Quantifying Parameter Identifiability Consider DAE d dt x ( t ) = f ( x , z , u ; θ ); x ( t 0 ) = x 0 ( θ ) (5) 0 = g ( x , z , u ; θ ) (6) y = h ( x , z , u ; θ ) (7) Sensitivity Equations: dt S 1 ( t ) = ∂ f ∂ xS 1 + ∂ f ∂ z S 2 + ∂ f d (8) ∂θ 0 = ∂ g ∂ x S 1 + ∂ g ∂ z S 2 + ∂ g (9) ∂θ S 3 ( t ) = ∂ h ∂ x S 1 + ∂ h ∂ z S 2 + ∂ h (10) ∂θ ∂θ ( θ 0 ) , S 3 = ∂ y where S 1 = ∂ x ∂θ ( θ 0 ) , S 2 = ∂ z ∂θ ( θ 0 ) Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 16
Quantifying Parameter Identifiability Consider DAE Define Fisher Information Matrix F ∈ R n θ × n θ d � t f dt x ( t ) = f ( x , z , u ; θ ); x ( t 0 ) = x 0 ( θ ) (5) S T 3 ( t ) Q − 1 S 3 ( t ) dt , F = (11) 0 = g ( x , z , u ; θ ) (6) 0 y = h ( x , z , u ; θ ) (7) Sensitivity Equations: dt S 1 ( t ) = ∂ f ∂ xS 1 + ∂ f ∂ z S 2 + ∂ f d (8) ∂θ 0 = ∂ g ∂ x S 1 + ∂ g ∂ z S 2 + ∂ g (9) ∂θ S 3 ( t ) = ∂ h ∂ x S 1 + ∂ h ∂ z S 2 + ∂ h (10) ∂θ ∂θ ( θ 0 ) , S 3 = ∂ y where S 1 = ∂ x ∂θ ( θ 0 ) , S 2 = ∂ z ∂θ ( θ 0 ) Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 16
Quantifying Parameter Identifiability Consider DAE Define Fisher Information Matrix F ∈ R n θ × n θ d � t f dt x ( t ) = f ( x , z , u ; θ ); x ( t 0 ) = x 0 ( θ ) (5) S T 3 ( t ) Q − 1 S 3 ( t ) dt , F = (11) 0 = g ( x , z , u ; θ ) (6) 0 y = h ( x , z , u ; θ ) (7) Cramér-Rao Bound characterizes param esti- mation accuracy. α − level confidence ellipsoid: Sensitivity Equations: � � ϑ | ( ϑ − ˆ θ ) T F − 1 ( ϑ − ˆ E = θ ) ≤ β (12) dt S 1 ( t ) = ∂ f ∂ xS 1 + ∂ f ∂ z S 2 + ∂ f d (8) ∂θ 0 = ∂ g ∂ x S 1 + ∂ g ∂ z S 2 + ∂ g (9) ∂θ S 3 ( t ) = ∂ h ∂ x S 1 + ∂ h ∂ z S 2 + ∂ h (10) ∂θ ∂θ ( θ 0 ) , S 3 = ∂ y where S 1 = ∂ x ∂θ ( θ 0 ) , S 2 = ∂ z ∂θ ( θ 0 ) Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 16
Quantifying Parameter Identifiability Consider DAE Define Fisher Information Matrix F ∈ R n θ × n θ d � t f dt x ( t ) = f ( x , z , u ; θ ); x ( t 0 ) = x 0 ( θ ) (5) S T 3 ( t ) Q − 1 S 3 ( t ) dt , F = (11) 0 = g ( x , z , u ; θ ) (6) 0 y = h ( x , z , u ; θ ) (7) Cramér-Rao Bound characterizes param esti- mation accuracy. α − level confidence ellipsoid: Sensitivity Equations: � � ϑ | ( ϑ − ˆ θ ) T F − 1 ( ϑ − ˆ E = θ ) ≤ β (12) dt S 1 ( t ) = ∂ f ∂ xS 1 + ∂ f ∂ z S 2 + ∂ f d (8) ∂θ Goal: Minimize (scalarization) of F − 1 : 0 = ∂ g ∂ x S 1 + ∂ g ∂ z S 2 + ∂ g (9) D-optimality : det ( F − 1 ) ∂θ S 3 ( t ) = ∂ h ∂ x S 1 + ∂ h ∂ z S 2 + ∂ h A-optimality : trace ( F − 1 ) (10) ∂θ E-optimality : λ max ( F − 1 ) ∂θ ( θ 0 ) , S 3 = ∂ y where S 1 = ∂ x ∂θ ( θ 0 ) , S 2 = ∂ z ∂θ ( θ 0 ) Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 16
Optimal Control Problem for Maximizing Parameter Identifiability � t f �� � − 1 � S T 3 ( t ) Q − 1 S 3 ( t ) minimize det dt (13) t = t 0 dt S 1 ( t ) = ∂ f ∂ xS 1 + ∂ f ∂ z S 2 + ∂ f d d subject to: dt x ( t ) = f ( x , z , u ; θ ); x ( t 0 ) = x 0 ( θ ) (14) ∂θ 0 = ∂ g ∂ x S 1 + ∂ g ∂ z S 2 + ∂ g 0 = g ( x , z , u ; θ ) (15) ∂θ S 3 ( t ) = ∂ h ∂ x S 1 + ∂ h ∂ z S 2 + ∂ h y = h ( x , z , u ; θ ) (16) ∂θ Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 17
