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein：bu、bθ、kU2P,kU2Q、kθ2P、kθ2QIt is symbol, represents the result of calculation of levoform；
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;D</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;Delta;P</mi> <mi>d</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>d</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>i</mi> <mi>c</mi> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>di</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>d</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>du</mi> <mi>d</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;DoubleRightArrow;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>du</mi> <mi>d</mi> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>d</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </mfrac> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>&amp;Delta;P</mi> <mi>d</mi> </msub> <mo>-</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;P</mi> <mi>d</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </mfrac> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;lsqb;</mo> <msub> <mi>C</mi> <mrow> <mi>d</mi> <mi>c</mi> </mrow> </msub> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>di</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>b</mi> <mi>u</mi> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow> <mi>u</mi> <mn>2</mn> <mi>i</mi> </mrow> </msub> <msub> <mi>di</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein：PdIt is invariant, CdcFor transverter sequence node to the transition matrix between DC network sequence node, Ku2iIt is Symbol, represents the result of calculation of levoform；
<mrow> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>d</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>u</mi> <mi>d</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>di</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <mi>M</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;delta;</mi> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <mi>M</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;delta;</mi> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>d</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <mi>M</mi> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>d</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;delta;</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>d</mi> <mi>M</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>&amp;delta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>d</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>U</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;lsqb;</mo> <msubsup> <mi>dU</mi> <mi>c</mi> <mi>s</mi> </msubsup> <mo>&amp;rsqb;</mo> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>u</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mi>d</mi> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;lsqb;</mo> <msub> <mi>du</mi> <mi>d</mi> </msub> <mo>&amp;rsqb;</mo> </mrow>
Replace
Wherein：M is the ontrol variables vector of transverter,With transverter exchange node voltage included in the dU in exchange Vector, which is replaced, to be eliminated；
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>k</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;i</mi> <mi>c</mi> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mrow> <mi>P</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mrow> <mi>Q</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>di</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>M</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>&amp;delta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>s</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;lsqb;</mo> <mi>d</mi> <mi>U</mi> <mo>&amp;rsqb;</mo> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>Y</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;lsqb;</mo> <msub> <mi>du</mi> <mi>d</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;DoubleRightArrow;</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>Y</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <msub> <mi>k</mi> <mrow> <mi>u</mi> <mn>2</mn> <mi>i</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <msub> <mi>k</mi> <mrow> <mi>U</mi> <mn>2</mn> <mi>P</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <msub> <mi>k</mi> <mrow> <mi>U</mi> <mn>2</mn> <mi>Q</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>dP</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>dQ</mi> <mi>c</mi> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>di</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>M</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>d</mi> <mi>&amp;delta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;i</mi> <mi>c</mi> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>Y</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>u</mi> <mi>c</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <msub> <mi>b</mi> <mi>u</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;P</mi> <mi>c</mi> <mrow> <mi>P</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mrow> <mi>Q</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Delta;Q</mi> <mi>c</mi> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msubsup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;Delta;Q</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>U</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mi>s</mi> <mo>_</mo> <mi>s</mi> <mi>e</mi> <mi>t</mi> </mrow> </msup> <msub> <mi>b</mi> <mi>U</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein：DU, dudLinear relation substitute into above formula, obtain the governing equation of AC network.
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