CN103792465A - Power distribution network one-phase grounding fault location method based on zero sequence voltage - Google Patents

Power distribution network one-phase grounding fault location method based on zero sequence voltage Download PDF

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CN103792465A
CN103792465A CN201310721987.1A CN201310721987A CN103792465A CN 103792465 A CN103792465 A CN 103792465A CN 201310721987 A CN201310721987 A CN 201310721987A CN 103792465 A CN103792465 A CN 103792465A
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CN103792465B (en
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梁睿
崔连华
高列
傅国庆
刘建华
王崇林
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China University of Mining and Technology CUMT
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Abstract

A power distribution network one-phase grounding fault location method based on a zero sequence voltage belongs to a power distribution network grounding fault location method. The fault location method starts from an overall zero sequence parameter of a single-end radial medium voltage power distribution network, analyzes a one-phase grounding fault while taking a distributed parameter model influence into consideration, measures a steady-state zero sequence voltage value and a zero sequence current of each feeder line at a bus position and at a tail end of each outlet line, and finds zero sequence voltage variation characteristics of a fault feeder line and non-fault feeder line. According to the invention, a large number of existing devices are used, the data sampling requirement is low in terms of being real-time, and the method of the invention is easy to realize; the simulation model analysis is established according to the actual parameters, so that the fault location can be realized in the system in which the neutral point is not grounded or the neutral point is grounded through an arc suppression coil; the precision is quite high; and the location method of the invention can be applied to the medium and low voltage power distribution network.

Description

A kind of method of the range finding of the one-phase earthing failure in electric distribution network based on residual voltage
Technical field:
The present invention relates to a kind of method of distribution net work earthing fault range finding, particularly a kind of method of the range finding of the one-phase earthing failure in electric distribution network based on residual voltage.
Background technology:
Power distribution network safe operation provides important guarantee to social production life, is difficult to estimate once there is significant trouble economic loss.Therefore, the fault of quick and precisely finding range, economy, safety and reliability to electric system are extremely important.Singlephase earth fault probability of happening maximum in power distribution network.
Along with the development of distribution network automated technology, the novel panel switches with measurement and communication function are installed on feeder line, can obtain a large amount of circuit information about power, then use electrician's network graph theory principle to set up adjacency matrix and the nodal information matrix of node, and then draw fault judgment matrix and to its ultimate analysis, can judge fault section.The identification of fault section can realize quick excision fault, but can not realize accurately fault localization, cannot meet follow-up work requirement.Mainly contain two kinds of methods for fault localization: traveling wave method and impedance method.Wherein traveling wave method has good accuracy, and impedance method has good stability, the advantage of comprehensive two kinds of methods, and utilize the phase relation of measurement point negative-sequence current and trouble spot negative sequence voltage to find range.But distribution line complex structure, branch are numerous, circuit distance is short, be difficult to solve the identification of fault wave head and the problem that mixed surge impedance of a line changes.Need many cover row ripple checkout equipments, financial cost is higher simultaneously.Therefore traveling wave method is difficult to be applicable to power distribution network.If employing impedance method, not only can overcome a range finding difficult problem for traveling wave method, and can utilize a large amount of existing equipment that puts into operation, hardware investment is little, easily realizes.For both-end impedance method, can adopt phase compensation method to eliminate error between asynchronous zero-sequence current, the voltage collecting to carry out fault distance calculating.But the impedance method adopting adopts lumped parameter model to calculate mostly in the past, owing to not considering the impact of distributed capacitance, error calculated is larger.Based on the fault distance-finding method of distributed parameter model, while having overcome based on lumped parameter model, ignore the drawback of distributed capacitance impact, can improve distance accuracy.Part computing method adopt fixing route parameter calculation, the larger error of same existence.China's medium voltage distribution network mostly is single-ended radial electrical network, the singlephase earth fault that mostly occurs, and failure line selection and fault section determine that research aspect has obtained great successes and obtained good application, the fault location difficulty that how to realize medium voltage distribution network is larger.When singlephase earth fault, transient state energy is little, and travelling wave signal measures difficulty; And traditional ranging technology based on impedance method is used for high-voltage fence, neutral grounding mode is different from low and medium voltage distribution network, when research, how to consider and to consider cannot directly apply to medium voltage distribution network by uniline more from positive order parameter.
