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ENE 2XX: Renewable Energy Systems and Control
LEC 04 : Case Studies in Optimal Energy Management:New Energy Vehicles
Professor Scott MouraUniversity of California, Berkeley
Summer 2017
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 1
Outline
1 Intro to Electric Drive Vehicles
2 Hybrid Electric Vehicle Energy Management
3 Optimal PEV Charge Scheduling
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 2
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 3
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 4
HEVs, EVs projected to dominate transportation market in China by 2050
Zhou, Nan, David Fridley, Michael McNeil, Nina Zheng, Jing Ke, and Mark Levine. “China’s Energy and Carbon
Emissions Outlook to 2050,” Lawrence Berkeley National Laboratory Tech Report LBNL-4472E (2011)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 5
Electric Drive Vehicle Basics
Hybrid ElectricVehicles (HEV)
Internal combustion(IC) engine is primaryenergy source
Battery serves asbuffer
Ex: Toyota Prius, Toyota
Camry, Honda Civic, Honda
Accord, Ford Fusion
Plug-in HybridElectric Vehicles
(PHEV)
IC engine & batteryare depletable stores
Fuel at station, chargewith plug
Ex: Chevy Volt, Prius PHEV
All-Electric Vehicle(EV)
Battery only, no engine
Requires charging to“re-fuel”
Ex: Nissan Leaf, Tesla Model S
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 6
Electric Drive Vehicle Basics
Hybrid ElectricVehicles (HEV)
Internal combustion(IC) engine is primaryenergy source
Battery serves asbuffer
Ex: Toyota Prius, Toyota
Camry, Honda Civic, Honda
Accord, Ford Fusion
Plug-in HybridElectric Vehicles
(PHEV)
IC engine & batteryare depletable stores
Fuel at station, chargewith plug
Ex: Chevy Volt, Prius PHEV
All-Electric Vehicle(EV)
Battery only, no engine
Requires charging to“re-fuel”
Ex: Nissan Leaf, Tesla Model S
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 6
Electric Drive Vehicle Basics
Hybrid ElectricVehicles (HEV)
Internal combustion(IC) engine is primaryenergy source
Battery serves asbuffer
Ex: Toyota Prius, Toyota
Camry, Honda Civic, Honda
Accord, Ford Fusion
Plug-in HybridElectric Vehicles
(PHEV)
IC engine & batteryare depletable stores
Fuel at station, chargewith plug
Ex: Chevy Volt, Prius PHEV
All-Electric Vehicle(EV)
Battery only, no engine
Requires charging to“re-fuel”
Ex: Nissan Leaf, Tesla Model S
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 6
Degrees of Hybridization
Guzzella, Lino, and Antonio Sciarretta. Vehicle propulsion systems. Vol. 2. Berlin: Springer,2005.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 7
Brake Specific Fuel Consumption (BSFC)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 8
Conclusions from BSFC Map
Operate in “sweet spot,” i.e. highest efficiency
Options to enhance enhance efficiency
Downsizing
Decouple vehicle velocity from engine velocity
Recuperate kinetic energy
Reduce (or eliminate) dependence on internal combustion engine
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 9
Hybrid Electric Vehicle (HEV)
We have random power demand (driver pedal positions)
We have an energy conversion device (engine) that has a single sweet spot
Add an energy storage device to buffer demand!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 10
Types of Hybrids
Gas-Electric
Diesel-Electric
Diesel-Hydraulic
Fuel Cell-Electric
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 11
Power-split HEV
Battery
Pack
Motor
(Generator)
Generator
(Motor)
Torque Coupler Final Drive
Inverter
Planetary Gear Set
Engine
Mechanical
Path
Electrical
Path
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 12
Power-Split HEV ModelEx: Toyota Prius, Ford Escape Hybrid
Control Inputs
EngineTorqueM1 Torque
State Variables
Engine speedVehicle speedBattery SOCVehicleacceleration(MarkovChain)
SUPERVISORY
CONTROLLER
PLANETARY
GEAR SET
BATTERY PACK
DRIVE
CYCLE
ENGINE
VEHICLE
M1
M2
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 13
Rule-based Energy Management
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 14
HEV Optimal Energy Management
Objective Function:
minu(k),k=0···kf
J =
kf∑k=0
[Fuel Consumption(k) + Emissions(k)]
Constraints:
subject to: x(k + 1) = f(x(k),u(k),w(k)),
SOC(0) = SOC(kf )
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 15
HEV Operation
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 16
Plug-in Hybrid Electric Vehicle (PHEV)
We have random power demand (driver pedal position)
We have an energy conversion device (engine) that has a single sweet spot
Add a second depletable energy store!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 17
Examples of PHEVs
Toyota Prius PHEV
Ford Fusion Energi
Chevrolet Volt
Fisker Karma
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 18
Two Depletable Energy Stores
Battery/
M/Gs
Mechanical
Energy
Chemical
Energy
Tank/
Engine
Wheel+
+
Ptank ωeng Teng=Peng
Pbatt
Pdemand
ωM/G1 TM/G1+
ωM/G2 TM/G2
η1
η2
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 19
All Electric Range
A misleading metric...