Optimal Control Problem for Maximizing Parameter Identifiability � t f �� � − 1 � S T 3 ( t ) Q − 1 S 3 ( t ) minimize det dt (13) t = t 0 dt S 1 ( t ) = ∂ f ∂ xS 1 + ∂ f ∂ z S 2 + ∂ f d d subject to: dt x ( t ) = f ( x , z , u ; θ ); x ( t 0 ) = x 0 ( θ ) (14) ∂θ 0 = ∂ g ∂ x S 1 + ∂ g ∂ z S 2 + ∂ g 0 = g ( x , z , u ; θ ) (15) ∂θ S 3 ( t ) = ∂ h ∂ x S 1 + ∂ h ∂ z S 2 + ∂ h y = h ( x , z , u ; θ ) (16) ∂θ Elegant formulation! However, computationally intractable: 2 weeks to generate 100 sec of optimized input signals Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 17
Optimal Control Problem for Maximizing Parameter Identifiability Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 18
Optimal Control Problem for Maximizing Parameter Identifiability Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 18
A completely different idea! Figure: Fixed menu of L inputs, index by j Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 19
A completely different idea! Figure: Fixed menu of L inputs, index by j Convex OED Pre-compute all sensitivities S j on menu − 1 L � η j S j Q − 1 S T minimize η log det j j = 0 L � subject to: η j ≥ 0 , η j = 1 j = 0 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 19
A completely different idea! Figure: Fixed menu of L inputs, index by j Convex OED Pre-compute all sensitivities S j on menu − 1 L � η j S j Q − 1 S T minimize η log det j j = 0 L � subject to: η j ≥ 0 , η j = 1 j = 0 Convex program → polynomial complexity Optimize 750 input profiles in 20 seconds Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 19
✶ ✶ Convex OED Fixed menu of ℓ inputs, indexed by j Execute m j ∈ { 0 , 1 } experiments for input j m 1 + m 2 + · · · + m ℓ = m Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 20
✶ ✶ Convex OED Fixed menu of ℓ inputs, indexed by j Execute m j ∈ { 0 , 1 } experiments for input j m 1 + m 2 + · · · + m ℓ = m calculate FIM from pre-computed sensitivities ¯ S j m ℓ � S i Q − 1 � m j ¯ S j Q − 1 ¯ S T S T F = i = j i j i = 1 j = 1 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 20
✶ ✶ Convex OED Fixed menu of ℓ inputs, indexed by j Execute m j ∈ { 0 , 1 } experiments for input j m 1 + m 2 + · · · + m ℓ = m calculate FIM from pre-computed sensitivities ¯ S j m ℓ � S i Q − 1 � m j ¯ S j Q − 1 ¯ S T S T F = i = j i j i = 1 j = 1 Combinatorial OED − 1 ℓ F − 1 = � m j ¯ S j Q − 1 ¯ S T minimize j j j = 1 subject to m j ≥ 0 , m 1 + · · · + m l = m m j ∈ { 0 , 1 } Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 20
✶ Convex OED Fixed menu of ℓ inputs, indexed by j Let η j = m j / m be fraction of total experiments Execute m j ∈ { 0 , 1 } experiments for input j to execute of type j . Relax integer constraint m 1 + m 2 + · · · + m ℓ = m ℓ � η j ¯ S j Q − 1 ¯ S T = F m calculate FIM from pre-computed sensitivities ¯ j S j j j = 1 − 1 m ℓ ℓ F − 1 = 1 � S i Q − 1 � m j ¯ S j Q − 1 ¯ S T S T F = i = � η j ¯ S j Q − 1 ¯ S T minimize j i j j j m i = 1 j = 1 j = 1 ✶ T η = 1 subject to η � 0 , Combinatorial OED − 1 ℓ F − 1 = � m j ¯ S j Q − 1 S T ¯ minimize j j j = 1 subject to m j ≥ 0 , m 1 + · · · + m l = m m j ∈ { 0 , 1 } Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 20
