This article deals with methods of numerical modeling of heterogeneous electric circuits with distributed parameters for their possible use in the method of partitioning integration. We propose the use of modified Lax (LW) method for calculating the state of components with distributed parameters at the transient simulation in high-Q electric circuits, for example a transmission line. A comparison of transient simulation results in such circuits with the simulation results by known software systems is presented

Korud, Vasyl

Rendzinyak, Serhiy

Advances in Electrical and Electronic Engineering

Serhiy Rendzinyak

Vasyl Korud

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POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBERThe choice of Numerical Integration Methods forthe Subsystems with the Distributed ParametersSerhiy RENDZINYAK, Vasyl KORUDDepartment of Theoretical Electrical Engineering, Institute of Electrical Power Engineering and ControlSystems, Lviv Polytechnic National University, 12 S. Bandera, 79013 Lviv, Ukraineemd@polynet.lviv.ua, wk@polynet.lviv.uaDOI: 10.15598/aeee.v13i5.1415Abstract. This article deals with methods of numeri-cal modeling of heterogeneous electric circuits with dis-tributed parameters for their possible use in the methodof partitioning integration. We propose the use of mod-ified Lax (LW) method for calculating the state of com-ponents with distributed parameters at the transientsimulation in high-Q electric circuits, for example atransmission line. A comparison of transient simula-tion results in such circuits with the simulation resultsby known software systems is presented.KeywordsHeterogeneous electric circuit, LW method,transient simulation, transmission line.1. IntroductionThe current level of development of electrical engineer-ing is characterized by a high degree of integration ofheterogeneous components and devices for functionsand principles of work. Electrical engineering technol-ogy is characterized by the complexity of the connec-tion between these components, and the wide use ofnon-traditional and non-electrical elements that carryone’s features. This leads to the formation of differ-ent types of individual components within a circuit.These complex electrical systems must be seen as aset of components with lumped parameters and com-ponents with distributed parameters. For their jointcalculation some special algorithms should be used, in-cluding methods of separation processes [1], [2]. Thesemethods are based on the decomposition of the primaryscheme to various subcircuits, or the system of differ-ential equations of different types into separate equa-tions of a certain type. However, the use of particularnumerical processes for analysing dynamic processes iscomplex, and modern software does not guarantee itsautomation.The problem of solving hyperbolic equations is quitedifficult, therefore a lot of original papers devoted tothis problem can be found [3], [4], [5], [6], [7]. Mathe-matical representation of differential equations usingdifference schemes is based on conservation laws [4].Accordingly, the whole research area is covered withuniform or non-uniform mesh with corresponding timeand space steps, and the aim is to find a solution inthe mesh points. The choice of mesh in extreme pointsis defined by boundary conditions. Depending on theform of replacing derivative we can use different meth-ods of solving equations that have certain stability andaccuracy properties.Researches [3], [4] have shown that the most famousones are explicit and implicit difference Crank-Nicolsonschemes providing a solution with the required accu-racy of the coordinate steps ∆x and on time ∆t. More-over, it is proved that the stability of solutions to linearhyperbolic equations depends on the smoothness of thecoefficients and their own local conditions of stability.A special place is occupied by numerical methods,based on the finite-difference methods (FDM). Build-ing a system of difference equations, with adequate con-tinuously variable arguments, essentially depends on apriori information about the original problem