CN104362625A - Optimization method for load flow calculation of active distribution network - Google Patents

Abstract

The invention discloses an optimization method for load flow calculation of an active distribution network. The optimization method comprises the following steps that S1, the number of branches of any tree of the active distribution network is expressed as the formula (1); S2, the minimum sum of transmission power of branches of the active distribution network is adopted as an objective, branch transmission power constraints are given, a radial network structure is determined, and node power balance is achieved; S3, the minimum active power transmission loss is adopted as an objective function; S4, equality constraints comprise a load flow constraint and a network radial constraint; S5, inequality constraints are composed of a node voltage constraint, an active and reactive force output constraint and a branch current constraint; S6, all the constraints and sij are multiplied, and only when sij=1, are the branches connected into the active distribution network for calculation. Transmission losses are reduced, meanwhile, security running of a system can be guaranteed, power grid control and distributed generation access are improved, and power grid voltage quality is improved. The complex load flow calculation of the active distribution network is simplified, and is convenient and easy to implement, and the control and adjusting method of the active distribution network is more scientific.

Description

A kind of optimization method of active distribution network Load flow calculation

Technical field

The present invention relates to the application of power system active power distribution network, particularly relate to a kind of optimization method of active distribution network Load flow calculation.

Background technology

Active distribution network (Active distribution network, ADN) is the system be made up of micro battery, load, energy-storage system and control device.It shows as a single controlled unit for bulk power grid, can realize the highly reliable supply to load various energy resources form.And the feature of active distribution network mainly contains two: one is containing local Blast Furnace Top Gas Recovery Turbine Unit (TRT), the distributed power source that the capacity that mostly is is less, distributed power source (Distributed Generation Resource, DGR) be often referred to mainly utilize renewable new forms of energy and generated output be a few kW to 50MW small modules formula, with the independent current source of environmental compatible, electric power system and user's particular requirement can be met.Two are active distribution network is controlled, and the function of active distribution network is effectively coupled together power supply and user's request, allows both sides jointly to determine how best real time execution.Reach this requirement, level of control will far above the level of conventional electrical distribution net.

Polytype distributed generation unit can be combined by active distribution network system, effectively plays the advantage of single energy resource system, realizes various energy resources complementation, improve the efficiency of whole micro-grid system, energy utilization rate and power supply reliability.Active distribution network access bulk power grid is incorporated into the power networks, and not only can make full use of the green regenerative energy sources of active distribution network inside, can also improve the fail safe of whole electrical network, be the important step that China builds up intelligent grid.Meanwhile, active distribution network system solves one of remote districts and the island effective means of powering.

Utilize mathematic programming methods to process Load flow calculation and network reconfiguration problem from Merlin, Bach etc. in 1975, people have attempted the problem that diverse ways solves distribution power flow and network reconfiguration.Load flow calculation and network reconfiguration are multi-target non-linear hybrid optimization problems, and the algorithm solved at present mainly contains several classes such as mathematical optimization theory, optimum stream method, branch exchange method and intelligent algorithm.Mathematics Optimization Method is described power distribution network reconfiguration problem by Mathematical Modeling, then by specific algorithm, do not relied on the optimum results of distribution network initial configuration, and then solve the loaded down with trivial details iterative problem of Load flow calculation.The method is long for computing time, and committed memory is large, solving for large-scale distribution network, and difficulty is very large, therefore generally first carries out abbreviation then approximate processing.Heuristic take intuitive analysis as foundation, and according to certain principle, progressive alternate is until obtain satisfied reconstruction result, and the method mainly contains optimal flow pattern algorithm and branch exchange method.Intelligent optimization algorithm developed rapidly in recent years, mainly contain artificial neural network algorithm, simulated annealing, tabu search algorithm, ant group algorithm, particle swarm optimization algorithm, genetic algorithm etc., intelligent algorithm has good global search performance, not easily be absorbed in local optimum, but it is large that its shortcoming is amount of calculation, and computing time is long.At present, run constraint in relevant active distribution network radiation and be implicitly included in algorithm document, also do not provide mathematical description and method accurately.

