ENE 2XX: Renewable Energy Systems and Control - Microgrids.pdf · Table:Notation and Parameter...

ENE 2XX: Renewable Energy Systems and Control

LEC 05 : Case Study: Microgrid Planning & Control

Professor Scott MouraUniversity of California, Berkeley

Summer 2017

Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 05 - Microgrids Slide 1

Outline

1 Microgrid Planning

2 Microgrid Control


Microgrids - A definition

U.S. DOE Microgrid Exchange Group:

group of interconnected loads & distributed energy resources (DERs)

clear electrical boundary w.r.t. grid

can connect and disconnect from grid

Basic components of a microgrid:

Generation: solar, wind, fuel cell, diesel generator, etc.

Loads: lighting, heating, A/C, etc.

Storage: batteries, supercaps, flywheels, thermal, hydraulic


Microgrid Configuration


Oakland EcoBlock

28 buildings, 110+ inhabitants, mixed ethnicity & economic levels


Problem Statement

Objective: Optimize size of solar PV and energy storage

Given:

Historical electricity loads

Forecast of potential PV generation

Electricity tariff (i.e. cost) structure


Problem Formulation

minimize cb · b + cs · s +N∑

k=0

cG(k) · G(k) CAPEX + OPEX (1)

subject to: s · S(k) + Bd(k) − Bc(k) + G(k) = L(k) Power balance (2)

E(k + 1) = E(k) +

[ηcBc(k) − 1

ηdBd(k)

]∆t battery dynamics (3)

0 ≤ E(k) ≤ b · Emax battery energy limits (4)

0 ≤ Bc(k) ≤ b · Bmax, 0 ≤ Bd(k) ≤ b · Bmax battery power limits (5)

− Gmax ≤ G(k) ≤ Gmax grid power limits (6)

smin ≤ s ≤ smax, bmin ≤ b ≤ bmax solar, batt scale limits (7)


Table: Notation and Parameter Values

Variable Value Units Descriptions,b optimize [-] Scale factors for solar size, battery sizeS(k) data provided [kW] Power generated from solarBd(k),Bc(k) optimize [kW] Batt charge / discharge powerG(k) optimize [kW] Power imported from gridL(k) data provided [kW] Power load of buildingE(k) optimize [kWh] Energy level of batterycb find [USD/kWh] Marginal levelized cost of scaling battcs find [USD/kW] Marginal levelized cost of scaling solarcG(k) data provided [USD/kW] Time-of-use cost of grid-imported power∆t 1 [hr] Time stepηc, ηd 0.9 [-] Battery charge, discharge efficiencyEmax 400 [kWh] Nominal battery energy capacityBmax 100 [kW] Nominal battery power capacityGmax 400 [kW] Maximum grid power (import & export)smin, smax [0,1] [-] Solar scale limitsbmin,bmax [0,5] [-] Battery scale limits


Program Reduction

Eliminate G(k) using G(k) = L(k) − s · S(k) − Bd(k) + Bc(k)


k=0

cG(k) · [L(k) − s · S(k) − Bd(k) + Bc(k)] (8)

subject to: E(k + 1) = E(k) +

[ηcBc(k) − 1

ηdBd(k)

]∆t (9)

0 ≤ E(k) ≤ b · Emax (10)

0 ≤ Bc(k) ≤ b · Bmax, 0 ≤ Bd(k) ≤ b · Bmax (11)

− Gmax ≤ L(k) − s · S(k) − Bd(k) + Bc(k) ≤ Gmax (12)

smin ≤ s ≤ smax, bmin ≤ b ≤ bmax (13)


Uncertain Load and Solar

Optimization variables are b, s,E(k),Bd(k),Bc(k). We still have a LP. In (8)and (12), the red vars denote random quantities: load L(k), solar power S(k).

