Current and Voltage Dependent Sources in EMTP-based Programs
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1 Current and Voltage Dependent Sources n EMTP-based Programs Benedto Donzet Bonatto * Department of Electrcal and Computer Engneerng The Unersty of Brtsh Columba 2356 Man Mall, Vancouer, B.C., V6T N4, Canada benedto.bonatto@eee.org Hermann Wlhelm Dommel Department of Electrcal and Computer Engneerng The Unersty of Brtsh Columba 2356 Man Mall, Vancouer, B.C., V6T N4, Canada Abstract - Ths paper presents the fundamental concepts for the mplementaton of current and oltage dependent sources n EMTP-based programs. These current and oltage dependent sources can be used to model many electronc and electrc crcuts and deces, such as operatonal amplfers, etc., and also deal transformers. As long as the equatons of the dependent sources are lnear, they could be added drectly to the networ equatons, but the matrx wll then become unsymmetrc. Another alternate dscussed here n more detal s based on the compensaton method, whch can also handle nonlnear effects wth a Newton-aphson algorthm. Nonlnear effects arse wth the ncluson of saturaton or lmts n the dependent sources. The mplementaton of ndependent sources, whch can also be connected between two ungrounded nodes, s also presented. A practcal example s gen to llustrate the soluton methods. Keywords: EMTP, current controlled oltage source (CCVS), current controlled current source (CCCS), oltage controlled oltage source (VCVS), oltage controlled current source (VCCS), dependent source, compensaton method. I. INTODUCTION EMTP-based programs are wdely used n the electrc power ndustry and n unerstes for the analyss of power system transents. Snce the publcaton of [], many hae contrbuted to the deelopment of models, whch hae been documented n [2] and elsewhere. As far as the authors now, dependent sources of all types hae not been mplemented n any EMTP-based programs, leadng to the motaton for the deelopment of ths wor. Dependent sources expand the capabltes of EMTP-based programs for modelng many electrc and electronc crcuts and deces. Wth a oltage controlled oltage source, for example, t becomes easy to smulate operatonal amplfers. These can then be used to set up control crcuts wth analog-computer bloc dagrams, whch s an alternate to the state-space representaton of control crcuts and ther smultaneous soluton wth the power networ dscussed n [4]. If the equatons of the dependent sources are lnear, they can be added drectly to the system of the nodal equatons used n EMTP-based programs, f a lnear equaton soler for unsymmetrc matrces s used. Another approach s based on the compensaton method, whch s chosen here because nonlnear equatons can easly be handled as well. Ths paper prodes the fundamental equatons for the mplementaton of dependent sources, as well as of ungrounded ndependent sources, n EMTP-based programs. II. COMPENSATION METHOD The compensaton method has long been used n EMTPbased programs for solng the equatons of nonlnear elements wth the Newton-aphson terate method. If the nonlnear elements are not too numerous, ths approach confnes the teratons to a relately small system of equatons, compared to the nodal equatons for the entre system. When there are M nonlnear elements n a crcut, the followng system of equatons, () to (6), allows the smultaneous soluton of the nonlnear equatons wth the rest of the lnear networ [2],[3], whch s then represented by ts M-phase Theenn equalent crcut, as llustrated n Fg.: [ ] [ r ] [ ] [ ] = () OPEN THEV OPEN OPEN2 [ OPEN] = OPENM [ r ] THEV r r r r r r r r r 2 M M = M M 2 MM (2) (3) * eceng a scholarshp from CAPES Brasíla / Brazl.
2 [ OPEN ] [r THEV ] [] [] M OPEN_ OPEN_ M M M s(m) s(4) s(3) s(2) s() Fg. M-phase Theenn equalent crcut. [] M 2 = (4) c [] M 2 = (5) d source() Fg. 2 epresentaton of branch equaton as a oltage source n seres wth a resstance. and (6) are the branch equatons of the nonlnear elements: ([ ],[ ], t, etc. ) = M f, = (6) c If the branch equatons n (6) are lnear, as n the case of dependent sources, they can be represented n the form of a oltage source behnd an mpedance, as llustrated n Fg. 2, or n the form of a current source n parallel wth an mpedance, as shown n Fg. 3. In ths paper, t s assumed that the branch mpedances are not coupled, and that they are resste ( ). d source() source() = source() / Fg. 3 epresentaton of branch equaton as a current source n parallel wth a resstance.
