CN103474993B - Quantitative analysis index of active bearing capacity of power network based on mapping elastic potential energy - Google Patents

Abstract

The active bearing capacity of a power network depends on such numerous factors as a power network structure, branch bearing capacity, size and distribution of power supply and load, and the like, and the quantitative analysis index of the active bearing capacity is still a difficult point in security analysis of the power network. The invention provides a quantitative analysis index of active bearing capacity of power network based on mapping elastic potential energy, comprising: mapping a branch state on the basis of a mapping elastic mechanics network model of the power network to obtain the mapping elastic potential energy of the power network branch; and adopting a potential energy superposition concept to obtain a calculation method of the mapping elastic potential energy of the power network. As shown in analysis and verified in examples, the mapping elastic potential energy can be used as the quantitative analysis index of the active bearing capacity of the power network; under a condition of constant total active load, the larger the value, the smaller the total active load margin of the power network is and the more unbalanced the active load of the branch is. The quantitative analysis index provided by the invention is used for enriching the theoretical basis of the security analysis of the power network and can be extensively applied to the aspects of planning, operation manner analysis, online scheduling of the power network, etc.

Description

Based on the electric network active bearing capacity quantitative analysis index mapping elastic potential energy

Technical field

Electric power system (electrical network) safety analysis.

Background technology

The most basic function of electrical network is active power (hereinafter referred to as meritorious) transmission, therefore meritorious bearing capacity (or claiming meritorious transmittability), be the leading indicator that electric network security is analyzed, it depends on the factors such as the size of electric network composition, branch road bearing capacity, power supply and load and distribution.At present, generally adopt the qualitative analysis that N-1 (even N-2) branch road limit for tonnage verifies in real work, the quantitative analysis method of electric network active bearing capacity and index remain the difficult point in electric network security analysis.

Statistics finds 15 years last century Mos, and U.S.'s large-scale blackout meets the self-organizing critical characteristic of complication system.Inspired, adopt the fail safe of Complex Networks Theory analysis and evaluation bulk power grid to become current study hotspot.But complex network model is the graph theory model based on wiring topology, do not comprise the component parameters of electrical network, more can not naturally embody grid nodes, membership amount and between physical relation, very difficult foundation comprises the composite target of electric network state amount and structural information simultaneously.

The structure of local indexes is relatively easy, but structure the whole network index is not easy thus, and desirable approach is formed by stacking by local indexes.What meet additivity at present has potential energy index, and as node potential energy, Branch Potential Energy etc., but these indexs are all the concepts in Power Network Transient Stability direct method, do not have clear and definite physical concept in static analysis; The meritorious bearing capacity of electrical network can be described after also failing to prove element potential superposition principle.Therefore its theoretical foundation characterizing electric network security is not enough.

Compare the definition in electrical network, " potential energy " has clear and definite, strict physical concept in mechanics, and tool uniqueness.Similar to the meritorious bearing capacity of electrical network, the bearing capacity of elastic network(s) also depends on grid structure, branch road intensity, stressed size and distribution.The patent " electrical network-elastic mechanics network topology mapping method " (application publication number: CN 102227084 A) of applying for before this, electrical network is mapped to Elasticity network (hereinafter referred to as the elastic network(s)) model of longitudinal stress, identical owing to mapping branch road direction in elastic network(s), stressedly in the same way also meet additivity, therefore always associate between potential energy with total load.So, by mapping the potential energy of elastic network(s), analyze the meritorious bearing capacity of corresponding electrical network.

Summary of the invention

The present invention's " electric network active bearing capacity quantitative analysis index based on mapping elastic potential energy ", based on the mapping elastic network(s) model of electrical network, by the state mapping of grid branch and elastic network(s) branch road, analyze the mapping elastic potential energy of grid branch, adopt potential superposition principle thinking, obtain the acquiring method that electrical network maps elastic potential energy.Analyze and find and Example Verification, this mapping elastic potential energy can characterize the overall meritorious loading allowance of electrical network and branch road and to gain merit the harmony of carrying: in certain total active load situation, its value is larger, then the overall meritorious loading allowance of electrical network is less, and the meritorious carrying of branch road is more unbalanced; Its value is less, then on the contrary.So can be used as the quantitative analysis index of electric network active bearing capacity.The present invention has enriched the theoretical foundation that electric network security is analyzed, and can be widely used in the aspects such as the planning of electrical network, operational mode analysis, on-line scheduling.