Summary of the invention

In order to solve the problem, the invention provides a kind of optimization method of active distribution network Load flow calculation, comprising the following steps:

S1: active distribution network can be regarded as containing n _bthe single system figure of individual node M bar branch road, the whole node of the radiation service requirement of power distribution network be communicated with and the concept without closed loop, being therefore summed up as " tree " so that this problem to be described.In electric network theory, the definition of " tree " is: the tree T of a connected graph G comprises whole node and the partial branch of G, and sets T itself and to be communicated with and not containing loop.Meanwhile, one has n _bthe connected graph of individual node, the tree number of its any one tree is formula (1):

M＝n _b-1 (1)

The network configuration of active distribution network according to the concept of " tree ", can form radial operating structure.Can draw according to above-mentioned analysis: distribution system is run radially and need be met following condition:

1. the number of branches that network packet contains is: M=n _b-1;

2. in network, all nodes must be communicated with.

But only strip part is 1. as the criterion that active distribution network radiation runs, and isolated island or looped network may appear in system, can not determine that power distribution network is that radiation runs.

S2: active distribution network branch road through-put power sum is minimum is target, meet Kirchhoff's first law, the constraint of given branch road through-put power, determine final network configuration radially and meet node power balance, formula is as (2) ~ (7):

$\min v=\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ij}}^2---\left(2\right)$

$s.t.\underset{\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ji}}-\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ij}}+g_i=d_i\∀i\∈{\mathrm{\Ω}}_b---\left(3\right)$

$\left|f_{\mathrm{ij}}\right|\≤{\stackrel{\‾}{f}}_{\mathrm{ij}}\·s_{\mathrm{ij}},\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l---\left(4\right)$

$0\≤g_i\≤{\stackrel{\‾}{g}}_i,\∀i\∈{{\mathrm{\Ω}}_b}_s---\left(5\right)$

$s_{\mathrm{ij}}\∈\left\{\mathrm{0,1}\right\},\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l---\left(6\right)$

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}=n_b-1---\left(7\right)$

Wherein, Ω _bfor the node set of system, Ω _lset of fingers, the set of node of exerting oneself, n _bfor total nodes, s _ijbeing 0-1 variable, is the state variable that control circuit cut-offs, if circuit disconnects, is 0, if closed, is then 1.F _ijthe meritorious through-put power between node i and node j, g _ithat the meritorious of balance node i is exerted oneself, d _ithe active load of node i, the branch road transmission maximum power of branch road ij, it is the maximum output of balance node i;

Described formula (7) is number of lines constraint, 1. corresponding with condition; The feasible solution meeting formula (2) must meet the load balancing of each node, thus makes must be communicated with between each node, namely may satisfy condition 2..Therefore, meet formula (2) and formula (7) simultaneously, whole network just can be made to be connected and not have loop to occur.Can obtain thus, constraints can make active distribution network final radially, this constraint is introduced in the optimization of active distribution network Load flow calculation below;

S3: minimum for target function with active power loss:

$\min v=\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}\left(g_{\mathrm{ij}}{s}_{\mathrm{ij}}(V_j^2+V_j^2-2V_i{V}_j\cos {\mathrm{\θ}}_{\mathrm{ij}})\right)---\left(8\right)$

S4: equality constraint is trend constraint and the radial constraint of network two kinds of constraints:

S4.1: trend retrains

${P_S}_i-{P_D}_i-\underset{j\∈{{\mathrm{\Ω}}_b}_i}{\mathrm{\Σ}}\left(s_{\mathrm{ij}}{P}_{\mathrm{ij}}\right)=0,\∀i\∈{\mathrm{\Ω}}_b---\left(9\right)$

${Q_S}_i-{Q_D}_i-\underset{j\∈{{\mathrm{\Ω}}_b}_i}{\mathrm{\Σ}}\left(s_{\mathrm{ij}}{Q}_{\mathrm{ij}}\right)=0,\∀i\∈{\mathrm{\Ω}}_b---\left(10\right)$

