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  • 1. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363Symbolic Simulation By Mesh Method Of Complementary CircuitryI. I. OKONKWO*, P. I. OBI *, G. C CHIDOLUE**, AND S. S. N. OKEKE** P. G. Scholars In Dept. Of Electrical Engineering, Anambra State University, Uli. Nigeria* Professors In Dept. Of Electrical Engineering, Anambra State University, Uli. Nigeria**Abstract Transient simulation of electric network circuits elements [2 – 4]. Symbolic formulation growswith non – zero initial values could be quite exponentially with circuit size and it limits thechallenging even in frequency domain, especially maximum analyzable circuit size and also makeswhen transient equation formulation involves more difficult, formula interpretation and its use invectorial sense establishment of the initialization design automation application [5 – 10]. This iseffect of the storage elements. In this paper, we usually improved by using semisymbolic formulationderived a robust laplace frequency transient mesh which is symbolic formulation with numericalequation which takes care of the vectorial sense of equivalent of symbolic coefficient. Other methods ofthese initialization effects by mere algebraic simplification include simplification beforeformulation. The result of the new derived generation (SBG), simplification during generationtransient mesh equation showed promising SDG, and simplification after generation (SAG) [11 –conformity with the existing simpowersystem 16].simulation tool with just knowledge of the steady Symbolic response formulation of electricalstate current and not the state variables. circuit can classified broadly as modified nodal analysis (MNA) [17], sparse tableau formulation andKeyword: transient simulation, mesh equation, state variable formulations. The state variable methodstate variables, and non – zero initialization. were developed before the modified nodal analysis, it involves intensive mathematical process and has1 INTRODUCTION major limitation in the formulation of circuit A single run of a conventional simulation equations. Some of the limitations arise because theprovide limited information about the behaviour of state variables are capacitor voltages and inductorelectrical circuit. It determines only how the circuit currents [18]. The tableau formulation has a problemwould behave for a single initial, input sequence and that the resulting matrices are always quite large andset of circuit parameter characterizing condition. the sparse matrix solver is needed. Unfortunately, theMany cad tasks require more extensive information structure of the matrix is such that coding thesethan can be obtained by a single simulation run. For routine are complicated. MNA despite the fact that itsexample the formal verification of a design requires formulated network equation is smaller than tableaushowing that the circuit will behave properly for all method, it still has a problem of formulating matricespossible initial start sequences that will detect a given that are larger than that which would have beenset of faults, clearly conventional simulation is of obtained by pure nodal formulation [19].little use for such task [1]. In this paper a new mesh analytical method Some of these tasks that cannot be solved is introduced which may be used on linear oreffectively by conventional simulation have become linearized RLC circuit and can be computertractable by extending the simulation to operate a applicable and user friendly. The simplicity of thesymbolic domain. Symbolic simulation involves new transient mesh formulation lies in the fact thatintroducing an expanded set of signal values and minimal mesh index is enough to formulate transientredefining the basic simulation functions to operate equation and also standard method of building steadyover this expanded set. This enables the simulator state mesh impedance bus is just needed to build theevaluate a range of operating conditions in a single two formulated impedance buses that are required torun. By linearizing the circuits with lumped formulate the new transient mesh equation.parameters at particular operating points and Simplicity, compactness and economy are theattempting only frequency domain analysis, the advantages of the newly formulated transient meshprogram can represent signal values as rational equation.functions in the s ( continuous time ) or z (discretetime) domain and are generated as sums of the 2 New Transient Mesh Equation.products of symbols which specify the parameters of Mesh analysis may not be as powerful as the 356 | P a g e
