CN105787191B - A kind of cable temperature quick calculation method based on parameter fitting - Google Patents

A kind of cable temperature quick calculation method based on parameter fitting Download PDF

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CN105787191B
CN105787191B CN201610145240.XA CN201610145240A CN105787191B CN 105787191 B CN105787191 B CN 105787191B CN 201610145240 A CN201610145240 A CN 201610145240A CN 105787191 B CN105787191 B CN 105787191B
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CN105787191A (en
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张宇娇
徐彬昭
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China Three Gorges University
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    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
    • G06F30/30Circuit design
    • G06F30/36Circuit design at the analogue level
    • G06F30/367Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10Complex mathematical operations
    • G06F17/11Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/12Simultaneous equations, e.g. systems of linear equations
    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10Complex mathematical operations
    • G06F17/11Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/13Differential equations
    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/08Thermal analysis or thermal optimisation

Abstract

A kind of cable temperature quick calculation method based on parameter fitting establishes the transient state cable temperature thermal circuit model for being used for parameter fitting using Re Lufa according to cable laying feature first, and column write the cable temperature differential equation;For a certain specific operating condition of cable, cable is in zero original state, rated current is applied to cable, measuring cable temperature reaches cable conductor and surface temperature in steady-state phase;According to actual measurement cable conductor temperature and cable surface temperature, using thermal capacitance, thermal resistance thermal parameter in the Trust Region Algorithm fitting cable differential equation;Finally according to fitting thermal parameter, solves cable temperature Transient Thermal Circuit and obtain cable temperature simplified solution formulas.A kind of cable temperature quick calculation method based on parameter fitting of the present invention, the transient state thermal parameter of conversion model can be comparatively fast solved in conjunction with the observed temperature under cable nominal transmission capacity, and obtain the cable core of cable and the simplified solution formulas of surface temperature, achieve the purpose that quickly to calculate cable temperature.

Description

A kind of cable temperature quick calculation method based on parameter fitting
Technical field
A kind of cable temperature quick calculation method based on parameter fitting of the present invention, is related to cable temperature calculating field.
Background technique
It calculates cable temperature judges whether transmission capacity is reasonable, is of great significance to cable economy with safe operation.Mesh The main method of preceding direct-buried cable temperature computation are as follows: the numerical value calculating side based on FInite Element, Finite Volume Method for Air, boundary element method Method and Re Lufa based on IEC60287, IEC60853 standard.But the above method is when calculating cable temperature all by cable insulation The thermal parameter of layer material and laying soil is as given value.However when reality calculating direct-buried cable temperature, cable insulation and soil The thermal parameter of earth is not easy to obtain, and cable insulation thermal parameter can change because of the aging of insulation.At this time using traditional Numerical method and Re Lufa are difficult to realize accurately calculating for cable temperature.
To realize that cable temperature accurately calculates, more true cable thermal parameter need to be obtained, it is quasi- that observed temperature can be used Close the thermal parameter of cable.Fitting for cable thermal parameter, be mostly at present according to the numerical value in direct-buried cable temperature field calculate, then Cable thermal parameter is obtained by optimization algorithm combination observed temperature.Singapore professor H.J.Li is for certain coastal dock 6.6kV cable provides a kind of by DFP algorithm combination temperature field because the operation reasons such as aging change the thermal parameter of cable The method that FEM calculation is fitted the thermal parameter of cable.It is proposed in North China Electric Power University Chang Wenzhi professor through actual measurement temperature The method that the numerical value of degree combination temperature field calculates to correct existing thermal parameter reaches to obtain more true cable thermal parameter To the purpose for accurately calculating temperature.It, need to be according to cable for the safe operation for guaranteeing cable however when cable band emergency loads Status quickly calculate the transient-state temperature of cable.Cable thermal parameter is fitted by the numerical method in temperature field at this time and there is solution Time longer problem.