that fur-ther defines methods for solving these equations.The quality difference schemes are determined bytheir stability and convergence. The family of explicitfinite-difference methods most satisfies these specifiedconditions.In general, the calculation of the transient dynamicmodes of systems described by hyperbolic equationscan be performed using the methods according to thefollowing classification:c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 466POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBER• direct numerical integration in time domain. It isadvisable to use the explicit and implicit Crank-Nicolson methods, Galerkin method and waveletsolutions,• solving equations in the frequency or the opera-tor domain with subsequent conversion results inthe time domain. Iterative algorithms of trans-formation are efficient for this class of methods.However, in this research these methods are usedfor illustrative purposes only because they do notprovide an effective solution of the problem – thecalculation of transient process taking into accountthe inner physical nature of the element.Based on the optimal construction of differentschemes, the so-called two-step Lax-Wendroff (LW)method, free of the drawbacks mentioned above, wasdeveloped [3]. In this method artificial units with halfstep are introduced. Such type modification greatlyimproves the stability and accuracy of the method.However, it should be noted that a detailed analysis ofthe methods of this class has shown that they are quiteeffective only for large-scale problems. In other cases,additional calculations by these methods are compen-sated gain, which is obtained with their application.2. Use of the LW Method forSimulation of TransmissionLineTo research the dynamic modes of nonlinear electriccircuits with components characterized by a wave ef-fect, let us consider the distribution parameters in thedirection of wave propagation in space. These compo-nents of heterogeneous electric circuits are described bypartial equations of hyperbolic type. In particular, itis a transmission line with mathematical model whichcan be represented as:−∂u (x, t)∂x= R0i (x, t) + L0∂i (x, t)∂t,−∂i (x, t)∂x= G0u (x, t) + C0∂u (x, t)∂t.(1)Boundary conditions are given values of voltages andcurrents that are consistent with the voltages and cur-rents of other components of the circuit at the inputand output of a transmission line, and can change overtime, which is taken into account when applying di-akoptical methods of calculation of transients in het-erogeneous electric circuits [8], [9], [10]. The most ef-fective method for solving hyperbolic equations is Laxmethod, in which the value of a variable in a node ofthe spatial grid is defined as unj =12(unj+1 + unj−1),the time derivative is presented asdudt=unj+1 − unj∆t,which results to an unstable computing process of thetransmission lines with high-frequency signals. There-fore, it is proposed to use a more accurate approxima-tion of the original method known as the Lax-Wendroffmethod [2], which improves the method in terms of cal-culation stability:dudt=unj − 0.5(un−1j−1 + un+1j−1)∆t;didt=inj − 0.5(in−1j−1 + in+1j−1)∆t.(2)Imposing a grid shown in Fig. 1 to the system ofEq. (1) gives the following scheme:−ux+1t−1 − ux−1t−12∆x=R0(ix+1t−1 + ix+1t−1)2++ L0(ixt − 0.5(ix−1t−1 + ix+1t−1)∆t),−ix+1t−1 − ix−1t−12∆x=G0(ux+1t−1 + ux+1t−1)2++ C0(uxt − 0.5(ux−1t−1 + ux+1t−1)∆t).(3)Fig. 1: A grid for the LW method.Therefore, the solution of required variables rela-tively to the grid can be written down using the nu-merical formulas from Eq. (4).The efficiency of the proposed approach Eq. (4)for the integration of hyperbolic equations was testedby the example of switching an unloaded transmis-sion line at a constant voltage of 220 V with pa-rameters R0 = 100 Ω·km−1, L0 = 490 mH·km−1,C0 = 1.66 µF·km−1, G0 = 0.5 µS·km−1. The simu-lation results are shown in Fig. 2, where [U ]=V, [t]=s.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 467POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBERuxt =(−∆(ix+1t−1 − ix−1t−1)−∆t∆xG0(ux+1t−1 + ux−1t−1)+ C0∆x(ux−1t−1 + ux+1t−1))/2C0∆x,ixt =(−∆(ux+1t−1 − ux−1t−1)−∆t∆xR0(ix+1t−1 + ix−1t−1)+ L0∆x(ix−1t−1 + ix+1t−1))/2C0∆x.