Accompanying drawing explanation

Fig. 1 electrical network-elastic network(s) Topological Mapping, (1) electrical network, (2) map elastic network(s)

Fig. 2 equivalence maps elasticity branch road

Fig. 3 longitudinally equivalence maps elastic network(s)

Fig. 4 New England 10 machine 39 node system

The mapping elastic network(s) structure of Fig. 5 New England 10 machine 39 node system

Mapping elastic network(s) structure after Fig. 6 New England 10 machine 39 node system line disconnection 21-22

Mapping elastic network(s) structure after Fig. 7 New England 10 machine 39 node system line disconnection 15-16

Figure 87 node system structure

Figure 94 plants the mapping elastic network(s) structure of sharing of load scheme, (1) outlet sharing of load scheme one, (2) outlet sharing of load scheme two, (3) outlet sharing of load scheme three, (4) outlet sharing of load scheme four

Embodiment

1. the state mapping of grid branch and elastic network(s) branch road

Negligible resistance, if AC network branch road L two end node is i, j, node voltage phase difference be θ _ij, reactance is X _l.When electric network reactive-load is abundant, U _i, U _jchange less, can make gaining merit of then transmitting is

P _L＝Csinθ _ij(1)

Make k _l=dP _l/ d θ _ij, be named as the mapping coefficient of elasticity of L, can be obtained by formula (1)

k _L＝Ccosθ _ij(2)

k _L＝P _L／tanθ _ij(3)

If F _l, x _lbe respectively active force and the elongation of elastic network(s) branch road l, coefficient of elasticity is k _l=dF _l/ dx _l.Set up the quantity of state mapping relations of L and l:

$\left\{\begin{array}{c}{P}_L=F_l\\ {\mathrm{\θ}}_{\mathrm{ij}}=x_l\\ k_L=k_l\end{array}\right.---\left(4\right)$

Corresponding with formula (1), (2), (3), the quantity of state of l closes and is

$\left\{\begin{array}{c}{F}_l=C\sin x_l\\ k_l=C\cos x_l\\ k_l=F_l/\tan x_l\end{array}\right.---\left(5\right)$

If θ _ijless, the quantity of state relation of L can approximate linearization, can be obtained by formula (1), (2), (3)

$\left\{\begin{array}{c}{P}_L=k_L\·{\mathrm{\θ}}_{\mathrm{ij}}\\ k_L=C\end{array}\right.---\left(6\right)$

According to formula (4), the quantity of state of corresponding linear l closes and is

$\left\{\begin{array}{c}{F}_l=k_l\·x_l\\ k_l=C\end{array}\right.---\left(7\right)$

2. the mapping elastic potential energy of grid branch

L stretch after, elastic potential energy be external force to its work, namely

E _l＝∫F _ldx _l(8)

According to formula (5), (8), obtaining elastic potential energy is

$E_l=F_l\tan \frac{x_l}{2}=k_l\frac{{1-\cos x}_l}{{\cos x}_l}=\frac{{\cos x}_l}{1+{\cos x}_l}\·\frac{{F_l}^2}{k_l}---\left(9\right)$

According to formula (4) mapping relations, the mapping elastic potential energy obtaining L is

$E_L=P_L\tan \frac{{\mathrm{\θ}}_{\mathrm{ij}}}{2}=k_L\frac{1-{\cos \mathrm{\θ}}_{\mathrm{ij}}}{{\cos \mathrm{\θ}}_{\mathrm{ij}}}=\frac{{\cos \mathrm{\θ}}_{\mathrm{ij}}}{1+{\cos \mathrm{\θ}}_{\mathrm{ij}}}\·\frac{{P_L}^2}{k_L}---\left(10\right)$

