CN107846023A - A kind of improved network loss duty value direct current optimal power flow computational methods - Google Patents

A kind of improved network loss duty value direct current optimal power flow computational methods Download PDF

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CN107846023A
CN107846023A CN201711265455.6A CN201711265455A CN107846023A CN 107846023 A CN107846023 A CN 107846023A CN 201711265455 A CN201711265455 A CN 201711265455A CN 107846023 A CN107846023 A CN 107846023A
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赵静波
刘建坤
卫志农
孙国强
周前
张清松
朱鑫要
王大江
解兵
徐珂
朱梓荣
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State Grid Corp of China SGCC
Electric Power Research Institute of State Grid Jiangsu Electric Power Co Ltd
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    • H02J3/04—Circuit arrangements for ac mains or ac distribution networks for connecting networks of the same frequency but supplied from different sources
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Abstract

The invention discloses a kind of improved network loss duty value direct current optimal power flow computational methods, comprise the following steps：(1) equivalent line loss resistance to earth is introduced at the circuit both ends of optimal load flow DC Model, forms network loss duty value model, and derive the resistance value of resistance to earth；(2) from alternating current optimal power flow model, the active loss formula of circuit is derived；(3) fitting of a polynomial is carried out to the trigonometric function item in active loss formula, and the voltage magnitude item in formula is eliminated using system performance；(4) by the equivalent equivalent resistance to earth for distributing to circuit both ends of the active loss of circuit, improved network loss duty value direct current optimal power flow model is formed.The accuracy and high efficiency of model of the present invention are verified in testing, and method provided by the invention effectively increases the computational accuracy of network loss duty value model, also ensure that the computational efficiency of model, improves the practical value of network loss duty value model.

Description

A kind of improved network loss duty value direct current optimal power flow computational methods
Technical field
The present invention relates to a kind of power system to linearize optimal load flow computational methods, more particularly to a kind of improved network loss etc. Duty value direct current optimal power flow computational methods.
Background technology
Optimal load flow calculates (optimal power flow, OPF) the 1960s by French scholar Carpentier is proposed first, is the important means for ensureing power system security economical operation.Alternating current optimal power flow model (alternating current optimal power flow, ACOPF) has very strong nonlinear characteristic, and between its variable Coupling it is very close, this causes the computational efficiency of the model relatively low, can not meet large scale system it is online in real time calculate need Ask.Therefore, suitable linearisation optimal load flow computation model is found to be particularly important.
Direct current optimal power flow (direct current optimal power flow, DCOPF) be current solving speed most Fast linearisation optimal load flow computation model.But, can not Efficient Characterization system because the model have ignored the influence of network loss factor Characteristics of tidal flow so that the difference of system generated energy and load all has balance nodes to undertake, and this will cause the power point of system Cloth is unreasonable, so as to cause the calculation error of model larger, can not put into practical application.
Research considers that the DC Model of network loss is advantageous to improve computational accuracy while model solution efficiency is ensured.At present More perfect DC Model is network loss duty value DC Model, and the model simulates network loss by way of introducing equivalent resistance Influence to system.But the relation approximate description line loss of existing model versatile system active power and apparent energy, the party Method is related to the calculating of circuit apparent energy amplitude and the active power amplitude proportion factor in equivalent process.Because the parameter exists It is a continually changing amount in iterative process, existing model is often set to definite value, certain deficiency be present, therefore can influence net Damage the computational accuracy of duty value model.
The content of the invention
Goal of the invention：For problem above, the present invention proposes a kind of improved network loss duty value direct current optimal power flow meter Calculation method.
Technical scheme：To realize the purpose of the present invention, the technical solution adopted in the present invention is：A kind of improved network loss etc. Duty value direct current optimal power flow computational methods, including step：
(1) equivalent line loss resistance to earth is introduced at the circuit both ends of optimal load flow DC Model, forms network loss duty value Model, and derive the resistance value of resistance to earth；
(2) from alternating current optimal power flow model, the active loss formula of circuit is derived；
(3) fitting of a polynomial is carried out to the trigonometric function item in active loss formula, and formula is eliminated using system performance In voltage magnitude item；
(4) by the equivalent equivalent resistance to earth for distributing to circuit both ends of the active loss of circuit, improved network loss etc. is formed Duty value direct current optimal power flow model.