Assume L(k),S(k) are independent Gaussian random variables:

L(k) ∼ N(L(k), σ2

L (k))

(14)

S(k) ∼ N(S(k), σ2

S(k))

(15)


Chance Constraints - I

Focus on (12) which includes random variables L(k),S(k). Relax into chanceconstraint:

Pr (L(k)− s · S(k) ≤ Bd(k)− Bc(k) + Gmax) ≥ α (16)

To express as second order cone constraint, let u = L(k)− s · S(k) with meanu = L(k)− s · S(k) and variance σ2

u = σ2L (k) + s2σ2

S(k). Then we can re-writeas

Pr

(u− u

σu≤ Bd(k)− Bc(k) + Gmax − u

σu

)≥ α (17)

Note (u− u)/σu is a zero mean, unit variance Gaussian random variable.The probability in (17) is given by normal CDF function

Φ

(Bd(k)− Bc(k) + Gmax − u

σu

)≥ α, where Φ(z) =

−1√2π

∫ z

−∞e−t

2/2dt

(18)


Chance Constraints - II

Consequently, (17) can be expressed as

u + Φ−1(α) · σu ≤ Bd(k)− Bc(k) + Gmax (19)

Replacing mean u = L(k)− s · S(k) and variance σ2u = σ2

L (k) + s2σ2S(k) we get

L(k)− s · S(k) + Φ−1(α) ·√σ2L (k) + s2σ2

S(k) ≤ Bd(k)− Bc(k) + Gmax (20)

Rearranging this expression and assuming Φ−1(α) > 0 (i.e. α > 0.5), weget a second order cone constraint with respect to optimization variabless,Bd(k),Bc(k):√

σ2S(k) · s2 + σ2

L (k) ≤ 1

Φ−1(α)

[s · S(k) + Bd(k)− Bc(k)− L(k) + Gmax

](21)

An identical process applied to lower bound in (12) yields√σ2S(k) · s2 + σ2

L (k) ≤ 1

Φ−1(α)

[−s · S(k)− Bd(k) + Bc(k) + L(k) + Gmax

](22)


SOCP Formulation

To summarize, we converted a stochastic LP into SOCP:


k=0

cG(k) · [L(k) − s · S(k) − Bd(k) + Bc(k)] (23)


[ηcBc(k) − 1

ηdBd(k)

]∆t (24)

0 ≤ E(k) ≤ b · Emax (25)

0 ≤ Bc(k) ≤ b · Bmax, 0 ≤ Bd(k) ≤ b · Bmax (26)√σ2S(k) · s2 + σ2

L (k) ≤ 1

Φ−1(α)

[s · S(k) + Bd(k) − Bc(k) − L(k) + Gmax

](27)√

σ2S(k) · s2 + σ2

L (k) ≤ 1

Φ−1(α)

[−s · S(k) − Bd(k) + Bc(k) + L(k) + Gmax

](28)



SOCP Formulation w/ Expected Operating Cost

Random vars still exist in the objective fcn. Take the expectation.


k=0

cG(k) ·[L(k) − s · S(k) − Bd(k) + Bc(k)

](30)


[ηcBc(k) − 1

ηdBd(k)

]∆t (31)

0 ≤ E(k) ≤ b · Emax (32)

0 ≤ Bc(k) ≤ b · Bmax, 0 ≤ Bd(k) ≤ b · Bmax (33)√σ2S(k) · s2 + σ2

L (k) ≤ 1

Φ−1(α)

[s · S(k) + Bd(k) − Bc(k) − L(k) + Gmax

](34)√

σ2S(k) · s2 + σ2

L (k) ≤ 1

Φ−1(α)

[−s · S(k) − Bd(k) + Bc(k) + L(k) + Gmax

](35)



Possible Extensions

Demand charges, i.e. costs associated with the maximum grid powercD ·maxk G(k)

Transformer sizing, i.e. add transformer cost to the objective functionand scale the grid power limits

EV charging, i.e. a load C(k) which is flexible subject to deliveringsufficient energy by a certain deadline

Islanded operation, i.e. optimize PV and storage size while ensuring atleast 72 hours of islanded operation (G(k) = 0)


Outline

1 Microgrid Planning

2 Microgrid Control


Motivation for Model Predictive Control (MPC)

What we learned so far: Optimization

Uses mathematical model.