A plug-in hybrid’s all-electric range is designated by PHEV-(miles)representing the distance the vehicle can travel on battery power alone. For
example, a PHEV-20 can travel 20 miles without using its internalcombustion engine.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 20
PHEV Optimal Energy Management
Objective Function:
minu(k),k=0···kf
J =
kf∑k=0
[Fuel Consumption(k) + Emissions(k)]
Constraints:
subject to: x(k + 1) = f(x(k),u(k),w(k)),
SOC(k) ≥ SOCmin, ∀ k = 0, · · · , kf
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 21
Charge Depletion-Charge Sustenance vs. Blending
500 1000 1500 2000 2500 3000
0.3
0.4
0.5
0.6
0.7
0.8
Timec(s)
SO
Cc
ChargecdepletingcandcchargecsustainingDynamiccprogramming
ChargecdepletingChargecsustaining
Optimal
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 22
PHEV Energy Management Summary
Depends critically on driving distance between charge events
Judiciously deplete, so you reach min SOC exactly when plugging in
Accurate forecasts of driving pattern are extremely useful
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 23
Use Real-time Traffic Data
Traffic Data
Energy Management
PHEV
Feedback
C. Sun, S. J. Moura, X. Hu, J. K. Hedrick, F. Sun, “Dynamic Traffic Feedback Data Enabled Energy
Management in Plug-in Hybrid Electric Vehicles,” IEEE Transactions on Control Systems
Technology, May 2015. DOI: 10.1109/TCST.2014.2361294
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 24
Approaching Optimal Performance
DP CDCS Static@T Static@S Dyn.@T Dyn.@S50
60
70
80
90
100
Fue
l2Opt
imal
ity2(
v)
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Ter
min
al2S
OC
Standard2Deviation
C. Sun, S. J. Moura, X. Hu, J. K. Hedrick, F. Sun, “Dynamic Traffic Feedback Data Enabled Energy
Management in Plug-in Hybrid Electric Vehicles,” IEEE Transactions on Control Systems
Technology, May 2015. DOI: 10.1109/TCST.2014.2361294
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 25
Electric Vehicle (EV)
We have random power demand (driver pedal position)
We have an energy conversion device (engine) that has a single sweet spot
Remove it! Replace with a battery & motor!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 26
Examples of EVs
Nissan Leaf
Ford Focus EV
Tesla Model S & Roadster
Renault Zoe
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 27
Vehicle-to-Grid (V2G) or Vehicle Grid Integration (VGI)
Plug-in electric vehicles (PEVs) communicate with the grid to providemutually beneficial services, such as demand response through throttled
charging, or selling power to the grid.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 28
Government Initiatives
CA Vehicle-Grid Integration Roadmap
– 1.5M zero emission vehicles in CA by 2025
“Electric vehicle charging creates a reciprocal re-lationship between battery-powered cars and thepower grid in a way that produces mutual ben-efits. Without compromising the driving habitsof consumers, incentives should be pursued asa way to aggregate vehicle charging to developvaluable grid services.”
China Ministry of Science & Technology
– 5 million new energy vehicles on China’s roadsby end of 2020
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 29
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 30
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 30
Source: C. Vlahoplus, G. Litra, P. Quinlan, C. Becker, “Revising the California Duck Curve: An
Exploration of Its Existence, Impact, and Migration Potential,” Scott Madden, Inc., Oct 2016.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 30
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 30
Many fascinating technical questions
How to optimally charge individual PEVs to minimize consumer cost?