Summary of the invention
When for calculating cable temperature, there are the thermal parameters of cable insulation and soil to be difficult to obtain, and passes through temperature field Numerical method fitting cable thermal parameter exist solve time longer problem.A kind of base is proposed on the basis of traditional Re Lufa In the cable temperature quick calculation method of parameter fitting, can comparatively fast be solved in conjunction with the observed temperature under cable nominal transmission capacity The transient state thermal parameter of model is converted, and obtains the cable core of cable and the simplified solution formulas of surface temperature, reaches and quickly calculates cable The purpose of temperature.
The technical scheme adopted by the invention is that:
A kind of cable temperature quick calculation method based on parameter fitting, comprising the following steps:
Step 1: the transient state cable temperature for being used for parameter fitting being established using Re Lufa according to cable laying feature first Thermal circuit model, column write the cable temperature differential equation;
Step 2: being directed to a certain specific operating condition of cable, cable is in zero original state, rated current is applied to cable, is surveyed Amount cable temperature reaches cable conductor and surface temperature in steady-state phase;
Step 3: micro- using Trust Region Algorithm fitting cable according to actual measurement cable conductor temperature and cable surface temperature Divide thermal capacitance, thermal resistance thermal parameter in equation;Finally according to fitting thermal parameter, solves cable temperature Transient Thermal Circuit and obtain cable temperature Simplified solution formulas.
A kind of cable temperature quick calculation method based on parameter fitting of the present invention, by cable core and the surface of surveying cable The method of temperature foh thermal circuit model parameter, and thus obtain the cable core of cable and the reduced mechanical model of surface temperature, relatively It is calculated in numerical value and traditional Re Lufa has the advantage that
(1), it using measured value fitted model parameters, solves cable insulation and soil thermal parameter is not easy to obtain or insulate The problems such as layer thermal parameter changes.
(2), the parameter of model is calculated using reduced value, the solution of practical heat source is avoided, in true property without misalignment Under the conditions of reduce the calculation amount of model parameter.
(3), the cable core of the cable obtained by fitting parameter model, surface temperature simplified solution formulas simplify cable transient state The calculating of temperature is of great significance for the reasonable distribution of current-carrying capacity of cable, cable temperature early warning.
(4), temporary when being established to other fever electrical equipments such as motor switch cabinet except for solving cable temperature Transient calculation When state Equivalent heat path model etc., the thermal parameter that can also be fitted by actual temperature in Transient Thermal Circuit model, and then realize its temperature The Transient calculation of degree.
(5), cable temperature transient state conversion thermal circuit model is established, the fitting of cable transient state thermal parameter is simplified.
(6), cable temperature is expressed using simplified solution formulas, cable temperature when facilitating different transmission capacities and initial temperature Calculating.
Detailed description of the invention
Fig. 1 is the flow chart of the method for the present invention.
Fig. 2 is the typical section figure of single-core cable;Wherein 1- conductor, 2- conductor shield, 3-XLPE insulating layer, 4- insulation Shielded layer, 5- buffer layer, 6- wrinkle aluminium layer, 7- external sheath layer, 8- semiconducting coating.
The cable Transient Thermal Circuit model that Fig. 3 is unfolded when being cable direct-burried by layers of material and soil different temperatures layer Figure.
Fig. 4 is three node Transient Thermal circuit diagrams in the present invention.
Fig. 5 is cable temperature rise experiment porch schematic diagram in the present invention.
Fig. 6 is that cable temperature calculates and experiment value curve graph in underload EXPERIMENTAL EXAMPLE of the present invention;
Fig. 7 is that cable calculates relative error curve graph in underload EXPERIMENTAL EXAMPLE of the present invention.
Fig. 8 is that cable temperature calculates and experiment value curve graph in present invention overload EXPERIMENTAL EXAMPLE;
Fig. 9 is that cable calculates relative error curve graph in present invention overload EXPERIMENTAL EXAMPLE.
Specific embodiment
A kind of cable temperature quick calculation method based on parameter fitting, comprising the following steps:
Step 1: by taking cable directly buried installation as an example, according to the hot road parameter of each insulating layer of cable and soil, establishing tradition electricity Cable Equivalent heat path model.Ignore cable insulation loss, is three sections by cable Equivalent heat path model abbreviation according to dual-port theory Point thermal circuit model, and it is equivalent using reduced value progress to the parameters such as calorific value, thermal resistance, thermal capacitance in model.To three node Re Lumo Type, column write the cable temperature differential equation.