Optimizes performance over finite-time horizon.

SDP models stochastic process as Markov chain.

Generates optimal control policy.

Some Questions/Gaps:

Model uncertainty?

Time horizon for my application is VERY long.

I want to forecast inputs with ML, instead of probability distributions


What is MPC?


Applications of MPC

Like Playing Chess!


Applications of MPC

Building Energy Management


Applications of MPC


Receding Horizon Philosophy

- At time t: Solve optimal control problem over a finite future horizon of Nsteps:

minimizeut,··· ,ut+N−1

N−1∑k=0

ct+k(xt+k,ut+k) + ct+N(xt+N),

subject to xt+(k+1) = f(xt+k,ut+k), k = 0, · · · ,N− 1

xt = x(t), [Measurement of States]

umin ≤ ut+k ≤ umax

xmin ≤ xt+k ≤ xmax

[HINT:] Can use DP, NLP, CP, SOCP, QP, LP...

- Only apply the first optimal move: u∗(t) = u∗k

- At time t + 1: Get new measurements, repeat the optimization...

Advantage of repeated online optimization: FEEDBACK!


Receding Horizon Philosophy

Advantage of repeated online optimization: FEEDBACK!


Learn more about MPC

Take ME 231A - Experiential Advanced Control Design

Read: Model Predictive Control, E. F. Camacho and C. Bordons, Springer,2007


SMART HOME ENERGYMANAGEMENT


The Building Solar+Storage Problem

Needs: Optimally manage energy flow between loads, solar, and storage.

Reality: Current controls are mostly heuristic - no models, no data.

Some Motivating Facts

Policy 50% renewables in CA by 2030, 100% in Hawaii by 2045

Climate 2011 Tsunami in Japan→ energy security and reliability

Costs Li-ion battery pack costs decreasing toward 125 USD/kWh

Data Over 50M (43%) of US homes have smart meters

Hybrid Vehicles

Photovoltaics/Grid↔ Engine

Home Demand↔ Driver Power Demand

Battery Storage↔ Battery Storage

PunchlineApply & Extend ∼10 years of HEV Energy Management

Control Research to Building Solar+Storage


Residential Buildings with Solar & Storage

Cloud

with information,

algorithms ...Photovoltaic

Arrays

Utility

Grid

Controller &

Converters

Battery

Load

Demand

BatteryPower

Electronic

PV

ArraysDC/DC

DC Bus AC Bus

AC Loads

DC Loads

Utility

Grid

DC/AC

(b)

(a)


Predictive Controller with Load/Weather Forecasting

Smart Home nGrid

Predic0ve Controller

PV Model

Load Forecaster

S T Pdem Hour of day Day of

Week

SOC

Pdem PPV

PbaB


Single-Family Home Energy Patterns in LA

Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar

0.5

1

1.5

2

2.5

3

3.5

4

Month

Ele

ctric

ityiL

oadi

HkW

Y

HourlyiLoadDailyiLoadMonthlyiLoadYeariAverage



12 14 16 18 20 22 240.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

WeeklyLAverageLTemperatureL(°C)

Wee

klyL

Ave

rage

LLoa

dL(k

W)

Noise

LoadLvs.LTempFitLResult


Artificial Neural Network (ANN)

w

w

w

······

···

····

··

X

H(X,C,σ)

Y=P dem

Ta

Dw

Td

Lh

Future load

demand.