  • 2. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363nodal analysis in power system because of a little bitof application complication in circuits with multiple E1 (s)  E 3 (s)  E5 (s)  [I1 (s)  I 2 (s)][R1branches in between two nodes, when power system 1 1is characterized with short line model, mesh analysis  sL1  ]  I1 (s)[R1  sL1  ]  [I1 (s) C1 C1become a faster option especially when using laplace 1transient analysis. The new mesh method sees a linear  I 3 (s)][R5  sL5  ]  [I1 (0)  I 2 (0)]L1RLC transient network in frequency domain as one C5that sets up complementary circuit by the initial dc Vc1(0)  [I1 (0)  I 3 (0)]L5  I1 (0)L3 quantities at transient inception. With this mesh smethod, the complimentary circuit sets its resultant Vc3 (0) Vc5 (0)residual quantities (voltage drop) which complement   0 3 s sthe mesh transient voltage source. but The constitutional effect of the initialquantities at the transient inception combine with the 1 Vck ( 0 )  i k ( 0 ) 4 voltage sources on the RLC linear circuits (1) is s1Cksetting up of two identifiable impedance diagrams. Where ik (0) is the initial branch current and s1 is theOne impedance diagram is the normal laplace steady state frequency.transformed impedance diagram of the original circuitelements, in this paper it is called the auxiliarytransient Impedance diagram. The other impedancediagram is due to non zero transient initializationeffect of the storage elements and it is called thecomplementary transient impedance diagram in thispaper.Z(s)I(s)  E(s)  ZC(s)IC(s) 1Where Z(s) is the Auxiliary impedance bus, s –domain equivalent of steady state mesh impedancematrix, Zc(s) is the s – domain complementaryimpedance bus, it is the storage element driving pointimpedance bus due to transient inception effect, I(s) isthe laplace mesh current vector and Ic(s) is the initial Figure 1: Three node, three mesh linear transientdc mesh current vector, equivalent to the steady state electric circuit.mesh current vector at the transient inception. substituting branch current in (4) with appropriate2.1 Derivation mesh currents to get branch capacitor voltage drops in The newly transient mesh equation may be terms of mesh currents,derived by considering a simple three node, three 1 Vc1(0)  [I1 (0)  I 2 (0)]mesh linearized RLC circuit, fig. 1, if Kirchhoff’s s1C1voltage law is applied on the various meshes then, Vc5 (0)  [I1 (0)  I 3 (0)] 1 5  s1C5From mesh 1 1 d Vc3 (0)  I1 (0)E1 (t)  {R1 [I1 (t)  I 2 (t)]  L1 [I1 (t)  I 2 (t)] s1C3 dt 1 C1  [I1 (t)  I 2 (t)]dt  E5 (t)  {R5 [I1 (t)  I 3 (t)] Substituting equation (5) in (3) and simplifying to get, I1 (s)[Z1 (s)  Z3 (s)  Z5 (s)]  I 2 (s)Z1 (s) d 1 L5 dt [I1 (t)  I 3 (t)]  C5  [I (t)  I 1 3 (t)]dt  E 3 (t)  I 3 (s)Z 5 (s)  [E1 (s)  E 3 (s)  E5 (s)] 1 1 1  I1 (0){[L1  ]  [L3  ]  [L5  ]} d {R 3 I1 (t)  L3 I1 (t)  1  I (t)dt  0 1 2 ss1C1 ss1C3 ss1C5 dt C3  I 2 (0)[L1  1 ]  I 3 (0)[L5  1 ] 6 ss1C1 ss1C5 then,taking the laplace transform of equation (2) to get, 357 | P a g e
  • 3. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363 I1 (s)[Z1 (s)  Z3 (s)  Z5 (s)]  I 2 (s)Z1 (s) Z( c )11 ( s )  Z( c )1 (s)  Z( c )3 (s)  Z( c )5 (s) I 3 (s)Z 5 (s)  [E1 (s)  E 3 (s)  E5 (s)] Z( c )22 ( s )  Z( c )1 (s)  Z( c )2 (s)  Z( c )4 (s)  I1 (0)[Z(C)1 (s)  Z(C)3 (s)  Z(C)5 (s)] Z( c )33 ( s )  Z( c )4 (s)  Z( c )5 (s)  Z( c )6 (s) 13   I 2 (0)Z(C)1 (s)  I 3 (0)Z(C)5 (s) 7  Z( c )12 ( s )  Z( c )21 ( s )  Z( c )1 ( s )where Z( c )13 ( s )  Z( c )31 ( s )  Z( c )5 ( s )Z( c )k  [Lk  1 ] 8 Z( c )23 ( s )  Z( c )32 ( s )  Z( c )4 ( s ) ss1CkZ(c)k(s) is the dc transient driving point impedance, Lk E(M)1 (s)  E1 (s)  E3 (s)  E5 (s)and Ck are the k – th branch inductance andcapacitance