S4.2: the radial constraint of network:

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}=n_b-1-\underset{j\∈{{\mathrm{\Ω}}_b}_T}{\mathrm{\Σ}}(1-y_i)---\left(11\right)$

$s_{\mathrm{ij}}\≤y_i,\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l,\∀j\∈{\mathrm{\Ω}}_{b_T}---\left(12\right)$

$s_{\mathrm{ji}}\≤y_i,\∀\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l,\∀j\∈{\mathrm{\Ω}}_{b_T}---\left(13\right)$

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}+\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}\≥2y_j,\∀j\∈{{\mathrm{\Ω}}_b}_T---\left(14\right)$

$y_i\∈\left\{\mathrm{0,1}\right\},\∀\left(j\right)\∈{\mathrm{\Ω}}_{b_T}---\left(15\right)$

Wherein, s _ijfor the state variable of circuit, if circuit closes, get s _ij=1, if circuit disconnects, then get s _ij=0; Contact point is the node of not on-load, be the set of contact point, formula (12) ~ (15) both can ensure do not have looped network to occur containing the power distribution network of contact point, also can ensure that contact point is not terminal node in a network.If but contact point in network all effectively time, then can access very little load on contact point, just can ensure that in network, whole node is all on-load node.Trend constraint can meet the load balancing of node, and the node that has namely in network is all be communicated with more; According to " tree " theory analysis, the radial constraint of network can guarantee that network can not form loop, therefore trend constraint and the respectively corresponding above-mentioned radial constraints of the radial constraint of network 1. and 2., so can determine that active distribution network is still radial structure and does not have isolated island after reconstruction in conjunction with two constraints;

S5: inequality constraints is node voltage constraint, meritorious, idle units limits and branch current retrain three kinds of constraint compositions:

S5.1: node voltage retrains

$\underset{\‾}{V}\≤V_i\≤\stackrel{\‾}{V},\∀i\∈{\mathrm{\Ω}}_b---\left(16\right)$

S5.2: meritorious, idle units limits:

${\underset{\‾}{P}}_{S_i}\≤{P_S}_i\≤{\stackrel{\‾}{P}}_{S_i},\∀i\∈{\mathrm{\Ω}}_{b_s}---\left(17\right)$

${\underset{\‾}{Q}}_{S_i}\≤{Q_S}_i\≤{\stackrel{\‾}{Q}}_{S_i},\∀i\∈{\mathrm{\Ω}}_{b_s}---\left(18\right)$

S5.3: branch current retrains:

${\underset{\‾}{I}}_{\mathrm{ij}}\≤s_{\mathrm{ij}}\·I_{\mathrm{ij}}\≤{\stackrel{\‾}{I}}_{\mathrm{ij}},\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l---\left(19\right)$

Wherein, Ω _bfor the node set of system, Ω _lset of fingers, the node set be connected with node i, the set of access point of exerting oneself, n _bfor total nodes, V _ifor the amplitude of node i voltage, with be respectively the bound of node i voltage magnitude, for meeting network power supply requirement, its perunit value respectively value is [1,0.9]; exert oneself meritorious, the idle power output of access point, the meritorious power output bound of access point, the idle power output bound of access point, node i active load and reactive load respectively; I _ijfor the line current between node i, j, be respectively the bound of line current between node i, j;

S6: in power flow equation all with s _ijbe multiplied, only have and work as s _ijwhen=1, branch road access active distribution network calculates, P _ij, Q _ijexpression formula as follows:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}=V_i^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}=-V_i^2{b}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(20\right)$

Wherein, g _ijand b _ijbe respectively the conductance between node i, j and susceptance.P _ij, Q _ijit is meritorious, the reactive power from i node-flow to j node.