Summary of the invention
The object of the invention is to provide a kind of method of the range finding of the one-phase earthing failure in electric distribution network based on residual voltage, solution distribution network line complex structure, the large problem of single-phase ground fault distance measuring error that branch is numerous, the short feature of circuit distance is brought.
The object of the present invention is achieved like this: this fault distance-finding method is from single-ended radial medium voltage distribution network entirety Zero sequence parameter, while analyzing singlephase earth fault, consider the impact of distributed parameter model, stable state residual voltage value and each feeder line zero-sequence current after measurement bus place and each outlet end fault, find out fault feeder and non-fault feeder residual voltage Variation Features.
Concrete steps are as follows:
(1), distributed parameter transmission line model
Power distribution network is carried out to fault localization, utilize distributed parameter model to carry out fault localization; By line parameter circuit value is calculated, when singlephase earth fault occurs, conduction current is much smaller than capacitance current over the ground, therefore can ignore the impact that electricity is led over the ground, zero sequence equivalent circuit distributed parameter model is reduced to line impedance and ground capacitance is evenly distributed along the line;
(2), non-fault line analysis
Feeder line is divided into n minizone, gets wherein arbitrary minizone and be denoted as [x, x+ Δ x], this minizone produce to vagabond current Δ I c=I cΔ x, I cfor the line mutual-ground capacitor electric current of unit length;
For non-fault line, the electric current Δ I that each minizone produces cthe scope flowing through is on the line the position x to place, minizone by bus, then flows to trouble spot through ground; If reference position corresponding to every minizone is x, the voltage that function of current produces is respectively Δ I cxx, the voltage that each interval function of current over the ground producing produces on circuit: Δ U=Δ I cxx, in formula: X is the resistance value of circuit unit length;
Known according to superposition theorem, the residual voltage at circuit two ends is poor is the result of zero-sequence current effect, and above formula both sides obtain x integration simultaneously: ∫ 0 l d U · ′ = ∫ 0 l I . C · X · x · dx , Calculate: U . 2 - U . 1 = 1 2 I C · X · l 2 - - - ( 1 ) , L is non-fault line length, non-fault line first and end residual voltage respectively;
(3), isolated neutral system faulty line is analyzed
For faulty line, trouble spot current flowing comprises non-fault capacitive earth current and faulty line capacitive earth current; Voltage, current distributions behind trouble spot are identical with non-fault line, minizone capacitive earth current flows to the earth by electric capacity, flow back to circuit by trouble spot again, to the circuit before trouble spot without effect, be equivalent to circuit, ground capacitance, and trouble spot between form loop checking installation; Minizone capacitive earth current flows to the earth by ground capacitance, then flows back to bus through trouble spot by circuit; Therefore minizone over the ground the function of current produce the scope of voltage be by minizone to trouble spot, minizone goes out one to bus section, busbar section electric current one and enters to cancel out each other; If fault distance is x, voltage variety Δ U '=Δ I that the function of current over the ground producing apart from the minizone of bus x ' produces on circuit cx (x-x '); The same, can obtain the voltage variety that bus produces to the trouble spot line-to-ground function of current in circuit: ∫ 0 x d U · ′ = ∫ 0 x - I . C · X · ( x - x ′ ) · dx ′ , ? U . ′ = - 1 2 I . C · X · x 2 ; Trouble spot also comprise non-fault line to vagabond current to bus section, busbar section zero-sequence current and voltage variety in this interval generation: U . ″ = I . C Σ · X · x ; Known according to superposition theorem: U . ′ + U . ″ = - ( 1 2 I . C · x - I . C Σ ) X · x = U . f - U . 1 ′ - - - ( 2 ) ; The change in voltage of trouble spot circuit is identical with non-fault line situation, therefore, wherein: l ' is faulty line total length, corresponding faulty line first and end residual voltage and fault point voltage respectively; (2), (3) formula is added: U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + I . C Σ · X · x - - - ( 4 ) ;
(4), neutral point is analyzed through arc suppression coil faulty line