Step 2: cable is initially located in zero original state, i.e., cable temperature is equal with environment temperature and application electric current is zero. Rated current is applied to cable, every one minute cable conductor of record and cable surface temperature, until cable conductor and table Face temperature reaches stable.
Step 3: according to actual measurement cable conductor and surface temperature, cable temperature differential side being fitted using Trust Region Algorithm Thermal parameter in journey solves the cable differential equation and obtains cable core and surface temperature simplified solution formulas.It is real finally to carry out cable temperature rise It tests, by cable overload and measurement temperature and simplified solution formulas temperature comparisons when underload, there is only lesser errors, it was demonstrated that this Method it is effective.
1. the foundation of three node Transient Thermal Circuit models:
When cable directly buried installation, main medium is soil around cable.After cable surface is greater than with a certain distance from, soil Temperature will not by cable operation influenced, thus can will remotely the soil moisture be considered as it is constant.When external environment and soil temperature When degree is approximate constant, the constant soil horizon of temperature can be analogous to voltage source in Transient Thermal Circuit, between cable surface The thermally conductive thermal resistance that can be considered that one impedance value of series connection is constant.Fig. 2 is the typical section figure of single-core cable, and Fig. 3 is pressed when being cable direct-burried The cable Transient Thermal Circuit model that layers of material and soil different temperatures layer are unfolded.
In actually calculating, the material thermal parameter (thermal resistance, thermal capacitance) of each layer of cable is often not easy to obtain, each insulating layer of cable Thermal parameter also certain change can occur because of the aging of operation, cause Transient Thermal Circuit thermal parameter solve have difficulties.It examines When considering cable operation, current-carrying capacity is determined by the core temperature of cable temperature highest point, and external heat dispersal situations are then by cable surface Temperature determines.Therefore the needs of general practical engineering calculation can be met by calculating the cable core of cable and cable surface temperature.Again by Be much larger than remaining each insulating layer in conductor temp.-elevating amount, insulating layer loss can be ignored, according in circuit two-port network it is theoretical by Fig. 3 Middle thermal circuit model is reduced to three node Transient Thermal Circuit models in Fig. 4.
Heat and the loading current that cable loss generates it is square approximate directly proportional.When ignoring insulating layer loss, it can incite somebody to action Cable internal loss is equivalent to single source Q.Under the premise of core temperature T1, the cable surface temperature T2 of guarantee cable are constant Heat source Q is calculated using reduced value, i.e.,
Q=I2/k (1)
In formula: Q is conversion calorific value;I is loading current;K is the conversion factor chosen.
Guarantee that node temperature T1, T2 are constant due to converting front and back, at this time thermal resistance R1, R2 in thermal circuit model, thermal capacitance C1, C2 Calculated value is the obtained reduced value on the basis of calorific value Q.The practical heat of solution is avoided using reduced value calculating for heat source Because of error caused by the factors such as skin effect, conductor indium when source.
2. column write node differential equation group:
The differential equation such as formula (2) is write to node T1, T2 column according to the cable Transient Thermal Circuit model in Fig. 4.For solution node T1, T2 temperature need to first be fitted thermal parameter R1, R2, C1, C2 in the differential equation.If directly quasi- by the analytic solutions of differential equation group Equation parameter is closed, will to obtain as unknown quantity be 4, the non-linear overdetermined equations that equation group number is 2n.When temperature sampling point is more, When expression formula is more complex, resulting Large Scale Nonlinear over-determined systems are not easy to solve.Therefore the numerical value of the differential equation is sought first Solution, then pass through numerical solution fit equation parameter.The numerical solution of differential equation group is solved using 4 grades of Runge-Kutta methods herein.
In formula: T1_0, T2_0 are node T1, T2 initial temperature;Q is conversion calorific value;T0 is the soil moisture.