3 III. DEPENDENT SOUCES Ths secton presents the necessary equatons for mplementng current and oltage dependent sources n EMTP-based programs. The followng assumptons are made:. A Theenn equalent crcut can be calculated where the dependent source s to be connected, and also where the controllng current or controllng oltage s to be measured. In cases where ths calculaton fals, the connecton of large resstors n parallel may mae a Theenn equalent crcut possble. 2. Proper precautons are taen to handle extremely large numbers and zero alues. The followng models are dered: Current Controlled Voltage Source (CCVS), Current Controlled Current Source (CCCS), Voltage Controlled Voltage Source (VCVS) and Voltage Controlled Current Source (VCCS). In all cases, the equatons from the Theenn equalent crcut are the same, namely, for the controllng branch as: n = (3) n =Ω + (4) =Input resstance of branch. =Output resstance of the dependent source n branch. Ω=Gan oer the controllng or measured current, appled as dependent source at branch. From (7), (8), (3) and (4), one can also obtan the followng equatons: OPEN + + r + + r + + r n M M (5) OPEN + + r + + r + M M (7) OPEN + + r +Ω + r r (6) M M and for the dependent source branch OPEN r + OPEN M M = oltage for (open crcut). r =Theenn resstance (self resstance of branch ). r =Theenn resstance (couplng or mutual resstance between branches and ). (8) For an deal current controlled oltage source, n and, from whch results: OPEN + + r + + r M M OPEN + + r +Ω + + r M M (7) (8) A. Current Controlled Voltage Source (CCVS) Assume that the controllng current s measured through a branch between nodes a and b n a crcut, such that s ts branch oltage and s ts branch current,.e., = (9) a = ab (0) and that the dependent source, CCVS, s connected between nodes c and d wth branch oltage = () and branch current c = cd. (2) Then the necessary equatons for the mplementaton of ths current controlled oltage source are (7) and (8) as well b d B. Current Controlled Current Source (CCCS) The necessary equatons for the mplementaton of a current controlled current source are (7) and (8) as well as: = (9) n = Β + (20) Β=Gan oer the controllng or measured current, appled as dependent source at branch. From the equatons aboe and from (7) and (8), one can also obtan the followng equatons: OPEN + + r + + r + + r n M M (2)
4 and OPENK + + r r r rm + +Β M (22) For an deal current controlled current source, n, resultng n: (29) OPEN + + r + + r M M (30) Equatons (29) and (30) can be used to model deal operatonal amplfers. + OPEN + r + + r M M (23) D. Voltage Controlled Current Source (VCCS) The necessary equatons for the mplementaton of a oltage controlled current source are (7) and (8) as well as: Β + (24) = (3) n C. Voltage Controlled Voltage Source (VCVS) The necessary equatons for the mplementaton of a oltage controlled oltage source are (7) and (8) as well as: = (25) n =Α + =Α n + (26) Α=Gan oer the controllng or measured oltage, appled as dependent source at branch. From the equatons aboe and from (7) and (8), one can also obtan the followng equatons: OPENJ + + n r n r + n r rm M n n n OPENK r OPEN + + r + Α Α r r + + r + r + Α Α rm + rm M Α If Α, n (27) (28), and 0 for an deal oltage controlled oltage source, we obtan: = Γ + (32) Γ=Gan oer the controllng or measured oltage, appled as dependent source at branch. From the equatons aboe and from (7) and (8), one can also obtan the followng equatons: OPEN r J + + n n r + n r rm M n n n OPEN r K Γ OPEN + + Γr + r r + + Γr + Γr + r M + ΓrM M n (33) (34) For an deal oltage controlled current source,, resultng n: and Γ OPEN +Γ r + (35) +Γ r + Γr + +Γ r M M (36)
5 IV. IDEAL TANSFOMES Een though an deal transformer model has already been mplemented n EMTP-based programs wth a specal connecton of 8 resstances and an extra node [2], t can also be mplemented as a specal dependent source. The necessary equatons for the mplementaton of an deal transformer are (7) and (8) as well as: n n = (37) = n (38) n = a = n =recprocal of the turns rato of the deal transformer. From the equatons aboe and from (7) and (8), one can easly obtan: + n (39) OPENK r OPEN + + r + n n r r + r + r + n n rm + rm M n (40) Equatons (39) and (40) can be used to model an deal transformer n EMTP-based programs. V. INDEPENDENT SOUCES It may be useful n a crcut or dece model to hae an ndependent current or ndependent oltage source connected between two ungrounded nodes. Ths can be accomplshed by the same technque used for the mplementaton