respectively. E(M)2 (s)  E1 (s)  E2 (s)  E4 (s) 14For mesh 2 E(M)3 (s)  E4 (s)  E5 (s)  E6 (s)Similarly, Kirchhoff’s voltage law may be applied inmesh 2 and simplified as in mesh 1 to get ,  I 2 (s)[Z1 (s)  Z2 (s)  Z4 (s)]  I1 (s)Z1 (s) K I 3 (s)Z 4 (s)  [ E1 (s)  E 2 (s)  E 4 (s)] E(M)m (s)   E (s)k 15  I1 (0)[Z(C)1 (s)  Z(C)2 (s)  Z(C)4 (s)] k 1 where m = 1,2 – – – M–th mesh and also k = 1,2 – –  I1 (0)Z(C)1 (s)  I 3 (0)Z(C)4 (s) 9 – K–th branch incident on the m–th mesh.For mesh 3Similarly, Kirchhoff’s voltage law may be applied inmesh 3 and simplified as in mesh 1 to get I 3 (s)[Z 4 (s)  Z5 (s)  Z6 (s)]  I 2 (s)Z 4 (s) I1 (s)Z 5 (s)  [ E 4 (s)  E5 (s)  E6 (s)]  I 3 (0)[Z(C)4 (s)  Z(C)5 (s)  Z(C)6 (s)]  I 2 (0)Z(C)4 (s)  I1 (0)Z(C)5 (s) 10 Equations (6), (8) and (8) may be combined to form s– domain mesh matrix equation as follows, Z11 ( s ) Z12 ( s ) Z13 ( s )  I1 ( s )   E( m )1 ( s )      Z21 ( s ) Z22 ( s ) Z23 ( s )  I 2 ( s )    E( m )2 ( s )  Z ( s ) Z ( s ) Z ( s )  I ( s )   E  31 32 33  3   ( m )3 ( s )  = Z( c )11 ( s ) Z( c )12 ( s ) Z( c )13 ( s )  I1 ( 0 )  Figure 2: s – domain auxiliary circuit diagram for   Z( c )21 ( s ) Z( c )22 ( s ) Z( c )23 ( s )  I 2 ( 0 )  11 transient nodal analysis.   Z( c )31 ( s ) Z( c )32 ( s ) Z( c )33 ( s )  I 3 ( 0 )  2.2 Generalized Matrix Form for Transient Nodal Equationwhere Equation (11) may be used to generalize anZ11 ( s )  Z1 (s)  Z3 (s)  Z5 (s) equation in the matrix form for transient mesh analysis of M th mesh electrical circuit, thusZ22 ( s )  Z1 (s)  Z 2 (s)  Z4 (s)  Z11 (s) Z12 (s)  Z1m (s)  I1 (s)   E1 (s) Z33 ( s )  Z4 (s)  Z5 (s)  Z6 (s) 12        Z21 (s) Z22 (s)  Z2m (s)  I 2 (s)   E 2 (s) Z12 ( s )  Z21 ( s )  Z1 ( s )           Z13 ( s )  Z31 ( s )  Z5 ( s )       Z (s) Z (s)  Z (s)  I (s)   E (s) Z23 ( s )  Z32 ( s )  Z 4 ( s )  m1 m2 mm  m   m  = 358 | P a g e
  • 4. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363 Z(C)11 (s) Z(C)12 (s)  Z(C)1m (s)  I1 (0)  Z(C)(s) is laplace frequency domain dc driving point   impedance bus, the impedance bus could be built Z(C)21 (s) Z(C)22 (s)  Z(C)2m (s)  I 2 (0)     16  from fig 3 using any standard method of building an      impedance bus when the branch dc driving point Z(C)m1 (s) Z(C)m2 (s)  Z(C)mm (s)  I (0)  impedance Z(C)k(s) of the circuit is evaluated from  m  equation (8). In this paper it is called the s – domain complementary admittance bus.The variables of equation (16) are defined in thecompact equation of section 2.  I1 (s)   E1 (s)       I 2 (s)   E 2 (s)  I(s)   , E(s)            I (s)   E (s)  m   m   I1 (0)     I 2 (0)  I c (s)  I(0)   20       I (0)  m  I(s) and E(s) are the vector of mesh transient current and mesh voltage source (vectorial sum) in laplace frequency domain respectively, while Ic(s)is equal to I(0) and is the steady state mesh current at the instant of fault inception. Hence Ic(s) is and I(0) will be used interchangeably in this paper.Figure 3: s – domain complementary circuit diagram 3 ANALYSIS PROCEDURESfor mesh analysis. 1. Solve for the initial dc mesh current from steady state for example2.3 Generalized Compact Form For Transient ZI  E 21Nodal Equation where Z is the steady state mesh impedance bus, I is The generalized compact form of the the steady mesh current vector, E is the mesh sumequation 16 is thus as follows, voltage source vector. 2. Transform all the branch voltage sources toZ(s)I(s)  E(s)  Z(C)(s)I(C)(s) 17 laplace equivalent and find the mesh sum (14) and eventually convert to vector form (20).  Z11 (s) Z12 (s)  Z1m (s)    3. Draw the auxiliary laplace impedance  Z21 (s) Z22 (s)  Z2m (s) Z(s)   18  diagram as in Fig 2. by converting all the branch     elements to laplace equivalent, then build the laplace    Z (s) Z (s)  Z (s)  impedance bus from the impedance diagram by using  m1 m2 mm  any of the standard method of building steady state Z(s) is laplace frequency domain impedance bus, the impedance bus.impedance bus have the same formulation with the 4. From the branch storage elements formulatecommon steady state mesh impedance bus only that the newly derived branch dc transient driving pointin this equation, the branch impedances are translated impedances (8), and then draw the complementaryto laplace frequency domain. In this paper it is called impedance diagram as in fig. 3. from the diagramthe s – domain auxiliary impedance bus. build the complementary impedance bus (19) with any of the standard method of building steady statealso impedance bus.  Z(C)11 (s) Z(C)12 (s)  Z(C)1m (s)  5. Form equation (16) and solve for I(s) using   Cramer’s rule.  Z(C)21 (s) Z(C)22 (s)  Z(C)2m (s) Zc (s)       19 6. Transform I(s) to time domain equivalent   using laplace inverse transform. Eg. in Matlab,  Z(C)m1 (s) Z(C)m2 (s)  Z(C)mm (s)    359 | P a g e