Convex OED Fixed menu of ℓ inputs, indexed by j Let η j = m j / m be fraction of total experiments Execute m j ∈ { 0 , 1 } experiments for input j to execute of type j . Relax integer constraint m 1 + m 2 + · · · + m ℓ = m ℓ � η j ¯ S j Q − 1 S T ¯ = F m calculate FIM from pre-computed sensitivities ¯ j S j j j = 1 − 1 m ℓ ℓ F − 1 = 1 � S i Q − 1 � m j ¯ S j Q − 1 ¯ S T S T F = i = � η j ¯ S j Q − 1 S T ¯ minimize j i j j j m i = 1 j = 1 j = 1 ✶ T η = 1 subject to η � 0 , Combinatorial OED − 1 Scalarize. Arrive at convex OED ℓ F − 1 = � m j ¯ S j Q − 1 ¯ S T minimize j j − 1 ℓ j = 1 � η j ¯ S j Q − 1 ¯ S T minimize log det subject to m j ≥ 0 , m 1 + · · · + m l = m j j j = 1 m j ∈ { 0 , 1 } ✶ T η = 1 η � 0 , subject to Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 20
Input Library Parameters of Interest 783 input profiles Pulses Sinusoids Dynamic drive cycles 112+ hours of experiments Compute sensitivities via cluster computing Selected 12 for OED Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 21
US06 UDDS SC04 LA92 DC2 DC1 Nominal Industry OED CVX 0 10 20 30 RMSE [mV] S. Park, D. Kato, Z. Gima, R. Klein, S. J. Moura, “Optimal Experimental Design for Parameterization of an Electrochemical Lithium-ion Battery Model,” Journal of the Electrochemical Society , DOI: 10.1149/2.0421807jes Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 22
� 2 � t f � y exp ( t ) − y mdl ( t ; ˆ Objective: J = θ ) dt 0 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 23
Outline MODEL IDENTIFICATION 1 STATE ESTIMATION 2 OPTIMAL SAFE-FAST CHARGING 3 FAULT DIAGNOSTICS 4 5 SUMMARY AND OPPORTUNITIES Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 24
Survey of SOC/SOH Estimation Literature Equivalent Circuit Model (ECM) Electrochemical Model (a) OCV-R - - sep sep + + 0 L 0 L L 0 x Cathode Anode Separator (b) OCV-R-RC c s - ( r ) c s + ( r ) e - Li + e - c ss - c ss + (c) Impedance r r c e ( x ) Li x C 6 Li 1-x MO 2 Electrolyte Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 25
Survey of SOC/SOH Estimation Literature What is new? What are the opportunities/challenges? EChem models provide unprecedented detail Computational challenges Observability/identifiability – i.e. is it possible? Provable convergence – i.e. mathematical certificate Want to capitalize on unprecedented detail of EChem models? We use a reduced EChem model Provable convergence? We mathematically prove estimation error convergence Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 25