How to aggregate PEVs so they can participate in power market?
Can PEVs mitigate variability of renewables?
How to participate in power market, w/o sacrificing user mobility?
Where to optimally locate charging station infrastructure?
Does V2G sacrifice battery life, and therefore affect warranty?
What are the economic benefits? To which stakeholders?
... and more!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 31
Outline
1 Intro to Electric Drive Vehicles
2 Hybrid Electric Vehicle Energy Management
3 Optimal PEV Charge Scheduling
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 32
Problem Statement
Objective: Optimize power flow of engine & battery to satisfy demand.Given:
Power demand time-seriesHEV powertrain model parameters
0 5 10 15 200
5
10
15
20
25
30
Speed [m
/s]
0 5 10 15 20−30
−20
−10
0
10
20
30
Time [min]
Pow
er
Dem
and [kW
]
0 5 10 15 200
5
10
15
20
25
30
Speed [m
/s]
0 5 10 15 20−30
−20
−10
0
10
20
30
Time [min]
Pow
er
Dem
and [kW
]
Power Demand for a Toyota Prius undergoing UDDS cycle
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 33
Modeling - I
Engine Vehicle
Ba-ery
Peng
Pba-
Pdem
Power Balance Peng(k) + Pbatt(k) = Pdem(k), ∀ k = 0, · · · ,N− 1
Battery dynamics E(k + 1) = E(k)− Pbatt(k) ∆t, ∀ k = 0, · · · ,N− 1
E(0) = E0
Net-zero batt energy
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 34
Modeling - I
Engine Vehicle
Ba-ery
Peng
Pba-
Pdem
Power Balance Peng(k) + Pbatt(k) = Pdem(k), ∀ k = 0, · · · ,N− 1
Battery dynamics E(k + 1) = E(k)− Pbatt(k) ∆t, ∀ k = 0, · · · ,N− 1
E(0) = E0
Net-zero batt energy
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 34
Modeling - I
Engine Vehicle
Ba-ery
Peng
Pba-
Pdem
Power Balance Peng(k) + Pbatt(k) = Pdem(k), ∀ k = 0, · · · ,N− 1
Battery dynamics E(k + 1) = E(k)− Pbatt(k) ∆t, ∀ k = 0, · · · ,N− 1
E(0) = E0
Net-zero batt energy E(N) = E(0)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 34
Modeling - I
Engine Vehicle
Ba-ery
Peng
Pba-
Pdem
Power Balance Peng(k) + Pbatt(k) = Pdem(k), ∀ k = 0, · · · ,N− 1
Battery dynamics E(k + 1) = E(k)− Pbatt(k) ∆t, ∀ k = 0, · · · ,N− 1
E(0) = E0
Net-zero batt energy 0.95 E(0) ≤ E(N) ≤ 1.05 E(0)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 34
Modeling - II
Engine Vehicle
Ba-ery
Peng
Pba-
Pdem
Batt energy lims Emin ≤ E(k) ≤ Emax, ∀ k = 0, · · · ,NBatt pwr lims Pmin
batt ≤ Pbatt(k) ≤ Pmaxbatt , ∀ k = 0, · · · ,N− 1
Eng pwr lims 0 ≤ Peng(k) ≤ Pmaxeng , ∀ k = 0, · · · ,N− 1
min. fuel consumption J =∑N−1
k=0 α · Peng(k) ∆t
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 35
Modeling - II
Engine Vehicle
Ba-ery
Peng
Pba-
Pdem
Batt energy lims Emin ≤ E(k) ≤ Emax, ∀ k = 0, · · · ,NBatt pwr lims Pmin
batt ≤ Pbatt(k) ≤ Pmaxbatt , ∀ k = 0, · · · ,N− 1
Eng pwr lims 0 ≤ Peng(k) ≤ Pmaxeng , ∀ k = 0, · · · ,N− 1
min. fuel consumption J =∑N−1
k=0 α · Peng(k) ∆t
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 35
Optimization Formulation
minPbatt(k),Peng(k),E(k)