In the step (1), in direct current optimal power flow model the active loss of equivalent resistance to earth be：
In formula, Pequ,i、Pequ,jFor node i, the active loss of the upper equivalent resistance to earth of j, Ui、UjFor node i, j voltage Amplitude, requ,ijFor the equivalent resistance to earth at branch road both ends；Ploss,ijFor branch road active loss；
Take Ui=Uj=1, obtain：
requ,ij=2/Ploss,ij (2)。
In the step (2), active power expression formula is in alternating current optimal power flow model：
In formula, i, j be circuit both ends node serial number, θij=θi-θjFor node i and node j phase difference of voltage, θiFor The voltage phase angle of node i, θjFor node j voltage phase angle, gij、bijThe respectively conductance and susceptance of circuit, PijFor node i stream To j active power；
The circuit active loss of alternating current optimal power flow model is：
In the step (3), fitting of a polynomial is carried out to the trigonometric function item in active loss formula and obtained：
Formula (5) is substituted into formula (4) and obtained：
In formula, Uij=Ui-UjFor the voltage amplitude value difference at circuit both ends；
Because of Uij＜ ＜ θij, ignore the U in formula (6)ijXiang get：
In the step (4), power balance equation is in direct current optimal power flow model：
In formula, Δ PiFor the active power amount of unbalance of node i, PGiContributed for i-th generated power, PDiFor node i Burden with power,For with the bus admittance matrix element reciprocal established of branch road reactance, θjFor node j voltage phase angle；
Convolution (1), formula (2) and formula (7) can obtain the network loss duty value P of node iequ,iFor：
The network loss duty value of each node is subtracted, power balance equation is in improved network loss duty value model：
In formula, f (x) be optimal load flow object function, nbFor node number, ngFor generator number, a2i、a1iAnd a0iFor I-th generator expends characterisitic parameter；θiFor the voltage phase angle of node i；θ i、For the voltage phase angle for node i lower limit and Higher limit；PGiContributed for i-th generated power；P Gi、The lower limit and higher limit contributed for i-th generated power.
Beneficial effect：The improved network loss duty value direct current optimal power flow model of the present invention, from ACOPF models to having Work(loss formula is derived again, and the voltage magnitude item in active loss formula is eliminated using system operating characteristics, makes it suitable For DCOPF models, it is accurate for apparent energy and active power scale factor value to eliminate existing network loss duty value model The dependence of exactness, the computational accuracy of model is effectively increased, also ensure that the computational efficiency of model, effectively increase network loss etc. The scope of application of duty value model.
Brief description of the drawings
Fig. 1 is network loss duty value model schematic.
Embodiment
Technical scheme is further described with reference to the accompanying drawings and examples.
It is network loss duty value model as shown in Figure 1, direct current optimal power flow is improved, improves direct current optimal power flow Computational accuracy.The main thought of network loss duty value model is on the basis of direct current optimal power flow model, considers line loss Factor, line loss is equally assigned into the loss of the equivalent resistance to earth in circuit both ends.In Fig. 1, requ,ijFor access leg both ends Equivalent resistance to earth；Ploss,ijFor branch road active loss.
In network loss duty value direct current optimal power flow model, U is takeni≈Uj≈ 1, Ui、UjThe voltage amplitude of respectively each node Value.Therefore r is worked asequ,ij=2/Ploss,ijWhen, the active power of each equivalent resistance to earth consumption is (1/2) Ploss,ij, now, Meet Pij=Pji+Ploss,ij, PijJ active power, the actual flow situations of coincidence circuit, as long as therefore seeking are flowed to for node i Rational line loss expression formula is looked for regard to the computational accuracy of direct current optimal power flow model can be effectively improved.
Existing methods are existed using the active power of circuit and the relation approximate description line loss of apparent energy, this method more It is related to the calculating of circuit apparent energy amplitude and the active power amplitude proportion factor in equivalent process.Because the parameter is in iteration During be a continually changing amount, and Existing methods more be set to definite value, certain deficiency be present, therefore network loss can be influenceed The computational accuracy of duty value model.