Non-fault line zero-sequence current and the residual voltage situation of change of neutral by arc extinction coil grounding system are identical with isolated neutral system; And faulty line zero-sequence current situation is equivalent to flow through in trouble spot stack the inductance current of arc suppression coil on isolated neutral system faulty line basis as system works in full compensating coefficient is, trouble spot reactive current is zero, if ignore system active current, the zero-sequence current recording from fault branch head end is the capacitance current being produced by this branch road self, and application residual voltage changes and cannot find range; But in arc suppression coil when operation, is for avoiding resonance appearance, generally operate in over-compensation state, also that the zero-sequence current character of fault branch occurs when single-phase earthing is similar to non-fault branch, because the capacitance current that self distribution parameter does not produce on year-on-year basis of over-compensation situation will be more greatly, and the gain of parameter of large I when tuning by arc suppression coil; Residual voltage acts on after the current hysteresis that arc suppression coil produces 90 °, and 90 ° of leading residual voltages of capacitive earth current, therefore with single spin-echo. the voltage variety producing in fault section: finally, draw the formula corresponding to compensated distribution network: U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + ( I . C Σ - I . L ) · X · x - - - ( 5 ) ;
Beneficial effect, has adopted such scheme, can utilize a large amount of existing equipment that puts into operation, and realizes simply, has stronger economy and good practical value; Employing distributed parameter model calculates, and has overcome in power distribution network tradition Fault Locating Method owing to using lumped parameter model to cause the problem that error is larger; The method is not only applicable to isolated neutral system, also be applicable to neutral by arc extinction coil grounding system, and only need obtain each branch road zero-sequence current and branch road end residual voltage steady-state value and the tuning situation of arc suppression coil after singlephase earth fault, on the basis of correctly selecting fault feeder, can accurately find range.
Accompanying drawing explanation
Fig. 1 zero sequence equivalent circuit distributed parameter model.
Fig. 2 non-fault line zero-sequence current distribution plan.
Fig. 3 isolated neutral system faulty line zero-sequence current forms feature.
Fig. 4 neutral point arc suppression coil earthing system faulty line zero-sequence current forms feature.
The method flow diagram of the one-phase earthing failure in electric distribution network range finding of Fig. 5 based on residual voltage.
Fig. 6 PSCAD realistic model.
Embodiment:
Embodiment 1: this fault distance-finding method is from single-ended radial medium voltage distribution network entirety Zero sequence parameter, while analyzing singlephase earth fault, consider the impact of distributed parameter model, stable state residual voltage value and each feeder line zero-sequence current after measurement bus place and each outlet end fault, find out fault feeder and non-fault feeder residual voltage Variation Features.
Concrete steps are as follows:
1, distributed parameter transmission line model
Utilize distributed parameter model to carry out fault localization to power distribution network; By line parameter circuit value is calculated, when singlephase earth fault occurs, conduction current is much smaller than capacitance current over the ground, therefore can ignore the impact that electricity is led over the ground, zero sequence equivalent circuit distributed parameter model is reduced to line impedance and ground capacitance is evenly distributed along the line;
2, non-fault line analysis
Feeder line is divided into n minizone, gets wherein arbitrary minizone and be denoted as [x, x+ Δ x], this minizone produce to vagabond current Δ I c=I cΔ x, I cfor the line mutual-ground capacitor electric current of unit length;
For non-fault line, the electric current Δ I that each minizone produces cthe scope flowing through is on the line the position x to place, minizone by bus, then flows to trouble spot through ground; If reference position corresponding to every minizone is x, the voltage that function of current produces is respectively Δ I cxx, the voltage that each interval function of current over the ground producing produces on circuit: Δ U=Δ I cxx, in formula: X is the resistance value of circuit unit length;
Known according to superposition theorem, the residual voltage at circuit two ends is poor is the result of zero-sequence current effect, and above formula both sides obtain x integration simultaneously: ∫ 0 l d U · ′ = ∫ 0 l I . C · X · x · dx , Calculate: U . 2 - U . 1 = 1 2 I C · X · l 2 - - - ( 1 ) , L is non-fault line length non-fault line first and end residual voltage respectively;
3, isolated neutral system faulty line is analyzed