J =N−1∑k=0
α · Peng(k) ∆t (1)
with equality constraints
Peng(k) + Pbatt(k) = Pdem(k), ∀ k = 0, · · · ,N− 1 (2)
E(k + 1) = E(k)− Pbatt(k) ∆t, ∀ k = 0, · · · ,N− 1 (3)
E(0) = E0 (4)
and inequality constraints
0.95 E(0) ≤ E(N) ≤ 1.05 E(0), (5)
Emin ≤ E(k) ≤ Emax, ∀ k = 0, · · · ,N (6)
Pminbatt ≤ Pbatt(k) ≤ Pmax
batt , ∀ k = 0, · · · ,N− 1 (7)
0 ≤ Peng(k) ≤ Pmaxeng , ∀ k = 0, · · · ,N− 1 (8)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 36
Optimization Formulation - reduced
minPbatt(k),E(k)
J =N−1∑k=0
α∆t · (Pdem(k)− Pbatt(k)) (9)
with equality constraints
E(k + 1) = E(k)− Pbatt(k) ∆t, ∀ k = 0, · · · ,N− 1 (10)
E(0) = E0 (11)
and inequality constraints
0.95 E(0) ≤ E(N) ≤ 1.05 E(0), (12)
Emin ≤ E(k) ≤ Emax, ∀ k = 0, · · · ,N (13)
Pminbatt ≤ Pbatt(k) ≤ Pmax
batt , ∀ k = 0, · · · ,N− 1 (14)
0 ≤ Pdem(k)− Pbatt(k) ≤ Pmaxeng , ∀ k = 0, · · · ,N− 1 (15)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 37
LP Formulation
minimizex cTx (16)
subject to: Ax ≤ b (17)
Aeqx = beq (18)
where the decision variable is given by
x = [Pbatt(0), Pbatt(1), · · · , Pbatt(N− 1),E(0),E(1), · · · ,E(N− 1),E(N)]T (19)
2N + 1 decision variables
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 38
Problem Data
UDDS drive cycle: N = 1369 time steps⇒ 2,739 optimization vars
∆t = 1 sec
α = 0.1 g/(s-kW)
E0 = 0.6 kWh = 2.16 MJ = 50%
Emin = 1.296 MJ = 30%, Emax = 3.024 MJ = 70%
Pminbatt = -15 kW, Pmax
batt = +15 kW
Pmaxeng = 35 kW
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 39
Results
0 5 10 15 200
100
200
300
Cum
m. F
uel C
ons. [g
]
0 5 10 15 200.35
0.4
0.45
0.5
0.55
Battery
Charg
e [%
]
0 5 10 15 20−20
−10
0
10
20
Pow
er
[kW
]
Time [min]
Batt Power
Eng Power
Batt Limits
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 40
Optimal Energy Management Problem
Power Flow Network
Supply = DemandPower Flow dynamics
& constraints
G1
G2
GNG
S1 S2 SNS
D1
D2
DND
Generators Demand
Storage
Applications• USElectricPowerGrid• DistributiongridonUCB
campus• Microgrid inKenyanvillage• Commercialbuildingwith
solar&storage• Ahybridvehicle(e.g.Prius)• Asolarcar/aircraft• Awirelesssensornode
withenergyharvesting
LARGE
small
Figure: Setup for the energy management problem
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 41
Outline
1 Intro to Electric Drive Vehicles
2 Hybrid Electric Vehicle Energy Management
3 Optimal PEV Charge Scheduling
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 42
Problem Statement
Objective: Optimize charge schedule to minimize electricity costGiven:
Time-varying price signalEV battery model parameters
00:00 04:00 08:00 12:00 16:00 20:00 24:000
5
10
15
20
25
30
35
Time of Day
Ele
ctr
icity C
ost
[ce
nts
/kW
h]
Hypothetical time-varying electricity price
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 43
Modeling - I
R
VT
I(t)
+ _Voc
+_
Integrator SOC(k + 1) = SOC(k) + ∆tQcap
I(k), k = 0, · · · ,N− 1
SOC(0) = SOC0
Kirchoff’s voltage law V(k) = Voc + RI(k), k = 0, · · · ,N