The present invention derives the active loss formula of circuit from alternating current optimal power flow model again, to alternating current optimal power flow Trigonometric function item in model is effectively equivalent by the way of fitting of a polynomial, and eliminates active loss using system operating characteristics Voltage magnitude item in formula, it is set to be applied to direct current optimal power flow model.By active loss calculation formula obtained by above-mentioned derivation Applied in network loss duty value direct current optimal power flow model, so as to eliminate existing network loss duty value model for apparent energy With the dependence of the active power scale factor value degree of accuracy, the computational accuracy of model is effectively increased.
Improved network loss duty value direct current optimal power flow computational methods proposed by the present invention, specifically include step：
(1) direct current optimal power flow model is derived, and equivalent line loss resistance to earth, shape are introduced at the circuit both ends of DC Model Into network loss duty value model, and derive resistance to earth requ,ijValue；
Such as Fig. 1, the active loss of equivalent resistance to earth can be written as：
In formula, Pequ,i、Pequ,jThe respectively active loss of the upper equivalent resistance to earth of node i, j, Ui、UjRespectively each node Voltage magnitude, requ,ijFor the equivalent resistance to earth at access leg both ends；Ploss,ijFor branch road active loss.
In DCOPF, U is takeni≈Uj≈ 1, therefore have 1/requ,ij=0.5Ploss,ij, therefore：
requ,ij=2/Ploss,ij (2)
(2) from alternating current optimal power flow model, the active loss formula of circuit is derived；
In ACOPF models, ignore line-to-ground impedance, then its active power can be expressed as：
In formula, i, j be circuit both ends node serial number, θij=θi-θjFor node i and node j phase difference of voltage, θiFor The voltage phase angle of node i, θjFor node j voltage phase angle, gij、bijThe respectively conductance and susceptance of circuit, PijFor node i stream To j active power.
Can be obtained according to the definition of line loss, in AC model circuit ij active loss be：
(3) fitting of a polynomial is carried out to the trigonometric function item in loss formula, and eliminated using system performance in formula Voltage magnitude item；
Contain part trigonometric function item in formula (4) so that the coupling of voltage magnitude and phase angle is very close, is not easy to equation Simplification.Therefore, the characteristic that the present invention is generally arrived between π/6 according to the phase angle difference at system line both ends in-π/6, is utilized MATLAB Fitting Toolbox is fitted to it.
Specific fit procedure is that 100 points of equidistant sampling between π/6 are arrived in-π/6, and its value is stored in matrix X, and The cosine of each point value is sought, is stored in matrix Y, matrix X and Y are imported into MATLAB Fitting Toolbox, it is following so as to can obtain Equivalent relation：
For convenience of follow-up statement, 0.49=C is made.Formula (5), which is substituted into formula (4), to be obtained：
In formula, Uij=Ui-UjFor the voltage amplitude value difference at circuit both ends.
Due to power system, node voltage is typically held at 1.0pu or so, therefore circuit first and last end in the process of running The numerical value of voltage amplitude value difference is smaller.By substantial amounts of Example Verification, U can be obtainedij＜ ＜ θij, therefore the U in negligible formula (6)ij , now, formula (6) can be written as：
(4) by the equivalent equivalent resistance to earth for distributing to circuit both ends of line loss, it is straight to form improvement network loss duty value Flow optimal load flow model；
In direct current optimal power flow model, power-balance constraint can be written as：
In formula, Δ PiFor the active power amount of unbalance of node i, PGiContributed for i-th generated power, PDiFor node i Burden with power,For with the bus admittance matrix element reciprocal established of branch road reactance, θjFor node j voltage phase angle.
As shown in Figure 1, network loss duty value model introduces network loss equivalent on the basis of DC Model on each node Duty value, therefore, the power balance equation in network loss duty value model also need to subtract the network loss duty value of each node.
Convolution (1), formula (2) and formula (7) can obtain the network loss duty value P of node iequ,iFor：
By above-mentioned derivation, improved network loss duty value direct current optimal power flow model can state following form as：
In formula, f (x) be optimal load flow object function, nbFor node number, ngFor generator number, a2i、a1iAnd a0iFor I-th generator expends characterisitic parameter；θiFor the voltage phase angle of node i；θ i、For the voltage phase angle for node i lower limit and Higher limit；PGiContributed for i-th generated power；P Gi、The lower limit and higher limit contributed for i-th generated power.
(5) accuracy and high efficiency of model are verified in test set.
