《电池与能量存储》课程教学课件（PPT讲稿）Basic Modeling Methods

Basic ModelingMethods

Basic Modeling Methods

References: Automatic Control Systems, 8th Edition, B.C.KuoF. Golnaraghi, John Wiley & Sons, 2002Matlab/Simulink

References • Automatic Control Systems, 8 th Edition, B.C. Kuo, F. Golnaraghi, John Wiley & Sons, 2002 • Matlab/Simulink

What are mathematics models forphysical systems?They are empirical representations of a physical system'sinput/output relationships and internal behavior by usingmathematics expressions.Themodels can be in differentforms:FunctionsDifferential equationsState space modelsTransfer functionsBlock diagramSimulation modulesThe models can be obtained by different methodsDerivations based on basic principles Experimental data and model fitting Real-time updatinglearning

What are mathematics models for physical systems? • They are empirical representations of a physical system’s input/output relationships and internal behavior by using mathematics expressions. • The models can be in different forms: • Functions • Differential equations • State space models • Transfer functions • Block diagram • Simulation modules • The models can be obtained by different methods: • Derivations based on basic principles • Experimental data and model fitting • Real-time updating • learning

Why do we need mathematics models?:They are cost effective in studying the main features ofphysical, environment, social systemsExamples: Battery SOC estimation, vehicle fuel economy andemission, power system security and reliability, ...They can be used to predict the future behavior of thephysical, environmental, and social systemExamples: Covid infection prediction, trafficpatterns, ... They can be used to evaluate different controls, designs andimpact of decisions.Examples: battery management systems, control systems formotors, autonomous vehicles, ... They can be used to coordinate component designs fromdifferent teams and companies.Examples: autonomous vehicles, battery managementsystems,

Why do we need mathematics models? • They are cost effective in studying the main features of physical, environment, social systems. Examples: Battery SOC estimation, vehicle fuel economy and emission, power system security and reliability, . • They can be used to predict the future behavior of the physical, environmental, and social system. Examples: Covid infection prediction, traffic patterns,. • They can be used to evaluate different controls, designs and impact of decisions. Examples: battery management systems, control systems for motors, autonomous vehicles, . • They can be used to coordinate component designs from different teams and companies. Examples: autonomous vehicles, battery management systems,

Derivation ofDifferential Equation Models

Derivation of Differential Equation Models

Derivation of Models for Electrical Systems(1) Start from the basic circuit component principle:BasicVoltage-CurrentRelationsv(t) = Ri(t)Resistor of R (ohms):V(s) = R I(s)cdv() =i(t)= C(sV(s) - v(0) = I(s)Capacitor of C (farads):dtI(s) + v(0) -[1(s) +Cv(0)]V(s) =SCsI di(t)Inductor ofL (henries):v(t)dt(0)V(s) = LsI(s) - Li(O) = Ls I(s)1S

Derivation of Models for Electrical Systems dt C dv(t) = i(t)  C(sV (s) − v(0)) = I (s) Resistor of R (ohms): Capacitor of C (farads): Cs s Cs Inductor of L(henries): V(s) = 1 I(s) + v(0) = 1 I(s) +Cv(0) s V (s) = LsI(s) − Li(0) = Ls  I (s) − i(0)     v(t) = Ri(t) V(s) = R I(s) dt v(t) = L di(t) (1) Start from the basic circuit component principle: Basic Voltage-CurrentRelations:

V=-RIVRV= RIVR1dtdtdlVdtdt

+V I -+V￾I -+V I R V = RI CL dt I = C dV dt V = L dI +V I -+V￾I -+V I R V = −RI CL dt I = − C dV dt V = − L dI

V=RIVRV=-RI1dvdtdtdlVdlVdtdt

-V I +-V+ I +-V I R V = −RI CL dV I = −C dt dt V = − L dI I -V+-V+ I +-V I R V = RI CL dt I = C dV dt V = L dI

(2)BuildUp Circuit InterconnectionsExample: The RC-Branch Model in a BatteryRpVocvVpR,dtpdvdv2PRdtdtRD

Rp R v i vocv vp + - Cp ip - + p p p p p p p p p dvp , vp R dvp dvp vp C dt R dt R C C i =C = i −i dt = i − vp  = − + 1 i (2) Build Up Circuit Interconnections Example: The RC-Branch Model in a Battery

Initial Condition Response (Zero-Input Response)i = O, the initial conditionis v, (O)dyVRpCp v(t)=v(0)e二dtR,C.pDT,= R,C,= Time ConstantIf C, is small, then the time constant is small= The initial consdition response will go down to zero relativelyfast= The RC branch will reach the steady state fastSteady State of theRC Branch(after the initial condition response diminishes)dyP= O and v(o) is now a constant.The steady state meansdtdyvFromwehave i, =0dt(8)VFromi-i, =i, we have v,(o)= R, iR,p

p p RpCp p p p p dv v dt R C i = 0, the initial condition is vp (0) − t = −  v (t) = v (0)e Initial Condition Response (Zero-Input Response) Tp = RpCp = Time Constant If Cp is small, then the time constant is small.  The initial consdition response will go down to zero relativelyfast  The RC branch will reach the steady state fast. Steady State of the RC Branch (after the initial condition response diminishes) p p p p p p p p dt dt R dv The steady state means p = 0 and v () is now a constant. Fromi = C dvp , wehave i = 0 v () From p = i −i = i, we have v () = R i