The present invention compared with the existing technology, has the following advantages and beneficial effect:

1. the present invention makes the Load flow calculation of active distribution network complexity simplify, convenient and easy, and active distribution network is controlled and control method more science;

2., compared to other method, the invention has the advantages that and do not need to calculate each road matrix, nodal information and node admittance matrix can be obtained by branch road information;

3. by active distribution network optimized calculation method, can give full play to the effect of Load flow calculation and control more comprehensively, effectively, quickly, guarantee the safe and reliable operation of electrical network, the present invention has saved the man power and material of debugging greatly, the production cost reduced;

4. emulated by embodiment and verify, the present invention can promote active distribution network safety and economic operation, reduces network loss simultaneously, improves node voltage level, the economy effectively raised and practicality.

Accompanying drawing explanation

Fig. 1 is example I EEE33 active distribution network system diagram.

Embodiment

An optimization method for active distribution network Load flow calculation, comprises the following steps:

S1: active distribution network can be regarded as containing n _bthe single system figure of individual node M bar branch road, the whole node of the radiation service requirement of power distribution network be communicated with and the concept without closed loop, being therefore summed up as " tree " so that this problem to be described.In electric network theory, the definition of " tree " is: the tree T of a connected graph G comprises whole node and the partial branch of G, and sets T itself and to be communicated with and not containing loop.Meanwhile, one has n _bthe connected graph of individual node, the tree number of its any one tree is formula (1):

M＝n _b-1 (1)

The network configuration of active distribution network according to the concept of " tree ", can form radial operating structure.Can draw according to above-mentioned analysis: distribution system is run radially and need be met following condition:

1. the number of branches that network packet contains is: M=n _b-1;

2. in network, all nodes must be communicated with;

But only strip part is 1. as the criterion that active distribution network radiation runs, and isolated island or looped network may appear in system, can not determine that power distribution network is that radiation runs;

S2: active distribution network branch road through-put power sum is minimum is target, meet Kirchhoff's first law, the constraint of given branch road through-put power, determine final network configuration radially and meet node power balance, formula is as (2) ~ (7):

$\min v=\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ij}}^2---\left(2\right)$

$s.t.\underset{\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ji}}-\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ij}}+g_i=d_i\∀i\∈{\mathrm{\Ω}}_b---\left(3\right)$

$\left|f_{\mathrm{ij}}\right|\≤{\stackrel{\‾}{f}}_{\mathrm{ij}}\·s_{\mathrm{ij}},\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l---\left(4\right)$

$0\≤g_i\≤{\stackrel{\‾}{g}}_i,\∀i\∈{{\mathrm{\Ω}}_b}_s---\left(5\right)$

$s_{\mathrm{ij}}\∈\left\{\mathrm{0,1}\right\},\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l---\left(6\right)$

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}=n_b-1---\left(7\right)$

Wherein, Ω _bfor the node set of system, Ω _lset of fingers, the set of node of exerting oneself, n _bfor total nodes, s _ijbeing 0-1 variable, is the state variable that control circuit cut-offs, if circuit disconnects, is 0, if closed, is then 1.F _ijthe meritorious through-put power between node i and node j, g _ithat the meritorious of balance node i is exerted oneself, d _ithe active load of node i, f _ijthe branch road transmission maximum power of branch road ij, it is the maximum output of balance node i;

Described formula (7) is number of lines constraint, 1. corresponding with condition; The feasible solution meeting formula (2) must meet the load balancing of each node, thus makes must be communicated with between each node, namely may satisfy condition 2.; Therefore, meet formula (2) and formula (7) simultaneously, whole network just can be made to be connected and not have loop to occur.Can obtain thus, constraints can make active distribution network final radially, this constraint is introduced in the optimization of active distribution network Load flow calculation below;

S3: minimum for target function with active power loss:

$\min v=\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}\left(g_{\mathrm{ij}}{s}_{\mathrm{ij}}(V_j^2+V_j^2-2V_i{V}_j\cos {\mathrm{\θ}}_{\mathrm{ij}})\right)---\left(8\right)$

S4: equality constraint is trend constraint and the radial constraint of network two kinds of constraints:

S4.1: trend retrains

${P_S}_i-{P_D}_i-\underset{j\∈{{\mathrm{\Ω}}_b}_i}{\mathrm{\Σ}}\left(s_{\mathrm{ij}}{P}_{\mathrm{ij}}\right)=0,\∀i\∈{\mathrm{\Ω}}_b---\left(9\right)$

${Q_S}_i-{Q_D}_i-\underset{j\∈{{\mathrm{\Ω}}_b}_i}{\mathrm{\Σ}}\left(s_{\mathrm{ij}}{Q}_{\mathrm{ij}}\right)=0,\∀i\∈{\mathrm{\Ω}}_b---\left(10\right)$

S4.2: the radial constraint of network:

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}=n_b-1-\underset{j\∈{{\mathrm{\Ω}}_b}_T}{\mathrm{\Σ}}(1-y_i)---\left(11\right)$

$s_{\mathrm{ij}}\≤y_i,\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l,\∀j\∈{\mathrm{\Ω}}_{b_T}---\left(12\right)$

$s_{\mathrm{ji}}\≤y_i,\∀\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l,\∀j\∈{\mathrm{\Ω}}_{b_T}---\left(13\right)$

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}+\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}\≥2y_j,\∀j\∈{{\mathrm{\Ω}}_b}_T---\left(14\right)$

$y_i\∈\left\{\mathrm{0,1}\right\},\∀\left(j\right)\∈{\mathrm{\Ω}}_{b_T}---\left(15\right)$

Wherein, s _ijfor the state variable of circuit, if circuit closes, get s _ij=1, if circuit disconnects, then get s _ij=0.Contact point is the node of not on-load, be the set of contact point, formula (12) ~ (15) both can ensure do not have looped network to occur containing the power distribution network of contact point, also can ensure that contact point is not terminal node in a network.If but contact point in network all effectively time, then can access very little load on contact point, just can ensure that in network, whole node is all on-load node.Trend constraint can meet the load balancing of node, and the node that has namely in network is all be communicated with more; According to " tree " theory analysis, the radial constraint of network can guarantee that network can not form loop, therefore trend constraint and the respectively corresponding above-mentioned radial constraints of the radial constraint of network 1. and 2., so can determine that active distribution network is still radial structure and does not have isolated island after reconstruction in conjunction with two constraints;

S5: inequality constraints is node voltage constraint, meritorious, idle units limits and branch current retrain three kinds of constraint compositions:

S5.1: node voltage retrains

$\underset{\‾}{V}\≤V_i\≤\stackrel{\‾}{V},\∀i\∈{\mathrm{\Ω}}_b---\left(16\right)$

S5.2: meritorious, idle units limits:

${\underset{\‾}{P}}_{S_i}\≤{P_S}_i\≤{\stackrel{\‾}{P}}_{S_i},\∀i\∈{\mathrm{\Ω}}_{b_s}---\left(17\right)$

${\underset{\‾}{Q}}_{S_i}\≤{Q_S}_i\≤{\stackrel{\‾}{Q}}_{S_i},\∀i\∈{\mathrm{\Ω}}_{b_s}---\left(18\right)$

S5.3: branch current retrains:

${\underset{\‾}{I}}_{\mathrm{ij}}\≤s_{\mathrm{ij}}\·I_{\mathrm{ij}}\≤{\stackrel{\‾}{I}}_{\mathrm{ij}},\∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l---\left(19\right)$

Wherein, Ω _bfor the node set of system, Ω _lset of fingers, the node set be connected with node i, the set of access point of exerting oneself, n _bfor total nodes, V _ifor the amplitude of node i voltage, with be respectively the bound of node i voltage magnitude, for meeting network power supply requirement, its perunit value respectively value is [1,0.9]; exert oneself meritorious, the idle power output of access point, the meritorious power output bound of access point, the idle power output bound of access point, node i active load and reactive load respectively; I _ijfor the line current between node i, j, be respectively the bound of line current between node i, j;

S6: in power flow equation all with s _ijbe multiplied, only have and work as s _ijwhen=1, branch road access active distribution network calculates, P _ij, Q _ijexpression formula as follows:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}=V_i^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}=-V_i^2{b}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(20\right)$

Wherein, g _ijand b _ijbe respectively the conductance between node i, j and susceptance.P _ij, Q _ijit is meritorious, the reactive power from i node-flow to j node.