Electric Power P(k) = I(k)V(k) = VocI(k) + RI2(k), k = 0, · · · ,N
Charging cost J =∑N−1
k=0 c(k)[Voc · I(k) + R · I2(k)
]∆t
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 44
Modeling - I
R
VT
I(t)
+ _Voc
+_
Integrator SOC(k + 1) = SOC(k) + ∆tQcap
I(k), k = 0, · · · ,N− 1
SOC(0) = SOC0
Kirchoff’s voltage law V(k) = Voc + RI(k), k = 0, · · · ,N
Electric Power P(k) = I(k)V(k) = VocI(k) + RI2(k), k = 0, · · · ,N
Charging cost J =∑N−1
k=0 c(k)[Voc · I(k) + R · I2(k)
]∆t
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 44
Modeling - I
R
VT
I(t)
+ _Voc
+_
Integrator SOC(k + 1) = SOC(k) + ∆tQcap
I(k), k = 0, · · · ,N− 1
SOC(0) = SOC0
Kirchoff’s voltage law V(k) = Voc + RI(k), k = 0, · · · ,N
Electric Power P(k) = I(k)V(k) = VocI(k) + RI2(k), k = 0, · · · ,N
Charging cost J =∑N−1
k=0 c(k)[Voc · I(k) + R · I2(k)
]∆t
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 44
Modeling - I
R
VT
I(t)
+ _Voc
+_
Integrator SOC(k + 1) = SOC(k) + ∆tQcap
I(k), k = 0, · · · ,N− 1
SOC(0) = SOC0
Kirchoff’s voltage law V(k) = Voc + RI(k), k = 0, · · · ,N
Electric Power P(k) = I(k)V(k) = VocI(k) + RI2(k), k = 0, · · · ,N
Charging cost J =∑N−1
k=0 c(k)[Voc · I(k) + R · I2(k)
]∆t
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 44
Modeling - II
R
VT
I(t)
+ _Voc
+_
SOC lims SOCmin ≤ SOC(k) ≤ SOCmax, k = 0, · · · ,NCurrent lims 0 ≤ I(k) ≤ Imax, k = 0, · · · ,N− 1
Final SOC SOC(N) ≥ EQcapVoc
+ SOCmin
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 45
Modeling - II
R
VT
I(t)
+ _Voc
+_
SOC lims SOCmin ≤ SOC(k) ≤ SOCmax, k = 0, · · · ,NCurrent lims 0 ≤ I(k) ≤ Imax, k = 0, · · · ,N− 1
Final SOC SOC(N) ≥ EQcapVoc
+ SOCmin
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 45
Optimization Formulation
minI(k),SOC(k)
J =N−1∑k=0
c(k)∆t VocI(k) + c(k)∆tR I2(k) (20)
with equality constraints
SOC(k + 1) = SOC(k) +∆t
QcapI(k), k = 0, · · · ,N− 1 (21)
SOC(0) = SOC0 (22)
and inequality constraints
SOCmin ≤ SOC(k) ≤ SOCmax, k = 0, · · · ,N (23)
0 ≤ I(k) ≤ Imax, k = 0, · · · ,N− 1 (24)
SOC(N) ≥ E
QcapVoc+ SOCmin (25)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 46
QP Formulation
minimizex1
2xTQx + RTx (26)
subject to: Ax ≤ b (27)
Aeqx = beq (28)
where decision variable is
x = [I(0), I(1), · · · , I(N− 1),SOC(0),SOC(1), · · · ,SOC(N− 1),SOC(N)]T (29)
2N + 1 decision variables
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 47
Problem Data
Given time-varying price: N = 96 time steps⇒ 193 optimization vars
∆t = 15 min
Qcap = 13.8 A-hr, Voc = 363V, R = 1.1 Ohms
SOC0 = 0.2
SOCmin = 0.1, SOCmax = 0.9
Imax = 9.66 A
E = 14.4 MJ
assume PEV plugged-in 16:00 - 24:00
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 48
Results
0 5 10 15 200
2
4
6
8
c(k ) [cents/kW h ]
P (k ) [kW ]
00:00 04:00 08:00 12:00 16:00 20:00 24:000
0.2
0.4
0.6
0.8
1
Battery
SO
C
Time of Day
Figure: Results for Optimal PEV Charge Schedule.Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - New Energy Vehicles Slide 49