Single Particle Model with Electrolyte (SPMe) V(t) Anode Separator Cathode Solid T(t) Cell Electrolyte Li + Li + I(t) c s - (r,t) c s + (r,t) I(t) Li + Solid r r R s - R s + c e (x,t) I(t) I(t) Electrolyte Li + Li + Li + x L - 0 sep L sep L + 0 - 0 + − (𝑆 𝑡 𝑡𝑓𝑞 (𝑦, 𝑢), 𝑑 𝑓 − , 𝑢), 𝑑 𝑡 + (𝑆 𝑡 + , 𝑢), 𝑑 𝑓 − (𝑦, 𝑢), 𝑑 𝑓 + (𝑦, 𝑢), 𝑈(𝑢), 𝐽(𝑢)) 𝑊(𝑢) = ℎ(𝑑 𝑡 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 26
SPMe - Physical Intuition c + s ( x, r, t ) /c + c − s ( x, r, t ) /c − Center s, max s, max 0.75 0.7 Space, r Space, r 0.65 0.6 0.55 0.5 Surface 0.45 2 Anode Separator Cathode c e ( x, t ) 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Space, x/L Approximate solid-phase concentration as uniform in x Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 27
SPMe Equations Subsystem Partial Differential Equation (PDEs) Boundary Conditions ∂ c ± ∂ r ( 0 , t ) = 0, s Solid phase ∂ c ± s ) · r 2 ∂ c ± � � r 2 ∂ 1 D ± s ( c ± ∂ t ( r , t ) = ∂ r ( r , t ) s s ∂ c ± Li diffusion ∂ r ∂ r ( R ± ± 1 s , t ) = s Fa ± L ± I ( t ) s D ± Electrolyte 1 − t 0 ∂ c − ∂ x ( 0 − , t ) = ∂ c + ∂ c e ∂ e ( c e ) ∂ c e ∂ x ( 0 + , t ) = 0 � D eff � ∂ t ( x , t ) = ∂ x ( x , t ) ∓ e FL ± I ( t ) c e e ε ± Li diffusion ∂ x Output Equation: � � � � RT − I ( t ) − RT I ( t ) α F sinh − 1 α F sinh − 1 V ( t ) = 2 a + L + ¯ 2 a − L − ¯ i + i − 0 ( t ) 0 ( t ) R + R − � � + U + ( c + s ( R + s , t )) − U − ( c − s ( R − f f s , t )) − a + L + + I ( t ) a − L − + L + + 2 L sep + L − ln c e ( 0 + , t ) − ln c e ( 0 − , t ) � � I ( t ) + k conc ( t ) 2 κ Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 28
Causal Structure of SPMe c + ss ( t ) ✲ ✲ c + s ( r , t ) c − ss ( t ) ✲ ✲ c − s ( r , t ) I ( t ) V ( t ) ✲ ✲ Output c + e ( 0 + , t ) ✲ c + e ( x , t ) ✲ c sep e ( x , t ) c − e ( 0 − , t ) ✲ c − e ( x , t ) Figure: Block diagram of SPMe. Note that the c + s , c − s , c e subsystems are all (i) quasilinear parabolic PDEs and (ii) independent of one another. Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 29
Model Comparison DFN - (line) � SPMe + (plus) � SPM ◦ (circle) 4 0.1C (a) 0.5C 3.9 1C 2C 3.8 5C 3.7 Voltage [V] 3.6 3.5 3.4 3.3 3.2 0 5 10 15 20 25 30 Discharged Capacity [Ah/m 2 ] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 30
Model Comparison Current [C−rate] 4 2 0 −2 4.2 DFN 4 SPMe Voltage [V] SPM 3.8 3.6 3.4 0 500 1000 1500 2000 2500 3000 Time [sec] 4 4 ZOOM ZOOM Voltage [V] 3.9 3.8 3.8 3.7 3.6 3.6 150 200 250 300 2600 2650 2700 2750 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 30
SPMe Conservation Properties Conservation of Solid Lithium d Moles of solid phase Li are conserved. Mathematically, dt ( n Li , s ( t )) = 0 where � R j ε j s L j s � 4 π r 2 c j n Li , s ( t ) = s ( r , t ) dr 3 π ( R j 4 s ) 3 0 j ∈{ + , −} Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 31
SPMe Conservation Properties Conservation of Solid Lithium d Moles of solid phase Li are conserved. Mathematically, dt ( n Li , s ( t )) = 0 where � R j ε j s L j s � 4 π r 2 c j n Li , s ( t ) = s ( r , t ) dr 3 π ( R j 4 s ) 3 0 j ∈{ + , −} Conservation of Electrolyte Lithium d Moles of electrolyte phase Li are conserved. Mathematically, dt ( n Li , e ( t )) = 0 where � L j � ε j 0 j c j n Li , e ( t ) = e ( x , 0 ) dx e j ∈{− , sep , + } Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 31