  • 5. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363 In this paper the transient mesh currents I( t )  ilap I( s )) ( 22 were simulated by using the described formulation (s – domain mesh equation by method of complementary circuitry). Analysis procedures ofFrom this branch currents could easily be obtained at section 3 were used to calculate the s – domainany instant of the transient. rational functions of the mesh currents I(s). The obtained s – domain rational functions were4 TEST CIRCUIT transformed to close form continuous time functions An earth faulted 100kV - double end fed using laplace inverse transformation. Discretization of70% series compensated 100km single transmission the close form continuous time functions were doneline was used for verification of the formulated s – to plot the mesh current response graphs.domain transient mesh equation. In this analysiscompensation beyond fault was adopted and fault 5.2 Simpower Simulation of Test Circuitposition was assumed to be 40%. To validate the formulated transient mesh equation, a simulation of the earth faulted line end series compensated single line transmission was performed using matlab simpowersystem software to obtain the circuit transient mesh current responses.Figure 4: earth faulted single line with compensation Results were compared with the responses obtainedbeyond fault from the simulations using the formulated transient mesh equation.Test Circuit ParameterGenerator 1 6 ResultsE1 (t)=10x104sin(t), ZG1=(6+j40), S=1MVA Mesh current response were simulated usingGenerator 2 the formulated mesh equation and also usingE2 (t)=0.8|E1|sin (t+450), ZG2=(4+j36), S=1MVA simpowersystem package, all simulation were doneLine Parameter using Matlab 7.40 mathematical tool. SimulatedRs=0.075 /km, Ls=0.04875 H/km responses by these methods for the earth faultedGs=3.75*10-8 mho/km, Cs=8.0x10-9F/km double end fed single line transmission were obtainedLine length=100 km and shown in fig 6 through fig 13. Mesh currentsFault position 40% C1=70%compensation. were taken for various simulating conditions. Simulating conditions included; zero initial condition,4.1 Modeling non – zero initial condition, high resistive (1000) A lumped parameter was adopted as a model fault but at zero initial condition, and 1 sec.for the test circuit. It was assumed that compensation simulation. All simulations were done, exceptprotection had not acted as such the compensation otherwise stated on 100km line at 40% fault positionwas of constant capacitance. More so, the model is and 5 earth resistive fault. Sampling interval for thecharacterized with constant parameter, shunt formulated equation simulation is 0.0005 sec, whilecapacitance and shunt conductance of transmission that of the simpowersystem simulation is at 0.00005line are neglected. The equivalent circuit of the test sec. The overall result showed almost 100%circuit is below fig 5. conformity between new mesh symbolic formulation and the simpowersystem simulation. 7 Conclusions Simulation software has been formulated for transient simulation of RLC circuits initiating from steady state. The simulation software is especially useful for power circuits that are modeled with shortFigure 5: single line with compensation beyond earth line parameter. The result of the simulation of thisfault equivalent circuit (short line model). new symbolic mesh software showed promising conformity with the existing simpowersystem5 Transient Simulation package and has the advantage of being able to5.1 Symbolic Simulation with Formulated simulate complex value initial conditions and alsoEquation sets the directions and the senses of the state variables automatically. 360 | P a g e