3. the objective function of constructing variable fitting:
When R1, R2, C1, C2 are considered as known constant, differential equation group (2) are solved in the time with 4 grades of Runge-Kutta methods Numerical solution at point t=[t1, t2 ..., tn] is T ' 1=[T ' a1, T ' a2 .., T ' an], T ' 2=[T ' b1, T ' b2 .., T ' bn].Objective function F (C1, C2, R1, R2) is constructed using nonlinear least square method, wherein F is from change with R1, R2, C1, C2 Amount, to calculate error sum of squares as functional value.F expression formula such as formula (3).Seek optimal fitting thermal parameter R1, R2, C1, C2 and makes F Value is minimum.
4. Trust Region solves fitting parameter:
Differential equation parameter fitting is converted to by construction objective function the optimization for solving non-linear unconfined condition Problem.For the globally optimal solution for finding thermal parameter, initial point is calculated by the way that the setting of Multi-Start algorithm is a large amount of first, then is led to It crosses Trust Region Algorithm and obtains multiple locally optimal solutions.Finally the corresponding functional value of multiple locally optimal solutions is compared, wherein The smallest solution of functional value is the approximation of globally optimal solution.The basic step of Trust Region Algorithm solution Unconstrained Optimization Problem It can be summarized as follows:
(1) setting solves initial value Xk=(C1k, C2k, R1k, R2k), iteration upper limit frequency n.Selection is initial to trust radius Rk, approximation to function degree judge parameter μ, η, and 0 < μ < η < 1, set k=1.
(2) F (Xk), F gradient are calculatedAnd judgeWhether allowable error ε is less than.As met, then Xk As locally optimal solution.It is such as unsatisfactory for, executes subsequent step.
(3) in Trust Region Radius rk, F approximate function G (x) such as formula (4) is constructed.Following iteration point is Xk+1, enables d=Xk +1-Xk.To seek Xk+1, second order subproblem function ψ (d) such as formula (5) is constructed.Solving dk is minimized ψ (d) in trusted zones.
(4) degree of approximation discriminant function ρ is constructedk(x) such as formula (6), judge in XkAnd Xk+1Between between G (X) and F (X) whether foot It is enough close.Work as ρk(x)≤μ when, the degree of approximation is unsatisfactory for requiring between G (X) and F (X), show approximate extents choose it is excessive, will trust Domain reduced radius is rk+1=0.5rk, and following iteration point remains unchanged i.e. Xk+1=Xk.As μ < ρk(x) < η shows the degree of approximation still It is acceptable, keep Trust Region Radius constant, setting an iteration point is Xk+1=Xk+dk.Work as ρk(x) >=η when, show the degree of approximation compared with It is good, expand and trusts radius rk+1=2rk, following iteration point is Xk+1=Xk+d.Iterate until meeting condition 2) or arrival iteration Upper limit number, X value is locally optimal solution of the F (X) near primary iteration point at this time.
5. cable temperature simplified solution formulas:
R1, R2, C1, C2 optimal fitting parameter are brought into differential side's group (2), calculating T1, T2 analytic solutions is cable The simplified solution formulas of cable core, surface temperature.Write equation (2) as matrix form such as formula (7), i.e.,
In formula:
Formula (7) both sides are carried out simultaneously pull-type to change to obtain formula (8).Formula (8) is the linear equation in two unknowns group under frequency domain S, meter Calculate cable cable core and surface temperature Frequency Domain Solution be T (S), such as formula (9).T (S) is finally subjected to counter pull type transformation, as To the simplified solution formulas under time domain of cable core and surface of cable.
ST (S)-T (0_)=AT (S)+B (8)
T (S)=(SE-A)-1(B+T(0_)) (9)
Wherein:
6. validation verification:
For the validity for verifying the cable temperature quick calculation method based on parameter fitting, the production of certain cable factory is chosen YJLW03 60/110kV type high-tension cable carries out direct-buried cable temperature rise experiment, and cable temperature changes when simulating actual motion, will count Calculation value is compared with measured value.
The basic device of experiment is as shown in figure 5, the peripheral equipment of cable can be divided into 2 regions, and wherein region one is high pressure Operation area, including rising current transformer, pressure regulator, compensating electric capacity, switchgear.Region two is low voltage control measured zone, including Test wooden case, temperature measurement probe, current transformer and control system.Wherein control system is come by adjusting the voltage of pressure regulator The electric current of cable loop is controlled, and the cable temperature of temperature measurement module acquisition and the electric current of current transformer acquisition can be recorded in real time.