(4)Fig. 2: The curve output voltage after switching transmissionline for the LW method.3. Comparison Simulations ofLow-Loss Transmission Lineby Known Software SystemsTo verify the adequacy of the results a comparativesimulation of a transmission line with small losses wascarried out using MATLAB/Simulink, PSpice, and theauthor’s program implemented by the method of trav-elling waves, taking into account nonzero initial condi-tions. The following models were compared:• transission line with distributed parameters de-veloped in MATLAB/Simulink in the form ofEq. (9) (Fig. 3),• model of Transission line with distributed param-eters from SimPowerSystem Blockset,• model proposed in [8].In [8] partial differential equations of a single-phaseoverhead line are given in Laplace operator form, tak-ing into account the initial conditions as follows:−du (p, x)dx= (R0 + pL0) i (p, x) + L0i (0, x) ,−di (p, x)dx= (G0 + pC0)u (p, x) + C0u (0, x) ,(5)where the u(p, x), i(p, x) are voltage and current oper-ators; u(0, x), i(0, x) are initial values of voltage andcurrent at any point of line in time t = 0.It is known that after differentiation Eq. (1) and re-placing the derivativesdu (p, x)dxanddi (p, x)dxby theexpressions obtained from Eq. (5):d2u (p, x)dx2− γ2u (p, x) == −C0 (R0 + pL0)u (0, x) + L0di (0, x)dx,d2i (p, x)dx2− γ2i (p, x) == −L0 (G0 + pC0) i (0, x) + C0du (0, x)dx,(6)where γ =√(R0 + pL0) (G0 + pC0) is the propagationconstant; Zc =√(R0 + pL0) / (G0 + pC0) is the waveresistance of a transmission line.Then the solution of Eq. (6) is presented in the formof: {u (p, x) = A1 (x) eγx +A2 (x) e−γx,u (p, x) = B1 (x) eγx +B2 (x) e−γx,(7)with the additional conditions:dA1 (x)dxeγx +dA2 (x)dxe−γx = 0,dB1 (x)dxeγx +dB2 (x)dxe−γx = 0.(8)If the line is without distortions, then γ = λ (p+ α),where α = R0/L0 = G0/C0, λ =√L0C0. Based onapplying of the inverse Laplace transform the originalsof voltages and currents at the beginning (index p) andreceiving end (index k) of the line can be obtained byEq. (9), where τ = λl is propagation time of an elec-tromagnetic wave along the transmission line; 1 (t− τ)is Heaviside step function.In the MATLAB/Simulink the simulation results oftransient on the unloaded transmission line, switchedto the voltage DC of 400 kV with the internal resis-tance, are shown in Fig. 4, the system SimPowerSys-tem – in Fig. 5, the system PSpice – in Fig. 6, theproposed scheme – in Fig. 7.Also to determine the adequacy of the obtained re-sults a comparative simulation of low-loss transmissionline was carried out. The Fig. 6 shows the results ofmathematical simulation of switching transmission lineat a constant voltage obtained using PSpice. The use ofthese softwares for the transient simulation of nonlin-ear electric circuits leads to the obvious deviations thatare directly connected with built-in models of compo-nents. Access to the correction of specific models isvery limited, if possible at all.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 468POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBER up (t)− Zcip (t) = (uk (t− τ)− Zcik (t− τ)) e−ατ + U (0) (1− 1 (t− τ)) e−ατ ,uk (t) + Zcik (t) = (up (t− τ) + Zcip (t− τ)) e−ατ + U (0) (1− 1 (t− τ)) e−ατ .(9)Fig. 3: Model of Transmission Line as Distributed Parameters Line in MATLAB/Simulink.Fig. 4: Result of Simulation of Output Voltage of TransmissionLine Obtained based on model Eq. (9) implemented intoSimulink.Fig. 5: Result of Simulation of Output Voltage of TransmissionLine Obtained using the model from SimPowerSystemBlockSet.A comparison of the transient simulation of the elec-tric circuit with a transmission line by Lax-Wendroffmethod shows that the proposed method for transientFig. 6: Result of Simulation of Output Voltage of TransmissionLine Obtained from PSpice.Fig. 7: The Curve of Output Voltage of Transmission Lines Ob-tained from the Simulation by Lax-Wendroff Method.simulation of heterogeneous high-Q circuits is more sta-ble.