For faulty line, trouble spot current flowing comprises non-fault capacitive earth current and faulty line capacitive earth current; Voltage, current distributions behind trouble spot are identical with non-fault line, minizone capacitive earth current flows to the earth by electric capacity, flow back to circuit by trouble spot again, to the circuit before trouble spot without effect, be equivalent to circuit, ground capacitance, and trouble spot between form loop checking installation; Minizone capacitive earth current flows to the earth by ground capacitance, then flows back to bus through trouble spot by circuit; Therefore minizone over the ground the function of current produce the scope of voltage be by minizone to trouble spot, minizone goes out one to bus section, busbar section electric current one and enters to cancel out each other; If fault distance is x, voltage variety Δ U '=Δ I that the function of current over the ground producing apart from the minizone of bus x ' produces on circuit cx (x-x '); The same, can obtain the voltage variety that bus produces to the trouble spot line-to-ground function of current in circuit: ∫ 0 x d U · ′ = ∫ 0 x - I . C · X · ( x - x ′ ) · dx ′ , ? U . ′ = - 1 2 I . C · X · x 2 ; Trouble spot also comprise non-fault line to vagabond current to bus section, busbar section zero-sequence current and voltage variety in this interval generation: U . ″ = I . C Σ · X · x ; Known according to superposition theorem: U . ′ + U . ″ = - ( 1 2 I . C · x - I . C Σ ) X · x = U . f - U . 1 ′ - - - ( 2 ) ; The change in voltage of trouble spot circuit is identical with non-fault line situation, therefore, wherein: l ' is faulty line total length, corresponding faulty line first and end residual voltage and fault point voltage respectively; (2), (3) formula is added: U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + I . C Σ · X · x - - - ( 4 ) ;
4, neutral point is analyzed through arc suppression coil faulty line
Non-fault line zero-sequence current and the residual voltage situation of change of neutral by arc extinction coil grounding system are identical with isolated neutral system; And faulty line zero-sequence current situation is equivalent to flow through in trouble spot stack the inductance current of arc suppression coil on isolated neutral system faulty line basis ; As system works in full compensating coefficient is, trouble spot reactive current is zero, if ignore system active current, the zero-sequence current recording from fault branch head end is the capacitance current being produced by this branch road self, and application residual voltage changes and cannot find range; But in arc suppression coil when operation, is for avoiding resonance appearance, generally operate in over-compensation state, also that the zero-sequence current character of fault branch occurs when single-phase earthing is similar to non-fault branch, because the capacitance current that self distribution parameter does not produce on year-on-year basis of over-compensation situation will be more greatly, and the gain of parameter of large I when tuning by arc suppression coil; Residual voltage acts on after the current hysteresis that arc suppression coil produces 90 °, and 90 ° of leading residual voltages of capacitive earth current, therefore with single spin-echo. the voltage variety producing in fault section: finally, draw the formula corresponding to compensated distribution network: U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + ( I . C Σ - I . L ) · X · x - - - ( 5 ) ;
As shown in Figure 5, specific implementation step is as follows for this method specific implementation process flow diagram:
From wave recording device, reading flow is crossed head end zero-sequence current and end residual voltage Wave data after arc suppression coil electric current (for neutral by arc extinction coil grounding system), busbar voltage, each feeder fault;
The data that read are carried out to Fast Fourier Transform (FFT) (FFT), obtain the effective value that each signal is corresponding, adopt power frequency component calculates can harmonic carcellation etc. the error of signal generation;
On the basis completing in route selection, determine fault branch;
To record the data substitution formula (1) that non-fault line is corresponding, online calculate fault moment circuit unit length zero sequence impedance value X;
Judge system neutral earthing mode;
For isolated neutral system, the circuit unit length zero sequence impedance value X substitution formula (4) that the data that faulty line is recorded and step 4 calculate calculates fault distance; For neutral by arc extinction coil grounding system, substitution formula (5) calculates.
(1) electric current and voltage distribution characteristics after fault
After fault, feeder line capacitive earth current distributes along the line, first all capacitive earth currents flow to the earth, flow back to faulty line through trouble spot, finally flow to bus, the circulation of vagabond current has been produced to effect to the variation of residual voltage, the rule that non-fault line residual voltage effective value along the line is increased by oblique line to line end gradually by bus distributes, the rule that faulty line residual voltage effective value is increased by oblique line to bus section gradually by trouble spot distributes, trouble spot distributes by the rule increasing gradually by oblique line to line end, it is trouble spot place residual voltage effective value minimum.