of dependent sources, but usng only one equaton n ths case. A. Independent Current Source Assumng that the ndependent current source s connected between nodes c and d wth branch oltage = (4) and branch current c = cd (42) then the necessary equatons for the mplementaton of an ndependent current source are: d source OPEN r + M M (43) = source + (44) = ndependent current source at branch, whch can be a lnear or nonlnear functon of tme, etc.. From the equatons aboe, one can also obtan the followng equaton: OPENK + + r r + rm M + source For the deal current source,, resultng n: (45) (46) + source Of course, there s a much easer way to represent an ndependent current source between nodes c and d drectly n the nodal equatons of the EMTP: nect the current source nto node c and wth a negate sgn nto node d [2]. B. Independent Voltage Source The necessary equatons for the mplementaton of an ndependent oltage source are: OPEN r + M M (47) = source + (48) = ndependent oltage source at branch, whch source can be a lnear or nonlnear functon of tme, etc.. From the equatons aboe, one can also obtan the followng equaton: OPEN + K + r r + M M source In an deal oltage source,, resultng n: + OPENK + + r + M M source (49) (50)
6 Another approach for oltage sources between ungrounded nodes frequently used n EMTP-based programs s the nserton of an deal transformer between the two ungrounded nodes, wth a oltage source to ground on the other sde. VI. POSSIBLE APPLICATIONS. Current and oltage sensors; 2. Operatonal amplfers; 3. Ideal Transformers; 4. User-defned coupled branches n a crcut; 5. Modelng of electronc components, where the physcal behaor would need to be represented by nonlnear equatons; 6. Instantaneous soluton of lnear and nonlnear control systems; 7. User-defned lnear and nonlnear functons; 8. User-defned modelng of lnear and nonlnear deces, lmted only by the creatty and ngenuty of the user. Fg. 4 and Fg. 5 llustrate the soluton method wth an example of a nonnertng amplfer crcut, whch conssts of a snusodal oltage source, an deal operatonal amplfer and 2 resstors ( and ). The deal operatonal f amplfer was modeled usng (29) and (30), whereas the snusodal oltage source and the resstors are part of the networ, represented through a Theenn equalent crcut. If = 2, then = 3, as shown n Fg. 5. f g put g nput Indeed, n theory ths nonnertng amplfer crcut should result n: put nput = + f g VII. CONCLUSIONS (5) The mplementaton of current and oltage dependent sources n EMTP-based programs has been presented, as well as the mplementaton of ndependent sources whch may be connected between two ungrounded nodes. It s based on the compensaton method, whch s already beng used to sole nonlnear equatons assocated wth nonlnear elements n electrc or electronc crcuts wth Newton- aphson teraton schemes. The method loos promsng for future wor n detaled modelng of crcuts and deces. Future mproements wll consder the mplementaton of saturaton or lmts for the elements or sources presented n ths paper. VIII. ACKNOWLEDGEMENTS The authors would le to than for the fnancal support of CAPES Brasíla / Brazl, whch granted a scholarshp to Mr. Benedto Donzet Bonatto, and also to acnowledge the help of Mr. Jesús Calño-Fraga for ndcatng the need for modelng operatonal amplfers n hs techncal report [5], whch motated the authors to deelop the methods presented n ths paper. IX. EFEENCES nput Voltage ( V ) Fg. 4 Crcut wth deal operatonal amplfer nput =.0 0 o f=60 (Hz) Vnput (V) Vput Tme ( ms ) put Fg. 5 Smulaton results of crcut wth deal operatonal amplfer (nonnertng amplfer crcut). f g [] H.W. Dommel, "Dgtal Computer Soluton of Electro magnetc Transents n Sngle- and Multphase Networs", IEEE Transactons on Power Apparatus and Systems, ol. PAS-88, Aprl 969, pp [2] H.W. Dommel, EMTP Theory Boo, Second Edton, Mcrotran Power System Analyss Corporaton, Vancouer, Brtsh Columba, Canada, 992, latest update 996. [3] McroTran eference Manual, Transents Program for Power and Power Electroncs Crcuts, Mcrotran Power System Analyss Corporaton, Vancouer, B.C., Canada, August 997. [4] A.E.A. Arauo, H.W. Dommel and J.. Martí, "Smultaneous Soluton of Power and Control Systems Equatons", IEEE Transactons on Power Systems, Vol. 8, No. 4, Noember 993, pp [5] Jesús Calño-Fraga, Modelng Operatonal Amplfers Usng McroTran, Techncal eport for the Graduate Course Adanced Power System Analyss, The Unersty of Brtsh Columba, Vancouer, Brtsh Columba, Canada, Aprl 998.
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