  • 6. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363Test Circuit Simulated Mesh Voltage ResponseGraphs: All graph are plotted except otherwisestated, 100 km Line, Compensation 70% FaultPosition=40%, And 5 Resistive Earth Fault. Figure 9: Simulation of Mesh Current versus Time; Initial Conditions, 0.013 Sec of Steady State Run.Figure 6: Simulation Of Mesh Current Versus time ;0% Initial Condition. Figure 10: Simulation Of Mesh Current Versus Time; 0% Initial Condition, and 1000 Resistive Earth Fault.Figure 7: Simulation Of Mesh Current Versus Time ;0% Initial Condition. Figure 11: Simulation Of Mesh Current Versus Time; 0% Initial Condition, and 1000 Resistive Earth Fault.Figure 8: Simulation of Mesh Current versus Time;Initial Conditions, 0.013 Sec of Steady State Run. 361 | P a g e
  • 7. I. I. Okonkwo, P. I. Obi, G. C Chidolue, S. S. N. Okeke / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.356-363 Rodríguez-Vázquez ―Comparison Of Matroid Intersection Algorithms For Large Circuit Analysis‖ Proc. Xi Design Of Integrated Circuits And Systems Conf., Pp. 199-204 Sitges (Barcelona), November 1996. [6] A. Rodríguez-Vázquez, F.V. Fernández, J.L. Huertas And G. Gielen, ―Symbolic Analysis Techniques And Applicaitons To Analog Design Automation‖, IEEE Press, 1996. [7] X. Wang And L. Hedrich, ―Hierarchical Symbolic Analysis Of Analog Circuits Using Two-Port Networks‖, 6th Wseas International Conference On Circuits, Systems, Electronics, Control & Signal Processing,Figure 12: Simulation Of Mesh Current Versus Time ; Cairo, Egypt, 2007.0% Initial Condition. [8] J. Hsu And C. Sechen, ―Dc Small Signal Symbolic Analysis Of Large Analog Integrated Circuits‖, IEEE Trans. Circuits Syst. I, Vol. 41, Pp. 817–828, Dec. 1994. [9] M. Galán, I. García-Vargas, F.V. Fernández And A. Rodríguez-Vázquez ―A New Matroid Intersection Algorithm For Symbolic Large Circuit Analysis‖ Proc. 4th Int. Workshop On Symbolic Methods And Applications To Circuit Design, Leuven (Belgium), October 1996 [10] Z. Kolka, D. Biolek, V. Biolková, M. Horák, Implementation Of Topological Circuit Reduction. In Proc Of Apccas 2010,Figure 13: Simulation Of Mesh Current Versus Time ; Malaysia, 2010. Pp. 951-954.0% Initial Condition. [11] Tan Xd, ―Compact Representation And Efficient Generation Of S-ExpandedReferences: Symbolic Network Functions For Computer [1] Randal E. Bryant‖ Symbolic Simulation— aided Analog Circuit Design‖. IEEE Techniques And Applications‖ Computer Transaction On Computer-Aided Design Of Science Department,. Paper 240. (1990). Integrated Circuits And Systems,. Pp. 20, [2] C.-J. Richard Shi And Xiang-Dong Tan 2001 ―Compact Representation And Efficient [12] R. Sommer, T. Halfmann, And J. Broz, Generation of S-Expanded Symbolic Automated Behavioral Modeling And Network Functions For computer-Aided Analytical Model-Order Reduction By Analog Circuit Design‖ IEEE Transactions Application Of Symbolic Circuit Analysis On Computer-Aided Design Of Integrated For Multi-Physical Systems, Simulation Circuits And Systems, Vol. 20, No. 7, July Modeling Practice And Theory, 16 Pp. 1024– 2001 1039. (2008). [3] F. V. Fernández, A.Rodríguez-Vázquez, J. L. [13] Q. Yu And C. Sechen, ―A Unified Approach Huertas, And G. Gielen, ―Symbolic Formula To The Approximate Symbolic Analysis Of Approximation,‖ In Symbolic Analysis Large Analog Integrated Circuits,‖ Ieee Techniques: Applications To Analog Design Trans. On Circuits And Systems – I: Automation, Eds. Piscataway, NJ: IEEE, Fundamental Theory And Applications, Vol. 1998, Ch. 6, Pp. 141–178. 43, No. 8, Pp. 656–669, 1996. [4] Gielen And W. Sansen, Symbolic Analysis [14] P. Wambacq, R. Fern´Andez, G. E. Gielen, For Automated Design Of Analog Integrated W. Sansen, And A. Rodriguez- V´Zquez, Circuits. Norwell, Ma: Kluwer, 1991. ―Efficient Symbolic Computation Of [5] Mariano Galán, Ignacio García-Vargas, Approximated Small-Signal Characteristics,‖ Francisco V. Fernández And Angel 362 | P a g e
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