If θ _ijless, the quantity of state of l, L is all approximately linear relationship.The elastic potential energy being obtained l by formula (7), (8) is

$E_l=\frac{1}{2}{F}_l{x}_l=\frac{1}{2}{k}_l{x_l}^2=\frac{{F_l}^2}{2k_l}---\left(11\right)$

According to formula (4) mapping relations, the mapping elastic potential energy of circuit L is

$E_L=\frac{1}{2}{P}_L{\mathrm{\θ}}_{\mathrm{ij}}=\frac{1}{2}{k}_L{{\mathrm{\θ}}_{\mathrm{ij}}}^2=\frac{{P_L}^2}{{2k}_L}---\left(12\right)$

Due to P _l, θ _ij, k _lwith F _l, x _l, k _lvalue correspondent equal, therefore E _lwith E _lalso mapping relations are met, namely

E _l＝E _L(13)

3. the mapping elastic potential energy of electrical network

" electrical network-elastic mechanics network topology mapping method " (application publication number: CN 102227084A) is applied for a patent according to what announce, electrical network is mapped to vertically stressed elastic network(s), and keep the incidence relation of node, branch road constant, as shown in Figure 1.

If map elastic network(s) to be made up of n bar branch road, no matter whether its branch road is linear elasticity branch road, and its total potential energy all meets superimposed characteristics, namely

$E_{\mathrm{l\Σ}}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{E}_{\mathrm{li}}---\left(14\right)$

Wherein E _{l ∑}and E _libe respectively the potential energy of elastic network(s) and wherein i-th branch road.Therefore the mapping elastic potential energy of electrical network is

$E_{\mathrm{L\Σ}}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{E}_{\mathrm{Li}}---\left(15\right)$

Wherein E _{l ∑}and E _libe respectively the mapping elastic potential energy of electrical network and wherein i-th branch road.

Because branch road in elastic network(s) is vertical in the same way, leg length is the difference in height of two end nodes, the phase place of the corresponding grid nodes voltage of height of node.If all branch roads are all linear elasticity branch road, according to formula (11), (14), the potential energy that can obtain linear elasticity net is

$E_{\mathrm{l\Σ}}=\frac{1}{2}\mathrm{\Σ}(F_{\mathrm{topi}}\·x_{\mathrm{topi}})-\frac{1}{2}\mathrm{\Σ}(F_{\mathrm{botj}}\·x_{\mathrm{botj}})---\left(16\right)$

Wherein, F _topi, x _topifor stressed size and the height of elastic network(s) top node, F _botj, x _botjfor stressed size and the height of load node.If ignore the active loss of electrical network, according to above formula, the mapping elastic potential energy of linear electrical network is

$E_{\mathrm{L\Σ}}=\frac{1}{2}\mathrm{\Σ}(P_{\mathrm{topi}}\·{\mathrm{\θ}}_{\mathrm{topi}})-\frac{1}{2}\mathrm{\Σ}(P_{\mathrm{botj}}\·{\mathrm{\θ}}_{\mathrm{botj}})---\left(17\right)$

In above formula, P _topi, θ _topifor the injection of power source bus node in electrical network is gained merit and phase place, P _botj, θ _botjfor burden with power and the phase place of load bus.

4. map elastic potential energy and electrical network totally to gain merit the relation of loading allowance

Identical owing to mapping all branch road Impact direction in elastic network(s), therefore available 1 and this net potential energy E _{l ∑}with total load F _{l ∑}all equal elasticity branch road equivalence, if the length of equivalent branch road is x _leq, as shown in Figure 2.Can be obtained by formula (9)

$E_{\mathrm{l\Σ}}=F_{\mathrm{l\Σ}}\tan \frac{x_{\mathrm{leq}}}{2}---\left(18\right)$

According to quantity of state mapping relations, have

$E_{\mathrm{L\Σ}}=P_{\mathrm{L\Σ}}\tan \frac{{\mathrm{\θ}}_{\mathrm{Leq}}}{2}---\left(19\right)$

Wherein, E _{l ∑}, P _{l ∑}and θ _leqfor the mapping potential energy of corresponding electrical network, total active load and equivalent branch road phase difference.Above formula is visible: work as P _{l ∑}during for a certain value, if E _{l ∑}larger, then θ _leqlarger; If E _{l ∑}less, θ _leqless.