Input Layer Hidden Layer Output Layer

Y = fANN(X), X = [Ta,Dw, Td, Lh] , Y = [Lk+1, · · · , Lk+m]

Y = fANN(X) =N∑i=1

ai · Hi (‖X − Ci‖)

Hi (‖X − Ci‖) = exp

[− 1

2σi2‖X − Ci‖2

]Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 05 - Microgrids Slide 28

Short-term Forecast of Home Load

10

20

30

° C

0 3 6 9 12 15 18 210

1

2

3

HourAofADayAonA2013 09 15 Tuesday

Load

AbkW

)

10

20

30

0 3 6 9 12 15 18 210

1

2

3

Hour of Day on 2013 12 06 Friday

AirATemperautreAonA09 15

RealALoadForecastAL

AirATemperautreAonA12 06

ba)-1

ba)-2

bb)-1


Short-term Forecast of Home Load

0 0y2 0y4 0y6 0y8 1 1y20

0y2

0y4

0y6

0y8

1

RMSE

Em

piric

altC

DF

24 21 18 15 12 9 6 3 00y4

0y43

0y46

0y49

0y52

InputtHistoricaltLoadtLength

Ave

rage

tRM

SE

RMSEtcdftoftLAtDataRMSEtcdftoftBerkeleytData

haA

hbA


Battery and Photovoltaic Cell Models

battery

d

dtSOC(t) = − Ibatt(t)

Q,

Pbatt(t) = VocIbatt(t)− I2batt(t)Rin,

(a)

(b)

(c)

OCV-R

OCV-R-RC

Impedance

X. Hu, S. Li, and H. Peng, “A comparative study of equivalent circuit modelsfor Li-ion batteries,” Journal of Power Sources, vol. 198, pp. 359-367, 2012.

photovoltaics

Vd = Vcell + IpvRs,

I = Isc − Is

[e

(qVd

AkTpv(t) ) − 1

]− Vd

Rp,

Is = Is,r

(Tpv(t)

Tr

)3

eqEbgAk

(1Tr− 1

Tpv(t)

),

Isc = [Isc,r + KI(Tpv(t)− Tr)]Spv(t)

1000,

Ppv(t) = ncellVcell(t)I(t)

Isc Vd

+

-

Rp

Rs

Vcell

+

-

I

S

G. Vachtsevanos and K. Kalaitzakis, “A hybrid photovoltaic simulator for utility inter-active studies,” IEEE Transactions on Energy Conversion, no. 2, pp. 227-231, 1987.


Internet-based Data Feeds

Smart Home nGrid


PV Model

Load Forecaster


Week

SOC

Pdem PPV

PbaB


Internet-based Data Feeds


Cloud-Enabled Control

Smart Home nGrid


PV Model

Load Forecaster


Week

SOC

Pdem PPV

PbaB


Nonlinear MPC Formulation

minimize Jk =

∫ (k+Hp)∆t

k∆t[λ1celecPrice(t)G(t) + λ2cCO2(t)G(t)] dt,

subject tod

dtSOC(t)=− Ibatt(t)

Q, [Battery]

0=Voc(SOC)Ibatt(t)− I2batt(t)Rin − B(t),

0 = hPV(P(t),S(t), T(t)), [Photovoltaic]

0 = G(t) + ηddηdaP(t) + ηdaB(t)− L(t), [Pwr Balance]