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 469POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBER4. ConclusionThe use of known software systems to simulate thedynamic modes of heterogeneous electric circuits withhigh adequacy is too complicated because simplifiedmodels of necessary built-in components do not alwaystake into account their topology, important parametersand nonlinearity of the elements.The choice of the numerical integration method ofhyperbolic equations is defined by the problem con-dition, i.e. initial and boundary conditions, and theaccuracy of the solution. To study the problem of dy-namic modes of nonhomogeneous electric circuits con-taining the components with lumped and distributedparameters, it is more expedient to use the Lax method[1], [2]. Applying this method makes it possible toconstruct difference equations of components with dis-tributed parameters. This method has the followingmain characteristics: sufficient accuracy and stabilityduring jumps of the boundary conditions. This is im-portant for the use of diakoptical calculation methods,as well as parallelization of computation of dynamicmodes of heterogeneous circuits.References[1] LAX, P. D. and R. D. RICHTMAYER. Sur-vey of the Stability of Linear Finite DifferenceEquations. Communications on Pure and AppliedMathematics. 1956, vol. 9, iss. 2, pp. 267–293.ISSN 1097-0312. DOI: 10.1002/cpa.3160090206.[2] MARCHUK, G. I. Splitting Methods. 1st ed.Moscow: Nauka, 1988.[3] LAX, P. D. and B. WENDROFF. On the Stabilityof Difference Schemes with Variable Coefficients.Communications on Pure and Applied Mathemat-ics. 1962, vol. 15, iss. 4, pp. 363–371. ISSN 1097-0312. DOI: 10.1002/cpa.3160150401.[4] MARCHUK, G.I. Methods of ComputationalMathematics. 1st ed. Moscow: Nauka, 1989.[5] LELARASMEE, E., A. E. RUEHLI and A.L. SANGIOVANNI-VINCENTELLI. The Wave-form Relaxation Method for Time DomainAnalysis of Large Scale Integrated Circuits.IEEE Transactions on Computer-Aided De-sign of Integrated Circuits and Systems. 1982,vol. 1, iss. 3, pp. 131–145. ISSN 0278-0070.DOI: 10.1109/TCAD.1982.1270004.[6] LINIGER, W. Multistep and One-Leg Methodto Implicit Mixed Differential Algebraic Systems.IEEE Transactions on Circuits and Systems. 1979,vol. 26, iss. 9, pp. 755–762. ISSN 0098-4094.DOI: 10.1109/TCS.1979.1084697.[7] MITCHELL, A. R. Computational Methodsin Partial Differential Equations. Internationaljournal for numerical methods in engineering.1970, vol. 2, iss. 2, pp. 299. ISSN 0029-5981.DOI: 10.1002/nme.1620020216.[8] STAKHIV, P. and O. HOHOLYUK. Mathemati-cal Modeling of Transmission Line Transients inMATLAB/Simulink Environment. Acta TechnicaCSAV. 2004, vol. 49, no. 1, pp. 89–105. ISSN 0001-7043.[9] RENDZINYAK, S., A. KORUD and B. MEL-NYK. Features of Subcircuits Matching withComponents Described by Discrete ParametersAnalysis of Electric Circuits Transient Processes.Przeglad Elektrotechniczny. 2007, vol. 2007, no. 2,pp. 154–156. ISSN 1731-6103.[10] STAKHIV, P., S. RENDZINYAK, O. HO-HOLYUK and A. KORUD. Analysis of Elec-trical Circuits Transient Processes with Compo-nents Described by Discrete Parameters. In: Proc.of Seventh International Conference in AdvancedMethods in the Theory of Electrical EngineeringApplied to Power Systems (AMTEE’05). Pilsen:IEEE, 2005, pp. 37–41. ISBN 80-7043-392-2.About AuthorsSerhiy RENDZINYAK was born in 1957 inUkraine. He received his D.Sc. from Lviv PolytechnicNational University, Ukraine in 2007 in electricalengineering. He is currently Professor of Departmentof Theoretical Electrical Engineering with Instituteof Electrical Power Engineering and Control Systems,Lviv Polytechnic National University. His researchinterests include problems of parallelization of compu-tational processes in the analysis of dynamic modesof electric circuits, research of transients in electriccircuits by diakoptical methods.Vasyl KORUD was born in 1949 in Ukraine.He received his Candidate of Science (Ph.D.) fromLviv Polytechnic Institute, USSR in 1984 in converterequipment. He is currently Associate Professor ofDepartment of Theoretical Electrical Engineering withInstitute of Electrical Power Engineering and ControlSystems, Lviv Polytechnic National University. His re-search interests include electronic functional elementsand applications.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 470