If equivalent branch road is linear characteristic, according to formula (11), (12), then have

$\left\{\begin{array}{c}{E}_{\mathrm{l\Σ}}=\frac{1}{2}{F}_{\mathrm{l\Σ}}{x}_{\mathrm{leq}}\\ E_{\mathrm{L\Σ}}=\frac{1}{2}{P}_{\mathrm{L\Σ}}\·{\mathrm{\θ}}_{\mathrm{Leq}}\end{array}\right.---\left(20\right)$

Equally, P is worked as _{l ∑}during for a certain value, if E _{l ∑}larger, then θ _leqlarger; If E _{l ∑}less, θ _leqless.

In identical total active load situation, different electrical network or same electrical network under different running method, if E _{l ∑}comparatively large, then θ _leqalso comparatively large, show that the meritorious loading allowance of electrical network is less on the whole, bearing capacity is poor, and the fail safe of merit angle is poor.

5. mapping elastic potential energy and grid branch are gained merit and are carried balanced relation

Because grid branch is gained merit unbalanced, even if electrical network entire load-carrying capacity is comparatively strong, some branch road also may be close to overload.If branch road transships and automatically excises, chain reaction may be caused, and cause electrical network local off-the-line, even disintegrate.Therefore, require that index from interior angle, can weigh the harmony of grid branch carrying.

The equilibrium of electric network active carrying, refers to and maps the larger circuit of coefficient of elasticity, should carry larger gaining merit.Owing to electrical network to be mapped to the elastic network(s) of longitudinal stress, can by the associated nodes of mains side and load side and between path carry out merging and equivalence, be simplified to the structure shown in Fig. 3.

As shown in Figure 3, be provided with n bar branch road, i-th corresponding branch road maps coefficient of elasticity, phase difference and is meritoriously respectively k _li, θ _liand P _li, total burden with power is there is following rule in discovery:

If P _{l ∑}=const., when branch road active distribute meets following formula

P _L1：P _l2：...：P _Ln=k _L1：k _L2：...：k _Ln(21)

Then always map elastic potential energy E _{l ∑}minimum.

Prove: can obtain according to formula (10), (15)

$E_{\mathrm{L\Σ}}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(P_{\mathrm{Li}}\tan -\frac{{\mathrm{\θ}}_{\mathrm{Li}}}{2})---\left(22\right)$

In above formula, θ _liit is the phase difference at i-th branch road two ends.Adopt Lagrangian extremum method, build following Lagrange's equation

$E=E_{\mathrm{L\Σ}}-\mathrm{\λ}(\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{P}_{\mathrm{Li}}-P_{\mathrm{L\Σ}})---\left(23\right)$

Wherein, λ is certain constant.E _{l ∑}condition for extreme value is above formula is zero to the partial derivative of branch power, namely

$\frac{\∂E}{{\∂P}_{\mathrm{Li}}}=\frac{{\∂E}_{\mathrm{L\Σ}}}{{\∂P}_{\mathrm{Li}}}-\mathrm{\λ}\frac{\∂}{{\∂P}_{\mathrm{Li}}}(\underset{i}{\overset{n}{\mathrm{\Σ}}}{P}_{\mathrm{Li}}-P_{\mathrm{L\Σ}})=0---\left(24\right)$

Therefore have

$\frac{{\∂E}_{\mathrm{L\Σ}}}{{\∂P}_{\mathrm{Li}}}-\mathrm{\λ}(1-0)=0---\left(25\right)$

Can be obtained by formula (22), (25)

$\tan \frac{{\mathrm{\θ}}_{L1}}{2}=\tan \frac{{\mathrm{\θ}}_{L2}}{2}=...=\tan \frac{{\mathrm{\θ}}_{\mathrm{Ln}}}{2}---\left(26\right)$

Known according to formula (3), formula (26) and formula (21) are of equal value each other.When namely meeting formula (21), E _{l ∑}for extreme value.