I ( t ) V ( t ) ✲ ✲ Battery Cell ❄ ✲ Output Fcn. c + ˆ ✲ Cathode Obs. ss Inversion c + ˆ s ( r , t ) ❄ ✻ ✻ ✻ c + − ˇ ✻ ✛ ss ♥ ˜ c + ss ❄ + ✲ c − ˆ Anode Obs. ss c − ˆ s ( r , t ) Electrolyte Obs. c + e ( 0 + ) ˆ ✲ c + ˆ e ( x , t ) c sep ˆ e ( x , t ) ˆ c − e ( 0 − ) ✲ c − ˆ e ( x , t ) Figure: Block diagram of SPMe Observer. Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 32
Stability Analysis Theorem 1 - Solid Phase Assume n Li , s is known. Then the anode & cathode solid Li concentration estimates converge asymptotically to the true values. ˆ c ± s ( r , t ) → c ± s ( r , t ) , as t → ∞ . Theorem 2 - Electrolyte Phase Assume n Li , e is known. Then electrolyte Li concentration estimates converge asymptotically to the true values. ˆ c e ( x , t ) → c e ( x , t ) , as t → ∞ . Theorem 3 - Output Inversion Assume −∞ < ∂ V /∂ c + ss < 0. Then the “processed” cathode surface concentration converges exponentially to its true value: ˇ c + ss ( t ) → c + ss ( t ) , as t → ∞ . Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 33
Test with Experimental Data Experimental voltage & current data obtained from our battery-in-the-loop facility Data used to fit full-order EChem model parameters offline “Truth data” generated from experimentally validated full-order EChem model TRUE initial condition: c − s ( r , 0 ) / c − s , max = 0 . 8224 c − s ( r , 0 ) / c − OBSERVER initial condition: ˆ s , max = 0 . 4 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 34
Constant 1C Discharge Cycle 2 Current [C−rate] (a) 1 0 1 (b) 0.8 Surface Conc. [−] OUTPUT NEARLY 0.6 NON−INVERTIBLE θ − 0.4 ˆ θ − θ + 0.2 ˆ θ + θ + ˇ 0 3.9 V (c) ˆ V 3.8 3.7 Voltage 3.6 3.5 3.4 0 500 1000 1500 2000 2500 3000 Time [sec] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 35
Constant 1C Discharge Cycle 2 5 2 Current [C−rate] (a) (a) U − ( θ − ) Cathode OCP [V] Anode OCP [V] U + ( θ + ) 1 1 4 0 1 (b) 0.8 Surface Conc. [−] OUTPUT NEARLY 0.6 0 3 NON−INVERTIBLE 1.5 θ − ∂h/∂c − ss ( θ − ) 0.4 (b) ˆ θ − 1 ∂h/∂c + ss ( θ + ) θ + 0.2 ˆ θ + [ × 10 − 6] 0.5 θ + ˇ 0 3.9 0 V (c) ˆ V 3.8 −0.5 LOW 3.7 Voltage SENSITIVITY −1 3.6 0 0.2 0.4 0.6 0.8 1 θ ± = c ± Normalized Surface Concentration , ss / c ± s , max 3.5 3.4 0 500 1000 1500 2000 2500 3000 Time [sec] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 35
EV Charge/Discharge Cycle: UDDSx2 0.2 Current [C−rate] θ − − ˆ 4 (a) (d) θ − Surface Conc. Error [−] θ + − ˆ θ + 2 θ + − ˇ θ + 0.1 0 −2 0 −4 (b) 0.8 −0.1 Surface Conc. [−] θ − 0.7 ˆ θ − −0.2 θ + (e) V − ˆ 0.6 ˆ θ + Voltage Error [mV] 20 V θ + ˇ 10 0.5 0 0.4 −10 V (c) ˆ V −20 4 0 500 1000 1500 2000 2500 3000 Voltage Time [sec] 3.8 SJM, F . Bribiesca Argomedo, R. Klein, A. Mirtabatabaei, M. 3.6 Krstic, “Battery State Estimation for a Single Particle Model with Electrolyte Dynamics,” IEEE Transactions on Control 0 500 1000 1500 2000 2500 3000 Time [sec] Systems Technology . DOI: 10.1109/TCST.2016.2571663 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 36