The choice of Numerical Integration Methods for the Subsystems with the Distributed Parameters

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POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBERThe choice of Numerical Integration Methods forthe Subsystems with the Distributed ParametersSerhiy RENDZINYAK, Vasyl KORUDDepartment of Theoretical Electrical Engineering, Institute of Electrical Power Engineering and ControlSystems, Lviv Polytechnic National University, 12 S. Bandera, 79013 Lviv, Ukraineemd@polynet.lviv.ua, wk@polynet.lviv.uaDOI: 10.15598/aeee.v13i5.1415Abstract. This article deals with methods of numeri-cal modeling of heterogeneous electric circuits with dis-tributed parameters for their possible use in the methodof partitioning integration. We propose the use of mod-ified Lax (LW) method for calculating the state of com-ponents with distributed parameters at the transientsimulation in high-Q electric circuits, for example atransmission line. A comparison of transient simula-tion results in such circuits with the simulation resultsby known software systems is presented.KeywordsHeterogeneous electric circuit, LW method,transient simulation, transmission line.1. IntroductionThe current level of development of electrical engineer-ing is characterized by a high degree of integration ofheterogeneous components and devices for functionsand principles of work. Electrical engineering technol-ogy is characterized by the complexity of the connec-tion between these components, and the wide use ofnon-traditional and non-electrical elements that carryone’s features. This leads to the formation of differ-ent types of individual components within a circuit.These complex electrical systems must be seen as aset of components with lumped parameters and com-ponents with distributed parameters. For their jointcalculation some special algorithms should be used, in-cluding methods of separation processes [1], [2]. Thesemethods are based on the decomposition of the primaryscheme to various subcircuits, or the system of differ-ential equations of different types into separate equa-tions of a certain type. However, the use of particularnumerical processes for analysing dynamic processes iscomplex, and modern software does not guarantee itsautomation.The problem of solving hyperbolic equations is quitedifficult, therefore a lot of original papers devoted tothis problem can be found [3], [4], [5], [6], [7]. Mathe-matical representation of differential equations usingdifference schemes is based on conservation laws [4].Accordingly, the whole research area is covered withuniform or non-uniform mesh with corresponding timeand space steps, and the aim is to find a solution inthe mesh points. The choice of mesh in extreme pointsis defined by boundary conditions. Depending on theform of replacing derivative we can use different meth-ods of solving equations that have certain stability andaccuracy properties.Researches [3], [4] have shown that the most famousones are explicit and implicit difference Crank-Nicolsonschemes providing a solution with the required accu-racy of the coordinate steps ∆x and on time ∆t. More-over, it is proved that the stability of solutions to linearhyperbolic equations depends on the smoothness of thecoefficients and their own local conditions of stability.A special place is occupied by numerical methods,based on the finite-difference methods (FDM). Build-ing a system of difference equations, with adequate con-tinuously variable arguments, essentially depends on apriori information about the original problem that fur-ther defines methods for solving these equations.The quality difference schemes are determined bytheir stability and convergence. The family of explicitfinite-difference methods most satisfies these specifiedconditions.In general, the calculation of the transient dynamicmodes of systems described by hyperbolic equationscan be performed using the methods according