The present invention can adopt IEEE33 node instance to carry out emulating and verifying.Below in conjunction with embodiment accompanying drawing, technical scheme of the present invention is clearly and completely described.

As shown in Figure 1, IEEE-33 example, this system contains 33 nodes, and a network electric power goes out force, is node 1,37 branch roads, wherein containing 5 interconnection switches, shown in dotted line.The total burden with power of system is 4000kW, and total load or burden without work is 2500.0kvar.The fiducial value of three phase power gets S _b=10000kVA, the voltage reference value of headend node gets U _b=12.57kV.Containing 37 0-1 variablees in this example, the method for the invention is adopted to be optimized and to solve.IEEE33 system active distribution network comparison of computational results is as follows: table 1 is the comparison of on off state change, network loss value, minimum voltage node voltage.

Table 1 IEEE33 system active distribution network result of calculation

Claims (1)

1. an optimization method for active distribution network Load flow calculation, is characterized in that, comprises the following steps:

S1: the tree number of any one tree of active distribution network is formula (1):

M＝n _b-1 (1)

Active distribution network is run radially and need be met following condition:

1. the number of branches that network packet contains is: M=n _b-1;

2. in network, all nodes must be communicated with;

S2: active distribution network branch road through-put power sum is minimum is target, the constraint of given branch road through-put power, network configuration radially and meet node power balance, formula is as (2) ~ (7):

$\min v=\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ij}}^2---\left(2\right)$

$s.t.\underset{\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ji}}-\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{f}_{\mathrm{ij}}+g_i=d_i\∀i\∈{\mathrm{\Ω}}_b---\left(3\right)$

$\begin{array}{cc}\left|f_{\mathrm{ij}}\right|\≤{\stackrel{\‾}{f}}_{\mathrm{ij}}\·s_{\mathrm{ij}}& \∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l\end{array}----\left(4\right)$

$\begin{array}{cc}0\≤g_i\≤{\stackrel{\‾}{g}}_i& \∀i\∈{\mathrm{\Ω}}_{b_s}\end{array}----\left(5\right)$

$\begin{array}{cc}{s}_{\mathrm{ij}}\∈\{0,\}& \∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l\end{array}---\left(6\right)$

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}=n_b-1---\left(7\right)$

Wherein, Ω _bfor the node set of system, Ω _lset of fingers, the set of node of exerting oneself, n _bfor total nodes, s _ijbeing 0-1 variable, is the state variable that control circuit cut-offs, if circuit disconnects, is 0, if closed, is then 1; f _ijthe meritorious through-put power between node i and node j, g _ithat the meritorious of balance node i is exerted oneself, d _ithe active load of node i, the branch road transmission maximum power of branch road ij, it is the maximum output of balance node i;

S3: minimum for target function with active power loss:

$\min v=\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}\left(g_{\mathrm{ij}}{s}_{\mathrm{ij}}(V_i^2+V_j^2-{2V}_i{V}_j\cos {\mathrm{\θ}}_{\mathrm{ij}})\right)---\left(8\right)$

S4: equality constraint is trend constraint and the radial constraint of network two kinds of constraints:

S4.1: trend retrains

$\begin{array}{cc}{P}_{S_i}-P_{D_i}-\underset{j\∈{\mathrm{\Ω}}_{b_i}}{\mathrm{\Σ}}\left(s_{\mathrm{ij}}{P}_{\mathrm{ij}}\right)=0& \∀i\∈{\mathrm{\Ω}}_b\end{array}---\left(9\right)$

$\begin{array}{cc}{Q}_{S_i}-Q_{D_i}-\underset{j\∈{\mathrm{\Ω}}_{b_i}}{\mathrm{\Σ}}\left(s_{\mathrm{ij}}{Q}_{\mathrm{ij}}\right)=0& \∀i\∈{\mathrm{\Ω}}_b\end{array}---\left(10\right)$