If alternating current circuit is simplified and is mapped as linear elasticity branch road, can obtain according to formula (12), formula (15)

$E_{\mathrm{L\Σ}}=\frac{1}{2}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{P}_{\mathrm{Li}}{\mathrm{\θ}}_{\mathrm{Li}}---\left(27\right)$

Equally, adopt Lagrangian extremum method, obtain E _{l ∑}extremum conditions be

θ _L1=θ _L2＝...＝θ _Ln(28)

Known according to formula (6), formula (28) and (21) are of equal value each other.

So no matter branch characteristic is linear or non-linear, when meeting formula (21), when namely the meritorious carrying of grid branch is the most balanced, E _{l ∑}for extreme value.

Because formula (21), (26), (28) are all of equal value, so E _{l ∑}extreme point is unique.Necessarily existed by the known minimum potential energy of problem itself, therefore when meeting formula (21), E _{l ∑}for minimum value.

Card is finished.

More than analyze and show, map elastic potential energy and can characterize the meritorious carrying harmony of grid branch.When total burden with power is constant, map elastic potential energy less, grid branch meritorious carrying harmony is better.

6. sample calculation analysis

6.1 electrical networks map the computational accuracy analysis of elastic potential energy

New England 10 machine 39 node system is as Fig. 4.Carry out Load flow calculation, its median generatrix 31 is balance node, and reference voltage is 345kV, and reference capacity is 100MVA, maps elastic network(s) structure as Fig. 5.

Map based on non-linear branch and linear branch, the electrical network using formula (15) and (17) to obtain this system respectively maps elastic potential energy perunit value (fiducial value is 100, disregards dimension), as shown in table 1.

The electrical network of table 1 New England 10 machine 39 node system maps elastic potential energy

In table 1, method 1 is theoretical strict acquiring method, and the error of method 2 only has 0.37%.Visible, when grid branch two ends phase difference is less (in this example, the phase difference at electrical network all branch roads two ends is maximum is no more than 10 °), adopt linear branch map and ignore the active loss of electrical network, the mapping elastic potential energy error of electrical network is very little.

Simulation result, demonstrates the feasibility of Section 3 theoretical method.

6.2 map elastic potential energys and electrical network totally gains merit the relationship analysis of loading allowance

Still for New England 10 machine 39 node system.Employing table 1 method 2 calculates, and the mapping elastic potential energy of electrical network is 4.536327, and equivalent branch road phase difference is 0.147520.

When do not excise power supply and load and keep size, distribute constant, N-1 is taken turns doing to grid branch, sorts from big to small according to potential variation, get front 6 circuits, as shown in table 2.

Table 2 electrical network maps elastic potential energy increment

Table 2 shows, after excising arbitrary grid branch, the equivalent branch road phase difference of electrical network all increases, and show that its loading allowance of totally gaining merit diminishes, bearing capacity dies down; In addition, map the same sequence of growth of elastic potential energy and equivalent branch road phase difference, show to map elastic potential energy larger, electrical network totally meritorious loading allowance is less.In table 1, after excision circuit 21-22, map elastic potential energy increment maximum, electrical network totally meritorious loading allowance is minimum.

For intuitively comparing, excise circuit 21-22 and 15-16 respectively, the mapping elastic network(s) structure of electrical network is as Fig. 6,7.Fig. 5 before relative excision circuit, visible, Fig. 6 increases more obvious than the overall elasticity elongation of Fig. 7, and structure is more fragile, and meritorious bearing capacity declines more.

The simulation results show theoretical analysis result of Section 4.

6.3 mapping elastic potential energys and branch road are gained merit and are carried balanced relationship analysis

As shown in Table 3, 4, from left to right, length is than being 600:400:300:240:200, and coefficient of elasticity ratio is approximately k in 5 outlets for 7 node system parameters shown in Fig. 8 ₁: k ₂: k ₃: k ₄: k ₅=1:1.5:2:2.5:3.