to thefollowing classification:c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 466POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBER• direct numerical integration in time domain. It isadvisable to use the explicit and implicit Crank-Nicolson methods, Galerkin method and waveletsolutions,• solving equations in the frequency or the opera-tor domain with subsequent conversion results inthe time domain. Iterative algorithms of trans-formation are efficient for this class of methods.However, in this research these methods are usedfor illustrative purposes only because they do notprovide an effective solution of the problem – thecalculation of transient process taking into accountthe inner physical nature of the element.Based on the optimal construction of differentschemes, the so-called two-step Lax-Wendroff (LW)method, free of the drawbacks mentioned above, wasdeveloped [3]. In this method artificial units with halfstep are introduced. Such type modification greatlyimproves the stability and accuracy of the method.However, it should be noted that a detailed analysis ofthe methods of this class has shown that they are quiteeffective only for large-scale problems. In other cases,additional calculations by these methods are compen-sated gain, which is obtained with their application.2. Use of the LW Method forSimulation of TransmissionLineTo research the dynamic modes of nonlinear electriccircuits with components characterized by a wave ef-fect, let us consider the distribution parameters in thedirection of wave propagation in space. These compo-nents of heterogeneous electric circuits are described bypartial equations of hyperbolic type. In particular, itis a transmission line with mathematical model whichcan be represented as:−∂u (x, t)∂x= R0i (x, t) + L0∂i (x, t)∂t,−∂i (x, t)∂x= G0u (x, t) + C0∂u (x, t)∂t.(1)Boundary conditions are given values of voltages andcurrents that are consistent with the voltages and cur-rents of other components of the circuit at the inputand output of a transmission line, and can change overtime, which is taken into account when applying di-akoptical methods of calculation of transients in het-erogeneous electric circuits [8], [9], [10]. The most ef-fective method for solving hyperbolic equations is Laxmethod, in which the value of a variable in a node ofthe spatial grid is defined as unj =12(unj+1 + unj−1),the time derivative is presented asdudt=unj+1 − unj∆t,which results to an unstable computing process of thetransmission lines with high-frequency signals. There-fore, it is proposed to use a more accurate approxima-tion of the original method known as the Lax-Wendroffmethod [2], which improves the method in terms of cal-culation stability:dudt=unj − 0.5(un−1j−1 + un+1j−1)∆t;didt=inj − 0.5(in−1j−1 + in+1j−1)∆t.(2)Imposing a grid shown in Fig. 1 to the system ofEq. (1) gives the following scheme:−ux+1t−1 − ux−1t−12∆x=R0(ix+1t−1 + ix+1t−1)2++ L0(ixt − 0.5(ix−1t−1 + ix+1t−1)∆t),− ix+1t−1 − ix−1t−12∆x=G0(ux+1t−1 + ux+1t−1)2++ C0(uxt − 0.5(ux−1t−1 + ux+1t−1)∆t).(3)Fig. 1: A grid for the LW method.Therefore, the solution of required variables rela-tively to the grid can be written down using the nu-merical formulas from Eq. (4).The efficiency of the proposed approach Eq. (4)for the integration of hyperbolic equations was testedby the example of switching an unloaded transmis-sion line at a constant voltage of 220 V with pa-rameters R0 = 100 Ω·km−1, L0 = 490 mH·km−1,C0 = 1.66 µF·km−1, G0 = 0.5 µS·km−1. The simu-lation results are shown in Fig. 2, where [U ]=V, [t]=s.