Differences and Common Aspects of POG and EMR Energy-Based Graphical Techniques

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1 Difference and Common Apect of POG and EMR -Baed Graphical echnique Roerto Zanai Dement of Information Engineering Univerity of Modena and Reggio Emilia Modena, Italy Federica Groi Dement of Information Engineering Univerity of Modena and Reggio Emilia Modena, Italy Atract hree main modeling technique have een propoed in the pat year: Bond Graph (BG), Power-Oriented Graph (POG) and Energetic Macrocopic Repreentation (EMR). hee modeling technique are energy-aed graphical technique which exploit the power interaction etween uytem a aic concept for modeling. In thi paper a comparion etween the POG and EMR technique i preented. hee technique have many apect in common and alo difference that are pointed out in the paper. he POG cheme have a welldefined mathematical tructure and ue vectorial compact form when the ytem complexity increae. he EMR contitute a graphical tool ueful for modeling, imulation and control of dynamical ytem. I. INRODUCION Nowaday, a lot of modelling method are availale to decrie complex dynamic ytem uch a Bond Graph []- [], Caual Ordering Graph [3], Power-Oriented Graph []- [5], Energetic Macrocopic Repreentation [6]- [7], etc. All thee technique ue power interaction etween uytem a the aic concept for modeling. he Bond Graph, introduced in 959, i the mot known technique and it i ued in many area. Some comparion among the BG, POG and EMR graphical technique can e found in [8]. he focu of thi paper i on the comparion etween the POG and EMR technique to highlight difference and common apect. he POG technique i aed on the ame energetic approach of the Bond Graph technique, ut it ue a different graphical notation (i.e. imple lock diagram) for modeling the phyical ytem. he POG lock cheme are eay to ue, eay to undertand and can e directly implemented in Simulink. For thee reaon the POG technique can e a ueful tool for promoting the ue of the energetic approach alo etween eginner and young reearcher. he EMR clearly how the power coupling among element and the energy flow through the ytem without howing the mathematical detail of the model. hi technique i mainly focued on giving a methodology to uild an effective control architecture that can e implemented in real-time. he tructure i eay to e read, however it doe not how the mathematical detail of the model and in icular the diipation are not explicitly hown. II. POWER-ORIENED GRAPHS AND ENERGEIC MACROSCOPIC REPRENAION BASIC FEAUR he Power-Oriented Graph (POG) are ignal flow graph comined with a icular modular tructure which eentially ue only two lock, ee [5]. he aic characteritic of thi modular tructure i the direct correpondence etween pair of ytem variale and real power flow: the product of the two variale involved in each dahed line of the graph ha the phyical meaning of power flowing through the ection. he two aic lock are named elaoration lock (e..) and connection lock (c..) and they are hown in Fig.. While the elaoration lock can tore and diipate energy (i.e. pring, mae and damper), the connection lock can only tranform the energy, that i, tranform the ytem variale from one type of energy-field to another (i.e. any type of gear reduction). he POG cheme are in direct correpondence with the tate pace notation of the ytem and ome rule are defined to graphically check the correctne of the model. Moreover POG cheme can e directly implemented in general purpoe imulator uch a Simulink. he Energetic Macrocopic Repreentation (EMR) i a graphical tool aed on the action-reaction principle [7],[9]. It allow to define a compact repreentation of complex ytem, repecting a functional decription. Specific pictogram, hown in Fig., are aociated to each power component depending on their power function: energy accumulation (rectangle with an olique ar), converion without energy accumulation (quare for monophyical converion, circle for multiphyical converion), interleaved pictogram for energy ditriution, ee [0] and []. hi decription point out the coupling device which ditriute the energy within the ytem. he EMR technique give a methodology to uild a control architecture tarting from the model of the ytem. he control tructure i otained y a tep y tep inverion of the ytem decompoed into elementary uytem. he EMR repreentation i planar, it clearly how the coupling among element and the energy flow through the ytem. he tructure i eay to e read. However it doe not how mathematical detail of the model.