(2) Fault Location Algorithm
After fault occurs and malfunction stable after, from wave recording device, read busbar voltage, after each feeder fault, head end zero-sequence current and end residual voltage Wave data also need to flow through the Wave data of arc suppression coil electric current for neutral by arc extinction coil grounding system, the data that read are carried out to Fast Fourier Transform (FFT) (FFT), obtain the effective value that each signal is corresponding, adopt power frequency component calculates can harmonic carcellation etc. the error of signal generation, in utilization, the existing ripe variant projects of location of step result of calculation basis is determined faulty line, the data substitution formula that non-fault line is corresponding will be recorded online calculate fault moment circuit unit length zero sequence impedance value X, the circuit unit length zero sequence impedance value X substitution formula that the data that faulty line recorded for isolated neutral system and upper step calculate calculate fault distance, for neutral by arc extinction coil grounding system, substitution formula U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + I . C Σ · X · x Calculate.
The effect assessment of scheme:
The present invention has very high position precision and possesses very high adaptability at Complicated Distribution Network, and for different neutral grounding modes, the present invention all can meet.Now take a model as example:
Utilize PSCAD/EMTDC Software tool to set up the realistic model of the single-ended radial network system of 35kV, as shown in Figure 6.Circuit Fault on Secondary Transformer connects arc suppression coil through isolating switch, can centering point earth-free and compensated distribution network emulation; System has three cables, and cable length is got respectively 18km, 16km and 20km, and cable adopts three single-phase cables that are embedded in underground 1m to be del and places the system of laying of (axle center spacing is 30mm), and cross-section of cable area is got 240mm2; Bus bar side adopts the 110kV of Υ-Δ connection to become 35kV transformer, and line end adopts the 35kV of Δ-Υ connection to become the transformer of 10kV; Load connects the three-phase balancing load of 0.35MW+0.08MVar.
With isolated neutral system, be that example is calculated there is metallic earthing apart from bus 2km place.Busbar voltage is 20469.2V, and each feeder line head end zero-sequence current is 7.33167A, 6.51593A, 13.8476A, each feeder terminal residual voltage 20493.9V, 20488.7V, 20483.4V.Utilize existing variant projects of location, it is circuit III that for example ratio phase comparing method can be judged faulty line.
Utilize circuit I unit of account length circuit capacitive earth current I cand unit length zero sequence impedance value X:I c=I c1/ l 1=0.407315A/km. X = 2 ( U 2 - U 1 ) I C · l 1 = 0.374327 Ω / km
For isolated neutral system, non-fault line capacitive earth current and equal zero-sequence current that faulty line head end measures: I C 3 = I C 1 + I C Σ = 13.8476 A .
Above each result of calculation substitution formula (4) is obtained to fault distance x=1.9791km.
Definition measuring error is: error is 0.104%.
For the validity of checking this method, table 1 has provided isolated neutral system singlephase earth fault and has betided diverse location, range finding result when the different transition resistance ground connection.
Table 1 isolated neutral single-phase ground fault distance measuring result
This algorithm is equally applicable to neutral by arc extinction coil grounding system as can be seen from Table 2, and all has higher range accuracy for different de-humorous degree.