6.1 temperature simplified solution formulas parametric solutions:
When experiment, cable is placed in the big wooden case of long 2.4m, width 0.8m, high 1.2m, drying is loaded in big wooden case The soil in somewhere simulates laying state when actual cable direct-burried.Soil aggregate depth 0.7m in big wooden case, cable bottom soil are deep 0.4m.In view of being influenced in the middle part of wooden case by the external world is smaller, with actual conditions when cable direct-burried more closely, when experiment measures Cable temperature in the middle part of wooden case.Thermocouple temperature measurement module is used for the measurement of cable temperature in this trial, wherein thermoelectricity The core temperature of even T1 measurement cable, T2 measure cable surface temperature.
Apply electric current before, cable is in zero original state, at this time cable and the soil moisture it is equal everywhere be 24.7 DEG C.To electricity Cable passes to rated current, and the core temperature T1 and cable surface temperature T2 of cable basically reach stable state after cable operation 10h. During this period every temperature data of one minute record, obtain T1=[Ta1, Ta2 ..Ta600], T2=[Tb1, Tb2, ...Tb600].Hot road parameter fit-procedure is write according to third section, by I, T1, T2 record value is brought into, is Pentiu m in CPU It is run on the computer of 2.0GHz about 7 minutes, obtains optimal fitting thermal parameter C1=18.0997, C2=62.6088, R1= 2.0790, R2=2.5556.Finally thermal parameter is brought into formula (2), when acquiring cable and being in zero original state, in laboratory institute Under the laying environment of simulation, the cable core and surface temperature T1 ' of cable, T2 ' simplified solution formulas such as formula (10) are shown.
T'1=A1+B1·e-0.239t+C1·e-0.00649t
T'2=A2+B2·e-0.239t+C2·e-0.00649t (10)
In formula:
A1=3.798Q+T0
A2=1.601Q+T0
B1=-0.00225Q+0.0214T1_0-0.0493T2_0+0.0279T0
B2=0.0447Q-0.4242T1_0+0.979T2_0-0.554T0
C1=-3.795Q+0.9786T1_0+0.0493T2_0-1.028T0
C2=-1.645Q+0.424T1_0+0.0214T2_0-0.446T0
Q=I2/ k, takes k=1.000 × 10 herein5
6.2 Test of accuracies:
(1) underload is tested.By soil moisture T0, the cable core and surface initial temperature T1_0, T2_0 of cable, loading current I The temperature for obtaining the cable when application electric current be 500A, the soil moisture be 25.7 DEG C, in the case where simulate laid condition is brought in formula (10) into Spend calculating formula such as formula (11).Cable is finally run to cable temperature calculated value T1 ', T2 ' and experiment value T1, T2 progress in 10h Comparison, as shown in Figure 6, Figure 7.And with two hours for the period, the cable core of cable and the maximum on surface are opposite during record operation Error and absolute error, as shown in table 1.
T1'=40.99-11.48e-0.00649t-0.00681e-0.239t
T2'=34.34-4.977e-0.00649t+0.135e-0.239t (11)
Max calculation error in 1 550A day part of table
As shown in Table 1 when cable is under above-mentioned running environment, using the method for hot road parameter fitting, there is only lesser meters Calculate error.In at runtime, the cable core of cable calculates relative error at 0.5 DEG C or so, calculates absolute error 1% or so; Maximum absolute error is 1.515 DEG C, and maximum relative error 5.318% was both present in 0~2h period.Cable surface meter Absolute error is calculated at 1 DEG C or so, for relative error 3% or so, maximum absolute error is 1.511 DEG C, maximum relative error is- 5.736%, it is both present in 0~2h sections.Show very close between calculated result and actual value, meets wanting for engineering calculation It asks, it was demonstrated that the validity of cable simplified solution formulas in underload.