2 POG lock EMR pictogram EMR Control Structure lock y y G() y K K Elaoration lock G(): calar or vectorial type. he lack pot repreent a minu ign Connection lock K: calar or vectorial type ource Element with energy accumulation Monophyical and Multiphyical converter (without energy accumulation) Multiphyical domain coupling device (energy ditriution) Control lock with controller Control lock with meaure without controller Control lock without meaure and controller Fig.. Synoptic of the POG and EMR graphical decription. A. Example: DC motor connected with an hydraulic pump A DC motor connected with an hydraulic pump i hown in Fig.. hi ytem involve three different energetic domain: electrical, mechanical and hydraulic. he correponding POG V a Ia L a R a I a E a Accumulator 6 J m m 5 3 Pump Filter Fig.. Q u 7 P 0 V Q 0 8 Q α α p 9 ank A DC motor connected with an hydraulic pump. graphical repreentation in hown in Fig. 3: the power ection preent in the POG cheme have a direct correpondence with the real phyical ection. he EMR of the ytem i hown in Fig. where the power ection in correpondence with the POG cheme are put in evidence. III. COMPARISON AMONG EMR AND POG BLOCKS A. Source In the EMR formalim the Source, ee Fig. 5, i an element collecting all that elong to the environment (thu not eing of the ytem) and exchanging energy with the correponding power ection of the ytem. In the POG technique uch element i imply decried y a dahed line which repreent an input/output power ection eparating the ytem from the environment, ee Fig. 5. B. Accumulation element he accumulation element in EMR, ee Fig. 6, repreent an energy toring etween two ource. It can e decried y a firt order tranfer function. It ha alway an integrative ource Fig. 5. y Input ection ource Output ection y y y EMR ource and POG input and output ection. relation etween output and input and it can conider loe (diipation) ut they are not hown explicitly in the graphical pictogram of the lock. It can not tand only for diipation, the integrative relationhip i mandatory. One can repreent thi in POG y an elaoration lock with a firt order tranfer function J+ where J>0 and 0, ee Fig. 6. Accumulation element y y Elaoration lock J+ Fig. 6. EMR accumulation element and POG elaoration lock (with J>0 and 0). In POG the elaoration lock can tore and/or diipate energy. In general it can e decried y a poitive real tranfer function matrix G() of any order. It can repreent alo pure diipation. A POG elaoration lock with a firt order tranfer function J+ can e repreented in an equivalent way y two elaoration lock a it i hown in Fig. 7: note that the lock repreent pure diipation. Some example of thi can e found in Fig. 3 etween ection - 3, - 6 and 7-9.

3 I a I a L a V a E m Motor inductance Motor reitance 3 Km Cm R a K m Converion EL-MR p J m Motor inertia ṗ 5 m Motor friction C p Kp 6 K p converion MR-ID 7 α p Q α Hydraulic leak Fig. 3. POG cheme of the DC motor with hydraulic pump of Fig.. Supply 3 Electrical EL-MR Converion 6 Mechanical MR-HY Converion 7 9 Hydraulic Envir. 8 P 0 C 0 Hydraulic accumulator 9 P 0 Q 0 V a I a C m I a E m C p Q P 0 P 0 Q 0 Fig.. EMR of the DC motor with hydraulic pump of Fig.. MS y J+ y J τ ω Fig. 9. τ ω { τ = αrτ ω = Rω () EMR of a reduction gear with efficiency and equation. Fig. 7. Equivalence of POG elaoration lock (with J>0 and 0). C. Converion element he EMR converion element, ee Fig. 8, allow an energy converion etween two uytem elonging to the ame energetic domain (monophyical converion) or to different domain (multiphyical converion). he relationhip etween input and output i a tatic relation. hi converion ha no energy accumulation, ut it can have loe, often conidered y mean of an efficiency parameter. he converion element can alo have a control input whoe power conumption i negligile with repect to the power converion. he eential difference etween the EMR accumulation and converion element i in the preence/aence of dynamic relation etween input and output. In the POG formalim the connection lock reditriute the power within the ytem without toring nor diipating energy, therefore the input power i alway equal to the output power: x y = x. D. Diipation In the EMR the diipation do not appear explicitly and can e preent oth in accumulation element and in converion element. Fig. 9 how a reduction gear repreented in EMR y a converion element and the correpondent equation defining it, where ω, τ, ω, τ are the peed and torque on the primary and on the econdary repectively. Parameter R i the gear ratio and α i an efficiency parameter (α <). he output power i P o = τ ω = ατ ω <τ ω = P i. hi i correct from an energetic perpective only if poitive power i flowing from left to right, ut if for example the converion element ha a downtream connection with an element impoing a change in the ign of the power, the converion element ecome a power generator