Interval PDE Observer under Parametric Uncertainty � � Map parameter uncertainty θ ∈ θ, θ to interval state estimates Problem Statement: � � ˆ ˆ c ( r , t ) , ˆ c ( r , t ) ∈ c ( r , t ) where c ( r , t ) is governed by PDE. Current [C−rate] 6 4 2 0 Diffusion −2 (a) PDE −4 0 4 10 20 30 40 50 x 10 S 1 (1 , t ) 2 S 2 (1 , t ) Sensitivity S 3 (1 , t ) S 4 (1 , t ) 1 Diffusion (b) 0 PDE Copy + 0 10 20 30 40 50 Output Inj. SOC ( t ) SOC ( t ) 1 0.6 ˆ ˆ 0.55 SOC ( t ) SOC ( t ) 0.8 Bulk SOC Sensitivity ˆ ˆ 0.5 SOC ( t ) SOC ( t ) 0.6 ˆ ˆ PDEs 0.45 SOC ( t ) SOC ( t ) 0.4 3.5 4 4.5 0.2 (c) 3.27 0 3.26 0 10 20 30 40 50 3.25 V ( t ) V ( t ) Interval 3.24 ˆ ˆ 3.4 V ( t ) V ( t ) 3.23 Estimator ˆ ˆ V ( t ) V ( t ) 3.7 3.8 3.9 Voltage ˆ ˆ 3.3 V ( t ) V ( t ) H. Perez, SJM, “Sensitivity-Based Interval PDE Observer for Battery SOC 3.2 Estimation,” 2015 American Control Conference, Chicago, IL, 2015. O. (d) 0 10 20 30 40 50 Hugo Schuck Best Paper & ACC Best Student Paper Awards Time [min] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 37
Outline MODEL IDENTIFICATION 1 STATE ESTIMATION 2 OPTIMAL SAFE-FAST CHARGING 3 FAULT DIAGNOSTICS 4 5 SUMMARY AND OPPORTUNITIES Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 38
Operate Batteries at their Physical Limits Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 39
Operate Batteries at their Physical Limits Commercialized Today iPhone 5S Macbook Pro 2015 Tesla Supercharger 0.64C 0.8C 1.4C – 1.8C + USB charger + 60W charger + Model S Defn: (C-rate) Capacity normalized current. C-rate = (current) / (charge capacity). Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 39
Operate Batteries at their Physical Limits Electrochemical model-based limits of operation ECM-based limits of operation ECM-based limits of operation Terminal Voltage Overpotential Cell Current Surface concentration Problem Statement x , ˆ Given accurate electrochemical state/parameter estimates (ˆ θ ) , govern the input current I ( t ) such that the EChem constraints are enforced. Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 39
Operate Batteries at their Physical Limits Measurements I r ( t ) I ( t ) V ( t ), T ( t ) EChem-based Battery Cell Controller + Innovations _ EChem-based Estimated ^ ^ x ( t ), θ ( t ) States & Params State/Param ^ ^ Estimator V ( t ), T ( t ) ElectroChemical Controller (ECC) Problem Statement x , ˆ Given accurate electrochemical state/parameter estimates (ˆ θ ) , govern the input current I ( t ) such that the EChem constraints are enforced. Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 39