S4.2: the radial constraint of network:

$\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}=n_b-1-\underset{j\∈{\mathrm{\Ω}}_{b_T}}{\mathrm{\Σ}}(1-y_i)---\left(11\right)$

$\begin{array}{cc}{s}_{\mathrm{ij}}\≤y_i& \∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l,\∀j\∈{\mathrm{\Ω}}_{b_T}---\left(12\right)\end{array}$

$\begin{array}{cc}{s}_{\mathrm{ji}}\≤y_i& \∀\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l,\∀j\∈{\mathrm{\Ω}}_{b_T}---\left(13\right)\end{array}$

$\begin{array}{cc}\underset{\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ij}}+\underset{\left(\mathrm{ji}\right)\∈{\mathrm{\Ω}}_l}{\mathrm{\Σ}}{s}_{\mathrm{ji}}\≥{2y}_j& \∀j\∈{\mathrm{\Ω}}_{b_T}---\left(14\right)\end{array}$

$\begin{array}{cc}{y}_j\∈\left\{\mathrm{0,1}\right\}& \∀\left(j\right)\∈{\mathrm{\Ω}}_{b_T}---\left(15\right)\end{array}$

Wherein: s _ijfor the state variable of circuit, if circuit closes, get s _ij=1, if circuit disconnects, then get s _ij=0; Contact point is the node of not on-load, be the set of contact point, formula (12) ~ (15) both can ensure do not have looped network to occur containing the power distribution network of contact point, also can ensure that contact point is not terminal node in a network;

S5: inequality constraints is node voltage constraint, meritorious, idle units limits and branch current retrain three kinds of constraint compositions:

S5.1: node voltage retrains

$\begin{array}{cc}\underset{\‾}{V}\≤V_i\≤\stackrel{\‾}{V}& \∀i\∈{\mathrm{\Ω}}_b\end{array}---\left(16\right)$

S5.2: meritorious, idle units limits:

$\begin{array}{cc}{\underset{\‾}{P}}_{S_i}\≤P_{S_i}\≤{\stackrel{\‾}{P}}_{S_i}& \∀i\∈{\mathrm{\Ω}}_{b_s}\end{array}---\left(17\right)$

$\begin{array}{cc}{\underset{\‾}{Q}}_{S_i}\≤Q_{S_i}\≤{\stackrel{\‾}{Q}}_{S_i}& \∀i\∈{\mathrm{\Ω}}_{b_s}\end{array}---\left(18\right)$

S5.3: branch current retrains:

$\begin{array}{cc}{\underset{\‾}{L}}_{\mathrm{ij}}\≤s_{\mathrm{ij}}\·I_{\mathrm{ij}}\≤{\stackrel{\‾}{I}}_{\mathrm{ij}}& \∀\left(\mathrm{ij}\right)\∈{\mathrm{\Ω}}_l\end{array}---\left(19\right)$

Wherein: Ω _bfor the node set of system, Ω _lset of fingers, the node set be connected with node i, the set of access point of exerting oneself, n _bfor total nodes, V _ifor the amplitude of node i voltage, with vbe respectively the bound of node i voltage magnitude, for meeting network power supply requirement, its perunit value respectively value is [1,0.9]; exert oneself meritorious, the idle power output of access point, the meritorious power output bound of access point, the idle power output bound of access point, node i active load and reactive load respectively; I _ijfor the line current between node i, j, i _ijbe respectively the bound of line current between node i, j;

S6: in power flow equation all with s _ijbe multiplied, only have and work as s _ijwhen=1, branch road access active distribution network calculates, P _ij, Q _ijexpression formula as follows:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}={V_i}^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}=-{V_i}^2{b}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(20\right)$

Wherein, g _ijand b _ijbe respectively the conductance between node i, j and susceptance; P _ij, Q _ijit is meritorious, the reactive power from i node-flow to j node.