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 467POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBERuxt =(−∆ (ix+1t−1 − ix−1t−1 )−∆t∆xG0 (ux+1t−1 + ux−1t−1 )+ C0∆x (ux−1t−1 + ux+1t−1 )) /2C0∆x,ixt =(−∆ (ux+1t−1 − ux−1t−1 )−∆t∆xR0 (ix+1t−1 + ix−1t−1 )+ L0∆x (ix−1t−1 + ix+1t−1 )) /2C0∆x. (4)Fig. 2: The curve output voltage after switching transmissionline for the LW method.3. Comparison Simulations ofLow-Loss Transmission Lineby Known Software SystemsTo verify the adequacy of the results a comparativesimulation of a transmission line with small losses wascarried out using MATLAB/Simulink, PSpice, and theauthor’s program implemented by the method of trav-elling waves, taking into account nonzero initial condi-tions. The following models were compared:• transission line with distributed parameters de-veloped in MATLAB/Simulink in the form ofEq. (9) (Fig. 3),• model of Transission line with distributed param-eters from SimPowerSystem Blockset,• model proposed in [8].In [8] partial differential equations of a single-phaseoverhead line are given in Laplace operator form, tak-ing into account the initial conditions as follows:−du (p, x)dx= (R0 + pL0) i (p, x) + L0i (0, x) ,−di (p, x)dx= (G0 + pC0)u (p, x) + C0u (0, x) ,(5)where the u(p, x), i(p, x) are voltage and current oper-ators; u(0, x), i(0, x) are initial values of voltage andcurrent at any point of line in time t = 0.It is known that after differentiation Eq. (1) and re-placing the derivativesdu (p, x)dxanddi (p, x)dxby theexpressions obtained from Eq. (5):d2u (p, x)dx2− γ2u (p, x) == −C0 (R0 + pL0)u (0, x) + L0 di (0, x)dx ,d2i (p, x)dx2− γ2i (p, x) == −L0 (G0 + pC0) i (0, x) + C0 du (0, x)dx ,(6)where γ =√(R0 + pL0) (G0 + pC0) is the propagationconstant; Zc =√(R0 + pL0) / (G0 + pC0) is the waveresistance of a transmission line.Then the solution of Eq. (6) is presented in the formof: {u (p, x) = A1 (x) eγx +A2 (x) e−γx,u (p, x) = B1 (x) eγx +B2 (x) e−γx, (7)with the additional conditions:dA1 (x)dxeγx +dA2 (x)dxe−γx = 0,dB1 (x)dxeγx +dB2 (x)dxe−γx = 0.(8)If the line is without distortions, then γ = λ (p+ α),where α = R0/L0 = G0/C0, λ =√L0C0. Based onapplying of the inverse Laplace transform the originalsof voltages and currents at the beginning (index p) andreceiving end (index k) of the line can be obtained byEq. (9), where τ = λl is propagation time of an elec-tromagnetic wave along the transmission line; 1 (t− τ)is Heaviside step function.In the MATLAB/Simulink the simulation results oftransient on the unloaded transmission line, switchedto the voltage DC of 400 kV with the internal resis-tance, are shown in Fig. 4, the system SimPowerSys-tem – in Fig. 5, the system PSpice – in Fig. 6, theproposed scheme – in Fig. 7.Also to determine the adequacy of the obtained re-sults a comparative simulation of low-loss transmissionline was carried out. The Fig. 6 shows the results ofmathematical simulation of switching transmission lineat a constant voltage obtained using PSpice. The use ofthese softwares for the transient simulation of nonlin-ear electric circuits leads to the obvious deviations thatare directly connected with built-in models of compo-nents. Access to the correction of specific models isvery limited, if possible at all.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 468POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBER up (t)− Zcip (t) = (uk (t− τ)− Zcik (t− τ)) e−ατ + U (0) (1− 1 (t− τ)) e−ατ ,uk (t) + Zcik (t) = (up (t− τ) + Zcip (t− τ)) e−ατ + U (0) (1− 1 (t− τ)) e−ατ .(9)Fig. 3: Model of Transmission Line as Distributed Parameters Line in MATLAB/Simulink.Fig. 4: Result of Simulation of Output Voltage of TransmissionLine Obtained based on model Eq. (9) implemented intoSimulink.Fig. 5: Result of Simulation of Output Voltage of TransmissionLine Obtained using the model from SimPowerSystemBlockSet.A comparison of the transient simulation of the elec-tric circuit with a transmission line by Lax-Wendroffmethod shows that the proposed method for transientFig. 6: Result of Simulation of Output Voltage of TransmissionLine Obtained from PSpice.Fig. 7: The Curve of Output Voltage of Transmission Lines Ob-tained from the Simulation by Lax-Wendroff Method.simulation of heterogeneous high-Q circuits is more sta-ble.