Constraints Variable Definition Constraint I ( t ) Current Power electronics limits c ± s ( x , r , t ) Li concentration in solid Saturation/depletion ∂ c ± ∂ r ( x , r , t ) Li concentration gradient Diffusion-induced stress s c e ( x , t ) Li concentration in electrolyte Saturation/depletion T ( t ) Temperature High/low temps accel. aging η s ( x , t ) Side-rxn overpotential Li plating, dendrite formation Each variable, y , must satisfy y min ≤ y ≤ y max . Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 40
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
Introduction to Reference Governors Idea of RG Given a desired reference r k , generate a modified applied reference v k which guarantees safety, while tracking r k as closely as possible. Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 41
The Algorithm: Modified Reference Governor (MRG) Modified I r I Battery V Reference Cell Governor x , z Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 42
The Algorithm: Modified Reference Governor (MRG) Modified I r I Battery V Reference Cell Governor x , z β ∗ ∈ [ 0 , 1 ] , I [ k + 1 ] = β ∗ [ k ] I r [ k ] , MRG Equations β ∗ [ k ] = max { β ∈ [ 0 , 1 ] | ( x ( t ) , z ( t )) ∈ O} Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 42
The Algorithm: Modified Reference Governor (MRG) Overpotential, y i y ( t+Ts ) y ( t ) Surface concentration, y j β ∗ ∈ [ 0 , 1 ] , I [ k + 1 ] = β ∗ [ k ] I r [ k ] , MRG Equations β ∗ [ k ] = max { β ∈ [ 0 , 1 ] | ( x ( t ) , z ( t )) ∈ O} Admissible Set O = { ( x ( t ) , z ( t )) : y ( τ ) ∈ Y , ∀ τ ∈ [ t , t + T s ] } f ( x ( t ) , z ( t ) , β I r ) ˙ x ( t ) = g ( x ( t ) , z ( t ) , β I r ) 0 = C 1 x ( t ) + C 2 z ( t ) + D · β I r + E y ( t ) = Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 42
Constrained Control of EChem States I ( t ) 3 Current [C−rate] 2 1 0 Side Rxn Overpotential [V] 0.15 η s ( L − , t ) 0.1 0.05 0 −0.05 −0.1 0 20 40 60 80 100 120 Time [sec] H. E. Perez, N. Shahmohammad, S. J. Moura, “Enhanced Performance of Li-ion Batteries via Modified Reference Governors & Electrochemical Models,” IEEE/ASME Transactions on Mechatronics , Aug 2015. DOI: 10.1109/TMECH.2014.2379695 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 43
Constrained Control of EChem States 3 Current [C−rate] 2 1 I ( t ) I r ( t ) 0 Side Rxn Overpotential [V] 0.15 0.1 0.05 0 −0.05 −0.1 η s ( L − , t ) 0 20 40 60 80 100 120 Time [sec] H. E. Perez, N. Shahmohammad, S. J. Moura, “Enhanced Performance of Li-ion Batteries via Modified Reference Governors & Electrochemical Models,” IEEE/ASME Transactions on Mechatronics , Aug 2015. DOI: 10.1109/TMECH.2014.2379695 Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 43
Application to Charging Current [C−rate] CCCV 1 MRG 0.5 0 4.4 Voltage [V] 4.2 4 3.8 3.6 1 SOC 0.8 0.6 0.2 Overpotential [V] Side Rxn 0.1 0 0 5 10 15 20 25 30 35 40 45 Time [min] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 44
Application to Charging Current [C−rate] CCCV 1 MRG 0.5 0 4.4 Voltage [V] 4.2 4 Exceed 4.2V “limit” 3.8 3.6 1 SOC 0.8 0.6 0.2 Overpotential [V] Side Rxn 0.1 0 0 5 10 15 20 25 30 35 40 45 Time [min] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 44