Outlet payload is as shown in table 5, and the feature of wherein burden with power is P ₁: P ₂: P ₃: P ₄: P ₅=1:1.5:2:2.5:3.Reference voltage is 500kV, and reference capacity is 100WM.Existing 4 kinds of sharing of load schemes, the corresponding elastic network(s) structure that maps is as Fig. 9.The corresponding grid nodes phase place of height of node in Fig. 9, branch road maps coefficient of elasticity (perunit value) and effective power flow (perunit value) is labeled in corresponding branch road side.

Employing formula (10), (15), the electrical network calculated respectively under 4 sharing of load schemes maps elastic potential energy, and fiducial value is 100, and result is as shown in table 6.

Table 3 main transformer basic parameter

Table 4 essential parameter of circuit

Table 5 goes out specific electric load (perunit value)

Table 6 outlet sharing of load scheme and the relation mapping elastic potential energy

Table 6 is visible, and from scheme 1 to scheme 4, the mapping elastic potential energy of outlet and electrical network all increases progressively.In scheme 1, the burden with power ratio of outlet is with mapping coefficient of elasticity than time identical, and when namely active power load dispatching is the most balanced, the mapping elastic potential energy of electrical network is minimum; In scheme 4, outlet burden with power ratio and mapping coefficient of elasticity, than time contrary, when namely active power load dispatching is least balanced, map elastic potential energy maximum.

Fig. 9 is visible, and in scheme 1, the elastic elongation of outlet is all equal, stressed the most balanced, shows that the active power load dispatching of outlet is the most balanced; In scheme 4, the elastic elongation of outlet is least balanced, and stressed least balanced, namely active power load dispatching is least balanced.Scheme 2,3, the harmony of active power load dispatching is between the 1st, between 4 kind of situation.

The simulation results show theoretical analysis result of Section 5.

7. conclusion

Under idle abundant condition, the meritorious bearing capacity of electrical network depends on the factors such as the size of electric network composition, branch road bearing capacity, power supply and load and distribution, is difficult to quantitative analysis.Elastic network(s) bearing capacity depend on factor and electrical network similar, in mechanical system, can the elastic potential energy of Applied Physics definite conception, the bearer properties of analysing elastic net.After electrical network being mapped to the elastic network(s) of longitudinal stress, due to the quantity of state correspondent equal in electrical network and elastic network(s), the relationship consistency between quantity of state, therefore the bearer properties of elastic network(s) is exactly the meritorious bearer properties of electrical network.

Analyze and find, from overall angle, mapping elastic potential energy can the meritorious loading allowance of quantitative measurement electrical network; From interior angle, can weigh grid branch gain merit carrying harmony; And when mapping elastic potential energy change, overall loading allowance is consistent with the balanced variation tendency of inner carrying.Therefore, the quantitative analysis index that elastic potential energy can be used as electric network active bearing capacity is mapped.Theory analysis and simulation analysis all show: in certain total active load situation, and map elastic potential energy larger, electric network active bearing capacity is more weak.

The present invention has enriched the theoretical foundation that electric network security is analyzed, and can be widely used in the aspects such as the planning of electrical network, operational mode analysis, on-line scheduling.

Claims (1)

1., based on the electric network active bearing capacity quantitative analysis index mapping elastic potential energy, this index feature is, comprises the steps:

1) negligible resistance, if AC network branch road L two end node is i, j, node voltage phase difference be θ _ij, reactance is X _l, when electric network reactive-load is abundant, U _i, U _jchange less, can make what then transmit gains merit for P _l=C sin θ _ij, make k _l=dP _l/ d θ _ij, k _lfor the mapping coefficient of elasticity of L, k can be obtained _l=C cos θ _ij=P _l/ tan θ _ij;

2) active force of elastic network(s) branch road l and elongation is established to be respectively F _l, x _l, coefficient of elasticity is k _l=dF _l/ dx _l, set up the quantity of state mapping relations of L and l, i.e. P _l=F _l, θ _ij=x _l, k _l=k _l, according to step 1), it is F that the quantity of state of l closes _l=C sinx _l, k _l=C cosx _l=F _l/ tan x _l;