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 469POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 | NUMBER: 5 | 2015 | DECEMBER4. ConclusionThe use of known software systems to simulate thedynamic modes of heterogeneous electric circuits withhigh adequacy is too complicated because simplifiedmodels of necessary built-in components do not alwaystake into account their topology, important parametersand nonlinearity of the elements.The choice of the numerical integration method ofhyperbolic equations is defined by the problem con-dition, i.e. initial and boundary conditions, and theaccuracy of the solution. To study the problem of dy-namic modes of nonhomogeneous electric circuits con-taining the components with lumped and distributedparameters, it is more expedient to use the Lax method[1], [2]. Applying this method makes it possible toconstruct difference equations of components with dis-tributed parameters. This method has the followingmain characteristics: sufficient accuracy and stabilityduring jumps of the boundary conditions. This is im-portant for the use of diakoptical calculation methods,as well as parallelization of computation of dynamicmodes of heterogeneous circuits.References[1] LAX, P. D. and R. D. RICHTMAYER. Sur-vey of the Stability of Linear Finite DifferenceEquations. Communications on Pure and AppliedMathematics. 1956, vol. 9, iss. 2, pp. 267–293.ISSN 1097-0312. DOI: 10.1002/cpa.3160090206.[2] MARCHUK, G. I. Splitting Methods. 1st ed.Moscow: Nauka, 1988.[3] LAX, P. D. and B. WENDROFF. On the Stabilityof Difference Schemes with Variable Coefficients.Communications on Pure and Applied Mathemat-ics. 1962, vol. 15, iss. 4, pp. 363–371. ISSN 1097-0312. DOI: 10.1002/cpa.3160150401.[4] MARCHUK, G.I. Methods of ComputationalMathematics. 1st ed. Moscow: Nauka, 1989.[5] LELARASMEE, E., A. E. RUEHLI and A.L. SANGIOVANNI-VINCENTELLI. The Wave-form Relaxation Method for Time DomainAnalysis of Large Scale Integrated Circuits.IEEE Transactions on Computer-Aided De-sign of Integrated Circuits and Systems. 1982,vol. 1, iss. 3, pp. 131–145. ISSN 0278-0070.DOI: 10.1109/TCAD.1982.1270004.[6] LINIGER, W. Multistep and One-Leg Methodto Implicit Mixed Differential Algebraic Systems.IEEE Transactions on Circuits and Systems. 1979,vol. 26, iss. 9, pp. 755–762. ISSN 0098-4094.DOI: 10.1109/TCS.1979.1084697.[7] MITCHELL, A. R. Computational Methodsin Partial Differential Equations. Internationaljournal for numerical methods in engineering.1970, vol. 2, iss. 2, pp. 299. ISSN 0029-5981.DOI: 10.1002/nme.1620020216.[8] STAKHIV, P. and O. HOHOLYUK. Mathemati-cal Modeling of Transmission Line Transients inMATLAB/Simulink Environment. Acta TechnicaCSAV. 2004, vol. 49, no. 1, pp. 89–105. ISSN 0001-7043.[9] RENDZINYAK, S., A. KORUD and B. MEL-NYK. Features of Subcircuits Matching withComponents Described by Discrete ParametersAnalysis of Electric Circuits Transient Processes.Przeglad Elektrotechniczny. 2007, vol. 2007, no. 2,pp. 154–156. ISSN 1731-6103.[10] STAKHIV, P., S. RENDZINYAK, O. HO-HOLYUK and A. KORUD. Analysis of Elec-trical Circuits Transient Processes with Compo-nents Described by Discrete Parameters. In: Proc.of Seventh International Conference in AdvancedMethods in the Theory of Electrical EngineeringApplied to Power Systems (AMTEE’05). Pilsen:IEEE, 2005, pp. 37–41. ISBN 80-7043-392-2.About AuthorsSerhiy RENDZINYAK was born in 1957 inUkraine. He received his D.Sc. from Lviv PolytechnicNational University, Ukraine in 2007 in electricalengineering. He is currently Professor of Departmentof Theoretical Electrical Engineering with Instituteof Electrical Power Engineering and Control Systems,Lviv Polytechnic National University. His researchinterests include problems of parallelization of compu-tational processes in the analysis of dynamic modesof electric circuits, research of transients in electriccircuits by diakoptical methods.Vasyl KORUD was born in 1949 in Ukraine.He received his Candidate of Science (Ph.D.) fromLviv Polytechnic Institute, USSR in 1984 in converterequipment. He is currently Associate Professor ofDepartment of Theoretical Electrical Engineering withInstitute of Electrical Power Engineering and ControlSystems, Lviv Polytechnic National University. His re-search interests include electronic functional elementsand applications.c© 2015 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 470

The choice of numerical integration methods for the subsystems with the distributed parameters

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