The advantage of prior art is compared for the checking present invention, the computational accuracy of model of the present invention is verified, together When with documents 1, (improvement direct current optimal power flows of the such as He Tianyu, Wei Zhinong, Sun Guoqiang based on network loss duty value model is calculated Method [J] Automation of Electric Systems, 2016,40 (6)：58-64) model and the DCOPF model is as a comparison.Chat for convenience State, it is AC, DCOPF DC to define ACOPF models, and model definition described in documents 1 is M_1, model definition of the present invention For M_2.
The present invention is solved using prim al- dual interior point m ethod to each model, realizes that algorithm is compiled on MATLAB 2014a platforms Journey.It is big to the node systems of IEEE 300, the node systems of Polish 2383, the node systems of Polish 2736 and 8304 nodes System carries out example test.
Table 1
Table 1 provides AC, DC, M_1 and M_2 result of calculation, and wherein relative error is referred between the model and AC models Relative error.As shown in Table 1, M_1 and M_2 can effectively improve the computational accuracy of DC models.Simultaneously as M_2 is more accurately The approximate formula of line loss is described, the model effectively increases the computational accuracy of network loss duty value model, successfully will meter Control errors are calculated within 5/1000ths, and for the node systems of IEEE 300 and the node systems of Polish 2736, M_2 is very Extremely by control errors 5/10000ths or so.
Table 2
Table 2, which provides, calculates the time needed for different model solution different system OPF problems, to verify that circuit of the present invention has Influence of the work(loss computing method to computation model solution efficiency.From the results, it was seen that have benefited from the powerful calculating of MATLAB Ability, AC models can effectively restrain when solving the OPF problems of most systems in 50 iteration, 7s.But for this hair The big system of 8304 nodes of bright test, AC models need iteration 796 times, used time about 165s are restrained, far beyond application on site pair In the requirement of computational efficiency, therefore the present invention has important practical significance to probing into for DCOPF models.In several models, Because DC models have done a large amount of simplification to system power equilibrium equation, therefore its iterations is minimum, computational efficiency highest, and right Lifting of the nodes less mini system model to efficiency is also fairly obvious.M_1 and M_2 iterations and calculating time base This is consistent, and this explanation the method for the invention can't influence the solution efficiency of network loss duty value model.Due to the two moulds Non-linear partial is all remained in type, therefore its iterations and calculating time are all slightly more than DC models.But nonetheless, its is right The lifting of OPF computational efficiencies still very significantly, even the big system of 8304 nodes, will calculate time control 2s with It is interior, reduce 99% with respect to AC models.
In summary, M_2 effectively improves the computational accuracy of network loss duty value model, and does not therefore influence its solution Speed, therefore there is higher actual application value compared with M_1.

Claims (5)

  1. A kind of 1. improved network loss duty value direct current optimal power flow computational methods, it is characterised in that：Including step：
    (1) equivalent line loss resistance to earth is introduced at the circuit both ends of optimal load flow DC Model, forms network loss duty value model, And derive the resistance value of resistance to earth；
    (2) from alternating current optimal power flow model, the active loss formula of circuit is derived；
    (3) fitting of a polynomial is carried out to the trigonometric function item in active loss formula, and is eliminated using system performance in formula Voltage magnitude item；
    (4) by the equivalent equivalent resistance to earth for distributing to circuit both ends of the active loss of circuit, it is equivalent negative to form improved network loss Lotus direct current optimal power flow model.
