The POG Modeling Technique Applied to Electrical Systems

Transcription

1 The POG Modeling Technique Applied to Electrical Systems Roberto ZANASI Computer Science Engineering Department (DII) University of Modena and Reggio Emilia Italy

2 Outline Main characteristics of the Power-Oriented Graphs (POG) modelling technique POG modelling examples: 1. DC motor connected to an hydraulic pump 2. Three-phase brushless motor 3. Three-phase asynchronous motor

3 POG Dynamic Modeling: Physical sections The physical elements (F.E.) interact with the external world through sections. Each section is characterized by two power variables v i e v o. v i P In POG a section is denoted by using a dashed line. Each power variable has its own positive direction. The power flowing through a section can be positive or negative. An arrow over the dashed line is used for denoting the positive direction of the power P. v o F.E. The power enters into the element: I I I P The power exits from the element: I I I P V Z V Z F.E. V Z V Z F.E. V V

4 POG Dynamic Modeling: Connections I Example: connection of two electrical 2 elements Z 1, Z 2. If the powers P 1, P 2 enter into the two electrical elements, the variables I 1, V 1, I 2, V 2 cannot have all the same positive direction. In this case a connection block is used for converting the power variables. Z 1 V 1 I 1 V 2 Z 2 P 1 P 2-1 I 1 I 2 Z 1 (s) 1/Z 2 (s) Equivalent description (scalar case) V 1 1 V 2

5 Dynamic Modeling: Electrical examples I R I I IR C V c R Kirchhoff s current law C V c R V O Kirchhoff s voltage law

6 Introduction Power-Oriented Graphs (POG) The Power-Oriented Graphs are ''block diagrams'' obtained by using a ''modular'' structure essentially based on the following two blocks: Positive power flows 3-Port Junctions: 0-Junction; 1-Junction; 1-Port Elements: Capacitor C; Inertia I; Resistor R; Elaboration block Connection block 2-Port Elements : Transformers TR; GiratorsGY; Modulated TR; Modulated GY; POG maintains a direct correspondence between pairs of system variables and real power flows: the product of the two variables involved in each dashed line of the graph has the physical meaning of ``power flowing through that section''. The Elaboration block can store and dissipate/generate energy. The Connection block can only ''transform'' the energy.

7 Dynamic Modeling of Physical Systems Different Energy domains: 1) Electrical; 2) Mechanical (tras./rot.); 3) Hydraulic; etc. The same dynamic structure: 2 dynamic elements D 1, D 2 that store energy; 1 static element R that dissipates (or generates) energy; 2 energy variables q 1 (t), q 2 (t) used for describing the stored energy; 2 power variables v 1 (t), v 2 (t) used for moving the energy; Across-variables Through-variables In POG the new symbols and the new definitions are minimal R. Morselli, R. Zanasi Modeling Automotive Systems using POG

8 Example of POG modeling: an electro-hydraulic clutch The electro-hydraulic clutch: Strongly nonlinear elements I c I c Control input R. Morselli, R. Zanasi Modeling Automotive Systems using POG

9 Example of POG modeling: DC electric motor with an hydraulic pump A DC motor connected to an hydraulic pump: There is a direct correspondence between the POG blocks and the physical elements L a R a E m The POG model: L a R a K m J m b m K p

10 Introduction Power-Oriented Graphs - LTI Systems Direct correspondence between POG and state space descriptions: Stored Energy: Dissipating Power: A power state space description of the DC motor with hydraulic pump: Which is the reduced model when J m ->0? Two possible solutions: 1) graphically inverting a path ; 2) using a congruent transformation

11 POG modeling reduction: graphically inverting a path POG reduced model!

12 POG modeling reduction: using a congruent transformation When an eigenvalue of matrix L goes to zero (or to infinity), the system degenerates towards a lower dynamic dimension system. The reduced system can be obtained by using a congruent transformation x=tz where T is a rectangular matrix: Dissipation 2D element

13 POG modeling of Electrical Motors Let us consider Electric Motors energetically characterized by: 1) the magnetic flux LI generated by the stator and/or rotor currents I s and I r ; 2) the magnetic flux j (q r ) of the permanent magnets (if present); 3) the momentum J r w r generated by rotor velocity w r ; The Energy K stored in the system can be expressed as follows: where and w r is the rotor angular position. The dynamic equations of the system are: Where R is a symmetric matrix (energy dissipation/generation ) and W is a skew-symmetric matrix (energy redistribution ):,

14 POG modeling of Electrical Motors Two different but equivalent POG graphical representations: The dynamic equations can be easily interpreted from a power point of view. Multiplying on the left of the first equation one obtains: Redistributed power Stored energy variation Entering power Dissipated power

15 Brushless motor: the three-phase stator circuit The constraint: The circuit: The dynamic equations: that is, expanded: Static dq By using a congruent transformation one obtains the reduced system :

16 Brushless motor: the rotating frame By using a orthonormal transformation one obtains the two-phase rotating dynamic model of the system. Expanded form where is the magnetic flux generated by the permanent magnets POG dynamic model of the brushless motor

17 Brushless motor: sinusoidal magnetic flux If the magnetic flux of the permanent magnets is sinusoidal Two-phase rotating Three-phase Two-phase static the dynamic equations of brushless motor strongly simplify d Dynamics q Dynamics Rotor as well as the POG graphical representation.

18 Asyncronous three-phase motor: the stator and rotor circuits The variables. Stator and rotor currents and voltages:,,, The constraints: The system: The dynamic equations: where and Stator and Rotor Self-Inductances Stator/Rotor Mutual-Inductances

19 Asyncronous motor: dynamic model Applying the two-phase static transformation (6->4): where and then the two-phase rotating transformation ( ): where one obtains the following full dynamic model: where,,,,,,,.

20 Asyncronous motor: dynamic model Dynamic model in a compact form: The energy matrix represents the stored energy: The symmetric part of the system matrix represents the energy dissipations: The skew symmetric part of the system matrix represents the energy redistribution: The active torque applied to the rotor:

21 Asyncronous motor: POG model The POG graphical representation: 3P->2P static 2P->2P rotating Energy storing element Dissipation element Connecton elements: the energy is redistributed Rotor s dynamics

22 Conclusions Power-Oriented Graphs (POG) are a simple and powerful graphical technique that can be used for modeling all types of physical systems involving power flows. POG are easily understandable, simple to use and suitable both for teaching and for research.