and thi i not phiically admiile. a) τ R τ ) ω R { τ = Rτ ω ω = Rω () ω τ ρ τ ω ω { τ = ρτ ω = ω (3) Fig. 0. a) POG lock cheme of a reduction gear with diipation and equation, ) Block cheme with no correponding phyical element. A reduction gear with diipation can e repreented y the POG cheme hown in Fig. 0.a, where i the friction coefficient. he output power P o i ( ) P o =τ ω =(Rτ ω )ω = τ ω }{{} P i ω R τ } {{ } ρ =ρp i. For R = it i ρ = ω τ. Note that the efficiency ρ depend on ratio ω τ therefore given the friction coefficient, then the efficiency ρ can e conidered contant only for ω and τ atifying ω τ = ρ. herefore a converion lock with efficiency correpond to a phyical element only in thi icular cae. One can tate that there i no phyical element which can e repreented a hown in Fig. 0.. he

4 y Monophyical Converion m y Multiphyical Converion m Connection lock K(m) y K (m) Connection and Elaoration lock K(m) y K (m) y Elaoration lock Fig. 8. EMR converter and POG connection lock. concept of efficiency i not admitted in the POG technique ecaue efficiency decrie only an average ehavior, while the POG are a dynamic decription in every condition. he POG formalim ha a connection lock which convert power without toring nor diipating, it tand only for connection. When a diipation i preent an elaoration lock mut e added. Since the diipation are conidered in different way y POG and EMR, for thi apect the ijective correpondence among POG and EMR lock cannot e given. Let conider a an example the electrical circuit given in Fig.. he EMR and POG lock cheme are given in the right of Fig. where alo the power ection - are hown. Note that in the EMR a converion element i ued to repreent the reitance R, while in the POG an elaoration lock i ued. E. Coupling element he EMR how the energy reition y mean of coupling element, ee Fig.. In general only the coupling element ueful for the control are hown, other coupling can e hidden within converion or accumulation element. herefore many EMR repreentation are poile for the ame ytem. In POG cheme a compact vectorial notation i preferred, however ome n-way node can alo e ued in order to how the erie/parallel connection of the element of the ytem, ee Fig.. EMR Coupling device y y 3 POG n-way node Fig.. EMR Coupling device and POG n-way node (n =). y y 3 IV. INVERSION BASED CONROL he EMR provide a methodology to deduce a control tructure for the ytem y a tep-y-tep inverion of each element of the ytem [5]. he POG have not any pecific tool aociated to the repreentation of the model to deduce a control tructure, ut the methodology provided y EMR can e applied alo to POG cheme. A. Exemple: five-phae permanent magnet ynchronou electrical motor Let conider a 5-phae permanent magnet ynchronou electrical motor. he EMR and POG cheme are hown in Fig. 3, ee [3] and []. Note that the power ection in EMR and POG cheme of Fig. 3 which are in correpondence are indicated y the ame numer. Let u introduce the following current and voltage tator vector: t I = [ I I I m ], t V = [ V V V m ]. Uing a coordinate tranformation it i poile to write the dynamic model of the motor in a rotating reference frame and it can e proved that the 5-phae motor can e conidered a a et of two (dq) eparated fictitiou machine M and M. he tranformed current and voltage vector ω I, ω V and ω I, ω V repectively for M and M are: ω I = ω I = [ ω ] I d ω, ω V I = q [ ω ] V d ω, V q [ ω ] [ I ω ] d ω, I ω V V = d ω. q V q he motor torque i τ m = τ m + τ m, the motor velocity i and the external torque i τ e. he control tructure for the control of the motor torque τ m can e deduced y the EMR following three tep (ee [5]): decompoition of the ytem into elementary uytem, determination of the tuning chain (from the tuning input to the gloal output to e controlled) and finally inverion tep y tep of the tuning chain uing the inverion rule. In thi cae the tuning input i the input voltage vector t V and the output to e controlled i the motor torque τ m. he tuning chain i highlighted in yellow in the EMR of Fig. 3. he maximal control tructure (MCS) repreented in Fig. 3 ha een deduced y a tep y tep inverion of the tuning chain: given the torque reference τ ref the control input t V reg i provided. Note that in the two control lock ome additional

5 Electric circuit EMR POG I R I I 3 I I R I Supply C R C Envir. R V C C V C R C I V I R V MS V V V I R V I C R C 3 Fig.. Electric circuit: EMR and POG cheme. POG t V t I t ω ω V t ω t ω ω L - ω I t ω ω V ω L - ω I ω R ω J ω L Electrical ω E ω Kτ ω K τ τm ω R ω J ω L ω E ω Kτ ω K τ τm 3 Converion J m τ m Mechanical 5 m τe 6 Supply Electrical coupling 3 Electrical converion Mechanical coupling 5 6 Mechanical EMR t V t I ω V ω I ω I ω E τ m τ m τ e MS ω V ω I τ m ω I ω E MCS ω V ref ω I ref τ ref t V reg ω V ref ω I ref τ ref τ ref Fig. 3. EMR and POG cheme of a five-phae ynchronou motor in rotating frame.