3) if θ _ijless, the quantity of state relation of L can approximate linearization, according to step 1), can P be obtained _l=k _lθ _ij, k _l=C, according to step 2), it is F that the quantity of state of corresponding linear l closes _l=k _lx _l, k _l=C;

4), after l stretches, elastic potential energy is that external force is to its work, i.e. E _l=∫ F _ldx _l, according to step 2) in the quantity of state relation of l, obtain elastic potential energy according to step 2) in the quantity of state mapping relations of L and l, obtain the mapping elastic potential energy of L and E _l=E _l;

5) if θ _ijless, the quantity of state of l, L is all approximately linear relationship, according to step 3), 4), obtain the elastic potential energy of l according to step 2) in quantity of state mapping relations, obtain the mapping elastic potential energy of L and E _l=E _l;

6) electrical network is mapped to vertically stressed elastic network(s), and keeps the incidence relation of node, branch road constant, be made up of n bar branch road if map elastic network(s), E _{l ∑}and E _libe respectively the potential energy of elastic network(s) and wherein i-th branch road, no matter whether its branch road is linear elasticity branch road, and its total potential energy all meets superimposed characteristics, namely therefore the mapping elastic potential energy of electrical network is e _{l ∑}and E _libe respectively the mapping elastic potential energy of electrical network and wherein i-th branch road;

7) if mapping all branch roads in elastic network(s) is all linear elasticity branch road, because branch road is vertical in the same way, leg length is the difference in height of two end nodes, and the phase place of the corresponding grid nodes voltage of height of node, if F _topi, x _topifor stressed size and the height of elastic network(s) top node, F _botj, x _botjfor load node stressed size and height, according to step 5), 6) in E _l, E _{l ∑}formula, the potential energy obtaining Linear Mapping elastic network(s) is therefore the mapping elastic potential energy of linear electrical network is p _topi, θ _topifor the injection of power source bus node in electrical network is gained merit and phase place, P _botj, θ _botjfor burden with power and the phase place of load bus;

8) identical owing to mapping all branch road Impact direction in elastic network(s), thus available 1 with the potential energy E of this elastic network(s) _{l ∑}with total load F _{l ∑}all equal elasticity branch road equivalence, if the length of equivalent branch road is x _leq, according to step 4), obtain if E _{l ∑}, P _{l ∑}and θ _leqfor the mapping potential energy of corresponding electrical network, total active load and equivalent branch road phase difference, according to quantity of state mapping relations, obtain if equivalent branch road is linear characteristic, according to step 5), obtain

9) step 8) show, in identical total active load situation, different electrical network or same electrical network under different running method, if E _{l ∑}comparatively large, then θ _leqalso comparatively large, the overall meritorious loading allowance of electrical network is less, and bearing capacity is poor, and the fail safe of merit angle is poor;

10) by the associated nodes of mains side and load side and between path carry out merging and equivalence, be simplified to longitudinally equivalence and map elastic network(s), be provided with n bar branch road, the mapping coefficient of elasticity that i-th branch road is corresponding, phase difference and be meritoriously respectively k _li, θ _liand P _li, total burden with power is find and prove rule, even P _{l ∑}=const., when branch road active distribute meets P _l1: p _l2: ...: P _ln=k _l1: k _l2: ...: k _lntime, total mapping elastic potential energy E _{l ∑}minimum;

11) equilibrium of electric network active carrying, refer to and map the larger circuit of coefficient of elasticity, larger gaining merit should be carried, step 10) show, map elastic potential energy and can characterize the meritorious carrying harmony of grid branch, namely when total burden with power is constant, map elastic potential energy less, grid branch meritorious carrying harmony is better;

12) step 9), 11) show from overall and interior angle respectively, map elastic potential energy can the meritorious loading allowance of quantitative measurement electrical network and grid branch to gain merit the harmony of carrying, and when mapping elastic potential energy change, overall loading allowance is consistent with the balanced variation tendency of inner carrying, therefore, map the quantitative analysis index that elastic potential energy can be used as electric network active bearing capacity, in certain total active load situation, map elastic potential energy larger, electric network active bearing capacity is more weak.