Application to Charging Current [C−rate] CCCV 1 MRG 0.5 0 4.4 Voltage [V] 4.2 4 Exceed 4.2V “limit” 3.8 3.6 1 SOC 0.8 0.6 0.2 Overpotential [V] Side Rxn Eliminate conservatism, Operate near limit 0.1 0 0 5 10 15 20 25 30 35 40 45 Time [min] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 44
Application to Charging Current [C−rate] CCCV 1 MRG 0.5 0 4.4 Voltage [V] 4.2 4 Exceed 4.2V “limit” 3.8 3.6 1 5% more charge capacity SOC 0.8 0.6 0.2 Overpotential [V] Side Rxn Eliminate conservatism, Operate near limit 0.1 0 0 5 10 15 20 25 30 35 40 45 Time [min] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 44
Application to Charging Current [C−rate] CCCV 1 MRG 0.5 0 4.4 Voltage [V] 4.2 4 Exceed 4.2V “limit” 3.8 3.6 20% reduction in 0-95% 1 SOC charge time 5% more charge capacity SOC 0.8 0.6 0.2 Overpotential [V] Side Rxn Eliminate conservatism, Operate near limit 0.1 0 0 5 10 15 20 25 30 35 40 45 Time [min] Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 44
Battery-in-the-Loop Test Facility Battery Tester Li-ion Cells in Chamber Measurements: I , V , T CAN bus Microcontroller Optimized w/ Algorithms Charge Cycle Estimates: concentrations, overpotentials, etc. Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 45
Battery-in-the-Loop Test Facility Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 45
Battery-in-the-Loop Test Facility Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 45
ElectroChemical Controller (ECC) Measurements I r ( t ) I ( t ) V ( t ), T ( t ) EChem-based Battery Cell Controller + Innovations _ EChem-based Estimated ^ ^ x ( t ), θ ( t ) State/Param States & Params ^ ^ V ( t ), T ( t ) Estimator ElectroChemical Controller (ECC) Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 46
l Chg l Dchg RG vs 1C vs 2C Capacity fade RG/1C/2C: l / p / n Cycling ¡ / r / o RPT -- RG Ref 2.8 2.6 Capacity [Ah] 2.4 2.2 2 1.8 20 40 60 80 100 120 Half cycle number Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 47
RG vs 1C vs 2C l Chg Charge time RG/1C/2C: l / p / n 1.8 1.7 1.6 Time [h] 1.5 1.4 1.3 1.2 1.1 20 40 60 80 100 120 Cycle number Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 47
l Chg RG vs 1C vs 2C Capacity per RG/1C/2C: l / p / n charge time 2.5 Capacity per charge time [Ah.h -1 ] 2 1.5 1 20 40 60 80 100 120 Cycle number Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 47
Outline MODEL IDENTIFICATION 1 STATE ESTIMATION 2 OPTIMAL SAFE-FAST CHARGING 3 FAULT DIAGNOSTICS 4 5 SUMMARY AND OPPORTUNITIES Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 48
The Battery Safety Problem Samsung Galaxy Note Boeing 787 Dreamliner Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 49
Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 50
Battery Fault Diagnostics Problem Objectives Detect fault Estimate fault size Boeing 787 Dreamliner Scott Moura | UC Berkeley Next-Gen BMS July 6, 2018 | Slide 51
Recommend
More recommend