6 input are conidered (the meaured current and the ack electromotive force) to provide a feedforward action. he approach of the inverion aed control can e applied alo to the POG, taking advantage from the correpondence etween power ection of EMR and POG, ee Fig. 3 where the correponding power ection are indicated y the ame numer. herefore the control tructure derived from the EMR can e applied to the POG model. In the POG cheme hown in Fig. 3 the correponding tuning chain i highlighted in yellow. [] F.A.Firetone A new analogy etween mechanical and electrical ytem, Journal of the acoutical ociety, Michigan, 933 [3] E. S , X. Ketelyn, A. Boucayrol, Right Harmonic Spectrum for the Back-Electromotive Force of a n-phae Synchronou Motor, Indutry Application Conference, 00, 39th IAS Annual Meeting. [] R. Zanai, F. Groi, Vectorial Control of Multi-phae Synchronou Motor uing POG Approach, IECON 009, 35th Annual Conference of the IEEE Indutrial Electronic Society, 009 Porto, Portugal. [5] P.J. Barre, A. Boucayrol, P. Delarue, E. Dumetz, F. Giraud, J.P. Hautier, X. Ketelyn, B. Lemaire-S , E. S , Inverion-aed control of electromechanical ytem uing caual graphical decription IECON 006-3nd Annual Conference of IEEE Indutrial Electronic, pp V. CONCLUSION In the paper a comparion etween POG and EMR technique i given to underline their common apect and difference and the poiility of cooperation etween the two technique in order to merge their different feature for the deired purpoe. he main ojective of the POG modeling technique i the analyi of phyical ytem, while EMR aim mainly at the control of the ytem. he POG cheme have a well-defined mathematical decription thu allowing to write the dynamic model of the conidered ytem and verify it correctne. he EMR provide a methodology to uild the control tarting from the repreentation of the conidered ytem. he main difference etween the two technique concern how diipation are taken into account. If the correpondence etween POG and EMR wa ijective, it would e poile to take advantage of oth POG and EMR feature: apply the inverion aed control alo to POG cheme and the exact mathematical decription to EMR. REFERENC [] Paynter, H.M., Analyi and Deign of Engineering Sytem, MI-pre, Cam., MA, 96. [] D. C. Karnopp, D.L. Margoli, R. C. Roemerg, Sytem dynamic - Modeling and Simulation of Mechatronic Sytem, Wiley Intercience, ISBN , 3rd ed [3] X. Guillaud, P. Degoert, J. P. Hautier, Modeling, control and cauality: the Caual Ordering Graph, 6th IMACS World Congre, CD-ROM, Lauanne (Switzerland), Augut 000. [] R. Zanai, Power-Oriented Modeling of Dynamical Sytem for Simulation, MCS 9, vol., pp. 3-35, Lille, France, May 99. [5] R. Zanai, he Power-Oriented Graph echnique: ytem modeling and aic propertie, 00 IEEE Vehicular Power and Propulion Conference (VPPC), pp.-6, Lille, France, Sept 00. [6] J. C. Mercieca, J. N. Verhille, A. Boucayrol, Energetic Macrocopic Repreentation of a uway traction ytem for a imulation model, IEEE-ISIE 0, Ajaccio (France), Ma00. [7] A. Boucayrol, B. Davat, B. de Fornel, B. Franoi, J. P. Hautier, F. Meiody-aar, M. Pietrzak-David, Multimachine Multiconverter Sytem: application for electromechanical drive, Eur. Phyic Journal - Appl. Phyic, vol. 0, no., pp. 3-7, Ma000. [8] R. Zanai, G.-H. Geitner, A. Boucayrol, W. Lhomme Different energetic technique for modelling traction drive, ElectrIMACS 008, Queec (Canada), June 008. [9] A. Boucayrol, Ph. Delarue, Weighted control of drive with erie connected DC machine, IEEE-IEMDC 03, Madion (USA), June 003, vol., pp [0] L. Boulon, D. Hiel, A. Boucayrol, O. Pape, M.C. Pera, Simulation Model of a Military HEV With a Highly Redundant Architecture, IEEE ranaction on Vehicular echnology, vol.59, no.6, pp , July 00. [] L. Boulon, D. Hiel, A. Boucayrol, M.C. Pera, From Modeling to Control of a PEM Fuel Cell Uing Energetic Macrocopic Repreentation, IEEE ranaction on Indutrial Electronic, vol.57, no.6, pp.88-89, June 00.