  2. 2. improved network loss duty value direct current optimal power flow computational methods according to claim 1, it is characterised in that：Institute State in step (1), in direct current optimal power flow model the active loss of equivalent resistance to earth be：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mfrac> <msubsup> <mi>U</mi> <mi>i</mi> <mn>2</mn> </msubsup> <msub> <mi>r</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mfrac> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <msubsup> <mi>U</mi> <mi>j</mi> <mn>2</mn> </msubsup> <msub> <mi>r</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    In formula, Pequ,i、Pequ,jFor node i, the active loss of the upper equivalent resistance to earth of j, Ui、UjFor node i, j voltage magnitude, requ,ijFor the equivalent resistance to earth at branch road both ends；Ploss,ijFor branch road active loss；
    Take Ui=Uj=1, obtain：
    requ,ij=2/Ploss,ij (2)。
  3. 3. improved network loss duty value direct current optimal power flow computational methods according to claim 1, it is characterised in that：Institute State in step (2), active power expression formula is in alternating current optimal power flow model：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>U</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>-</mo> <msub> <mi>g</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>U</mi> <mi>i</mi> </msub> <msub> <mi>U</mi> <mi>j</mi> </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>U</mi> <mi>i</mi> </msub> <msub> <mi>U</mi> <mi>j</mi> </msub> <msub> <mi>sin&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    In formula, i, j be circuit both ends node serial number, θij=θi-θjFor node i and node j phase difference of voltage, θiFor node i Voltage phase angle, θjFor node j voltage phase angle, gij、bijThe respectively conductance and susceptance of circuit, PijFlow to j's for node i Active power；
    The circuit active loss of alternating current optimal power flow model is：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>U</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mi>j</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>g</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>U</mi> <mi>i</mi> </msub> <msub> <mi>U</mi> <mi>j</mi> </msub> <msub> <mi>cos&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
  4. 4. improved network loss duty value direct current optimal power flow computational methods according to claim 1, it is characterised in that：Institute State in step (3), carrying out fitting of a polynomial to the trigonometric function item in active loss formula obtains：
    <mrow> <msub> <mi>cos&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>C&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Formula (5) is substituted into formula (4) and obtained：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>Cg</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    In formula, Uij=Ui-UjFor the voltage amplitude value difference at circuit both ends；
    Because of Uij＜ ＜ θij, ignore the U in formula (6)ijXiang get：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>Cg</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
  5. 5. improved network loss duty value direct current optimal power flow computational methods according to claim 1, it is characterised in that：Institute State in step (4), power balance equation is in direct current optimal power flow model：
    <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>i</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>n</mi> <mi>b</mi> </msub> </munderover> <msubsup> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mi>&amp;theta;</mi> <mi>j</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    In formula, Δ PiFor the active power amount of unbalance of node i, PGiContributed for i-th generated power, PDiFor having for node i Workload,For with the bus admittance matrix element reciprocal established of branch road reactance, θjFor node j voltage phase angle；
    Convolution (1), formula (2) and formula (7) can obtain the network loss duty value P of node iequ,iFor：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>&amp;Element;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>&amp;NotEqual;</mo> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mo>,</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>&amp;Element;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>&amp;NotEqual;</mo> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <msub> <mi>Cg</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>&amp;theta;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    The network loss duty value of each node is subtracted, power balance equation is in improved network loss duty value model：
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>o</mi> <mi>b</mi> <mi>j</mi> </mrow> </mtd> <mtd> <mrow> <mi>min</mi> <mi> </mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>n</mi> <mi>g</mi> </msub> </munderover> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </msub> <msubsup> <mi>P</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <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>i</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>n</mi> <mi>b</mi> </msub> </munderover> <msubsup> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mi>&amp;theta;</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2...</mn> <msub> <mi>n</mi> <mi>b</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <munder> <mi>P</mi> <mo>&amp;OverBar;</mo> </munder> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;le;</mo> <msub> <mi>P</mi> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>&amp;le;</mo> <msub> <mover> <mi>P</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>G</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2...</mn> <msub> <mi>n</mi> <mi>g</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <munder> <mi>&amp;theta;</mi> <mo>&amp;OverBar;</mo> </munder> <mi>i</mi> </msub> <mo>&amp;le;</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>&amp;le;</mo> <msub> <mover> <mi>&amp;theta;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2...</mn> <msub> <mi>n</mi> <mi>b</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    In formula, f (x) be optimal load flow object function, nbFor node number, ngFor generator number, a2i、a1iAnd a0iFor i-th Platform generator expends characterisitic parameter；θiFor the voltage phase angle of node i；θ i、Lower limit for the voltage phase angle for node i and upper Limit value；PGiContributed for i-th generated power；P Gi、The lower limit and higher limit contributed for i-th generated power.
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CN110867861A (en) * 2019-12-04 2020-03-06 广西大学 Method for accurately delimiting theoretical line loss of power distribution network
CN112531718A (en) * 2020-12-21 2021-03-19 陕西航空电气有限责任公司 Load flow calculation method suitable for multi-electric-plane power system
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CN114221555B (en) * 2021-11-05 2023-10-10 国网浙江省电力有限公司嘉兴供电公司 Multi-terminal flexible soft switch with power flow controller
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