(2) overload experiment: when in view of cable band emergency load, current-carrying capacity may overrate.It is applied to cable It increases and carries out the temperature rise experiment of cable overload in 20% value of rated current, for examining the effective of cable overload chronothermometer formula Property.In 4.2, starts cable and be in zero original state, it is 25.3 DEG C that cable is equal everywhere with the soil moisture.Apply to cable The cable core of cable and surface temperature reach substantially to stable state after 850A electric current is run 10 hours, record cable core and the surface of cable The measured value of temperature is T1, T2.
It is similar to (1), by soil moisture T0, loading current I, the cable core and surface initial temperature T1_0, T2_0 of cable, It brings into formula (10) and obtains the cable in the thermometer formula such as formula (12) under testing laid condition, when application electric current is 850A.Meter It calculates temperature value T1 ', the T2 ' in cable operation 10h and is compared with measured value T1, T2, as shown in Figure 8, Figure 9.In same (1), It is every two hours period, the cable core of cable and the maximum relative error on surface and absolute error during record operation, such as table 2.
T1'=56.44-27.42e-0.00649t-0.0163e-0.239t
T2'=40.56-11.89e-0.00649t+0.323e-0.239t (12)
Worst error in 2 850A day part of table
As shown in Table 2, when cable cable under above-mentioned running environment cable core calculate absolute error at 1.5 DEG C or so, meter Relative error is calculated 3% or so.Wherein maximum absolute error is 1.944 DEG C, appears in for 6~8 periods;Maximum relative error is 4.375%, appeared in for 0~2 small period.Cable surface calculates absolute error at 3.5 DEG C or so, and relative error is 10% or so. Wherein maximum absolute error is 1.996 DEG C, appears in for 2~4 periods;Maximum relative error is 6.059%, and it is small to appear in 1~2 When the period in.Calculating error when showing cable overload meets the needs of engineering calculation, it was demonstrated that cable temperature when overload The validity of calculating formula.

Claims (1)

1. a kind of cable temperature quick calculation method based on parameter fitting, it is characterised in that the following steps are included:
Step 1: the hot road of transient state cable temperature for being used for parameter fitting being established using Re Lufa according to cable laying feature first Model, column write the cable temperature differential equation;
Step 2: being directed to the specific operating condition of cable, cable is in zero original state, rated current, measuring cable temperature are applied to cable Reach cable conductor and surface temperature in steady-state phase;
Step 3: according to actual measurement cable conductor temperature and cable surface temperature, cable differential side being fitted using Trust Region Algorithm Thermal capacitance, thermal resistance thermal parameter in journey;Finally according to fitting thermal parameter, solves cable temperature Transient Thermal Circuit and show that cable temperature simplifies Calculating formula;
Steps are as follows for specific calculating:
1), the foundation of three node Transient Thermal Circuit models:
According to the theory of two-port network in circuit, by what is be unfolded when cable direct-burried by layers of material and soil different temperatures layer Cable Transient Thermal Circuit model simplification is three node Transient Thermal Circuit models;
Heat and the loading current that cable loss generates it is square approximate directly proportional, when ignoring insulating layer loss, inside cable Loss is equivalent to single source Q;Under the premise of core temperature T1, the cable surface temperature T2 of guarantee cable are constant, to heat source Q It is calculated using reduced value, it may be assumed that
Q=I2/k (1)
In formula: Q is conversion calorific value;I is loading current;K is the conversion factor chosen;
Guarantee that node temperature T1, T2 are constant due to converting front and back, at this time thermal resistance R1, R2 in thermal circuit model, the calculating of thermal capacitance C1, C2 Value is the obtained reduced value on the basis of calorific value Q;
2), column write node differential equation group:
According to cable Transient Thermal Circuit model, the differential equation such as formula (2) is write to node T1, T2 column;For solution node T1, T2 temperature, Thermal parameter R1, R2, C1, C2 in the differential equation need to be first fitted;If directly passing through the analytic solutions fit equation parameter of differential equation group, To obtain as unknown quantity be 4, the non-linear overdetermined equations that equation group number is 2n;When temperature sampling point is more, expression formula is more multiple When miscellaneous, resulting Large Scale Nonlinear over-determined systems are not easy to solve;Therefore the numerical solution of the differential equation is sought first, then pass through number Value solution fit equation parameter;The numerical solution of differential equation group is solved using 4 grades of Runge-Kutta methods;
In formula: T1_0, T2_0 are node T1, T2 initial temperature;Q is conversion calorific value;T0 is the soil moisture;
3), the objective function of constructing variable fitting:
When R1, R2, C1, C2 are considered as known constant, differential equation group (2) are solved in time point t with 4 grades of Runge-Kutta methods Numerical solution at=[t1, t2 ..., tn] is T ' 1=[T ' a1, T ' a2 .., T ' an], T ' 2=[T ' b1, T ' b2 .., T ' bn]; Objective function F (C1, C2, R1, R2) is constructed using nonlinear least square method, wherein F is using R1, R2, C1, C2 as independent variable, with Calculating error sum of squares is functional value;F expression formula such as formula (3);Seek optimal fitting thermal parameter R1, R2, C1, C2 and makes F value It is minimum;
4), Trust Region solves fitting parameter:
Differential equation parameter fitting is converted to by construction objective function the optimization problem for solving non-linear unconfined condition; For the globally optimal solution for finding thermal parameter, initial point is calculated by the way that the setting of Multi-Start algorithm is a large amount of first, then passes through trust Domain algorithm obtains multiple locally optimal solutions;Finally the corresponding functional value of multiple locally optimal solutions is compared, wherein functional value The smallest solution is the approximation of globally optimal solution;The basic step that Trust Region Algorithm solves Unconstrained Optimization Problem is concluded such as Under:
1., setting solve initial value Xk=(C1k, C2k, R1k, R2k), iteration upper limit frequency n;Selection is initial to trust radius rk, Approximation to function degree judges parameter μ, η, and 0 < μ < η < 1, sets k=1;
2., calculate F (Xk), F gradientAnd judgeWhether allowable error ε is less than;As met, then Xk is Locally optimal solution;It is such as unsatisfactory for, executes subsequent step;
3., in Trust Region Radius rk, construct F approximate function G (x) such as formula (4);Following iteration point is Xk+1, enables d=Xk+1- Xk;To seek Xk+1, second order subproblem function ψ (d) such as formula (5) is constructed;Solving dk is minimized ψ (d) in trusted zones;
4., construction degree of approximation discriminant function ρk(x) such as formula (6), judge in XkAnd Xk+1Between whether connect enough between G (X) and F (X) Closely;Work as ρk(x)≤μ when, the degree of approximation is unsatisfactory for requiring between G (X) and F (X), show approximate extents choose it is excessive, by trusted zones half Diameter is reduced into rk+1=0.5rk, and following iteration point remains unchanged i.e. Xk+1=Xk;As μ < ρk(x) < η shows that the degree of approximation still connects By holding Trust Region Radius is constant, and setting an iteration point is Xk+1=Xk+dk;Work as ρk(x) >=η when, show that the degree of approximation is preferable, expand It is big to trust radius rk+1=2rk, following iteration point is Xk+1=Xk+d;Iterate until meet condition 2. or reach the iteration upper limit Number, X value is locally optimal solution of the F (X) near primary iteration point at this time;
5), cable temperature simplified solution formulas:
R1, R2, C1, C2 optimal fitting parameter are brought into differential side's group (2), calculate T1, T2 analytic solutions be cable cable core, The simplified solution formulas of surface temperature;Write equation (2) as matrix form such as formula (7), it may be assumed that
In formula:
Formula (7) both sides are carried out simultaneously pull-type to change to obtain formula (8);Formula (8) is the linear equation in two unknowns group under frequency domain S, is calculated The cable core of cable and the Frequency Domain Solution of surface temperature are T (S), such as formula (9);T (S) is finally subjected to counter pull type transformation, as arrives electricity Simplified solution formulas of the cable core and surface of cable under time domain;
ST(S)-T(0-)=AT (S)+B (8)
T (S)=(SE-A)-1(B+T(0-)) (9)
Wherein:
CN201610145240.XA 2016-03-15 2016-03-15 A kind of cable temperature quick calculation method based on parameter fitting Active CN105787191B (en)

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