Claims (1)

1. the method for the range finding of the one-phase earthing failure in electric distribution network based on residual voltage, it is characterized in that: this fault distance-finding method is from single-ended radial medium voltage distribution network entirety Zero sequence parameter, while analyzing singlephase earth fault, consider the impact of distributed parameter model, stable state residual voltage value and each feeder line zero-sequence current after measurement bus place and each outlet end fault, find out fault feeder and non-fault feeder residual voltage Variation Features;
Concrete steps are as follows:
(1), distributed parameter transmission line model
Power distribution network is carried out to fault localization, utilize distributed parameter model to carry out fault localization; By line parameter circuit value is calculated, when singlephase earth fault occurs, conduction current is much smaller than capacitance current over the ground, therefore can ignore the impact that electricity is led over the ground, zero sequence equivalent circuit distributed parameter model is reduced to line impedance and ground capacitance is evenly distributed along the line;
(2), non-fault line analysis
Feeder line is divided into n minizone, gets wherein arbitrary minizone and be denoted as [x, x+ Δ x], this minizone produce to vagabond current Δ I c=I cΔ x, I cfor the line mutual-ground capacitor electric current of unit length;
For non-fault line, the electric current Δ I that each minizone produces cthe scope flowing through is on the line the position x to place, minizone by bus, then flows to trouble spot through ground; If reference position corresponding to every minizone is x, the voltage that function of current produces is respectively Δ I cxx, the voltage that each interval function of current over the ground producing produces on circuit: Δ U=Δ I cxx, in formula: X is the resistance value of circuit unit length;
Known according to superposition theorem, the residual voltage at circuit two ends is poor is the result of zero-sequence current effect, and above formula both sides obtain x integration simultaneously: ∫ 0 l d U . = ∫ 0 l I . C · X · x · dx , Calculate: U . 2 - U . 1 = 1 2 I C · X · l 2 - - - ( 1 ) , L is non-fault line length, non-fault line first and end residual voltage respectively;
(3), isolated neutral system faulty line is analyzed
For faulty line, trouble spot current flowing comprises non-fault capacitive earth current and faulty line capacitive earth current; Voltage, current distributions behind trouble spot are identical with non-fault line, minizone capacitive earth current flows to the earth by electric capacity, flow back to circuit by trouble spot again, to the circuit before trouble spot without effect, be equivalent to circuit, ground capacitance, and trouble spot between form loop checking installation; Minizone capacitive earth current flows to the earth by ground capacitance, then flows back to bus through trouble spot by circuit; Therefore minizone over the ground the function of current produce the scope of voltage be by minizone to trouble spot, minizone goes out one to bus section, busbar section electric current one and enters to cancel out each other; If fault distance is x, voltage variety Δ U '=Δ I that the function of current over the ground producing apart from the minizone of bus x ' produces on circuit cx (x-x '); The same, can obtain the voltage variety that bus produces to the trouble spot line-to-ground function of current in circuit: ∫ 0 x d U · ′ = ∫ 0 x - I . C · X · ( x - x ′ ) · dx ′ , ? U . ′ = - 1 2 I . C · X · x 2 ; Trouble spot also comprise non-fault line to vagabond current to bus section, busbar section zero-sequence current and voltage variety in this interval generation: U . ″ = I . C Σ · X · x ; Known according to superposition theorem: U . ′ + U . ″ = - ( 1 2 I . C · x - I . C Σ ) X · x = U . f - U . 1 ′ (2); The change in voltage of trouble spot circuit is identical with non-fault line situation, therefore, wherein: l ' is faulty line total length, corresponding faulty line first and end residual voltage and fault point voltage respectively; (2), (3) formula is added: U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + I . C Σ · X · x - - - ( 4 ) ;
(4), neutral point is analyzed through arc suppression coil faulty line
Non-fault line zero-sequence current and the residual voltage situation of change of neutral by arc extinction coil grounding system are identical with isolated neutral system; And faulty line zero-sequence current situation is equivalent to flow through in trouble spot stack the inductance current I L of arc suppression coil on isolated neutral system faulty line basis; As system works in full compensating coefficient is, trouble spot reactive current is zero, if ignore system active current, the zero-sequence current recording from fault branch head end is the capacitance current being produced by this branch road self, and application residual voltage changes and cannot find range; But in arc suppression coil when operation, is for avoiding resonance appearance, generally operate in over-compensation state, also that the zero-sequence current character of fault branch occurs when single-phase earthing is similar to non-fault branch, because the capacitance current that self distribution parameter does not produce on year-on-year basis of over-compensation situation will be more greatly, and the gain of parameter of large I when tuning by arc suppression coil; Residual voltage acts on after the current hysteresis that arc suppression coil produces 90 °, and 90 ° of leading residual voltages of capacitive earth current, therefore single spin-echo. the voltage variety producing in fault section: finally, draw the formula corresponding to compensated distribution network: U . 2 ′ - U . 1 ′ = 1 2 I . C · X · ( l ′ 2 - 2 l ′ · x ) + ( I . C Σ - I . L ) · X · x - - - ( 5 ) ;
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