SOCmin ≤ SOC(t) ≤ SOCmax, Iminbatt ≤ Ibatt(t) ≤ Imax

batt ,

Bmin ≤ B(t) ≤ Bmax, Gmin ≤ G(t) ≤ Gmax,

d̂ ((k + n)∆t)=fforecast(d(k∆t), · · · ,d((k − Hh)∆t)), n = 1, · · · ,Hp

‘state’ = SOC(t), ‘control’ = G(t),B(t), ‘exo. input’ = [L(t),S(t), T(t)]T

∆t = 1 hr, Hp = 6 hrs, Hh = 6 hrsSolved via Dynamic Programming


Model Predictive Control w/ Cloud-enabled Forecasts

0f40f60f8

1

SO

C

2

0

2

4

Pow

er7y

kWG

Monf Tuef Wedf Thuf Frif Satf Sunf15161718

Cen

ts&k

Wh

Load7DemandPV7PowerBattery7PowerGrid7Power

Electric7Rate7from7PGVE

Optimize for Grid Electricity Cost


Model Predictive Control w/ Cloud-enabled Forecasts

0I40I60I8

1

SO

C

2

0

2

4

Pow

ergG

kWb

MonI TueI WedI ThuI FriI SatI SunI0I4

0I450I5

0I55

kg/k

Wh

LoadgDemandPVgPowerBatterygPowerGridgPower

CarbongEmissiongfromgCAISO

Optimize for Marginal CO2 Produced from Power Plants


Control Horizon Length?Short-Term Greedy vs. Long-Term Planning

1 5 9 13 17 21 2555

60

65

70

75

80

85

90

95

100

105

Control Horizon Length (Hour)

Nor

mal

ized

Cos

t (%

)

MPC with PB

Without PBLower bound


Load Forecasting - how accurate is accurate enough?

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

62

64

66

68

70

72

LoadiDemandiForecastiRMSE (kW)

Nor

mal

ized

iCos

tip)

UDPiOptimalUniformiDistributionRBF−NNiResult


Battery Health Aware

minimize Jk =

∫ (k+Hp)∆t

k∆t[λ · cElecPrice(u, t) + (1− λ) · Qloss(u)]2 dt

J.Wang, et al., Journal of PowerSources (2010), doi:10.1016/j.jpowsour.2010.11.134


http://dx.doi.org/10.1016/j.jpowsour.2010.11.134


70 75 80 85 90 95 100 105 1100

0.2

0.4

0.6

0.8

1

1.2

Monthly Electric Cost (USD)

Mon

thly

Bat

tery

Cap

acity

Los

s (%

) λ=1, Electric Cost Emphasis

Battery Health

Emphasis, λ=0

λ=0.89

λ=0.45

Utopia



0.3 0.45 0.6 0.75 0.9−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Battery SOC

Cel

l C−

rate

(a)

Off−peak On−peak0

50

100

150

200

250

300

350

400

450

Period

Cum

ulat

ive

Grid

Pow

er (

kW)

(b)λ=1λ=0.83λ=0

λ=1λ=0.83λ=0

C. Sun, F. Sun, S. J. Moura, “Nonlinear predictive energy management of residential buildings withphotovoltaics & batteries” Journal of Power Sources. Sept 2016, DOI: 10.1016/j.jpowsour.2016.06.076



Excited about this research?Reading Materials

C. Sun, F. Sun, S. J. Moura, “Nonlinear predictive energy management ofresidential buildings with photovoltaics & batteries” Journal of Power Sources.Sept 2016, DOI: 10.1016/j.jpowsour.2016.06.076

X. Wu, X. Hu, S. J. Moura, X. Yin, V. Picket, “Stochastic Control of Smart HomeEnergy Management with PEV Energy Storage and Photovoltaic Array,” Journalof Power Sources, Nov 2016. DOI: 10.1016/j.jpowsour.2016.09.157

X. Wu, S. J. Moura, X. Hu, X. Yin, “Stochastic Optimal Energy Management ofSmart Home with PEV Energy Storage,” to appear in IEEE Transactions on SmartGrid. DOI: 10.1109/TSG.2016.2606442

E. Burger, S. J. Moura, “Gated Ensemble Learning Method for Demand-SideElectricity Load Forecasting,” Energy and Buildings, Dec 2015. DOI:10.1016/j.enbuild.2015.10.019.

E. Burger, S. J. Moura, “Building Electricity Load Forecasting via StackingEnsemble Learning Method with Moving Horizon Optimization,” eScholarship,Dec 2015. http://escholarship.org/uc/item/6jc7377f.




http://dx.doi.org/10.1109/TSG.2016.2606442

http://dx.doi.org/10.1016/j.enbuild.2015.10.019

http://dx.doi.org/10.1016/j.enbuild.2015.10.019

http://escholarship.org/uc/item/6jc7377f

ENE 2XX: Renewable Energy Systems and Control - Microgrids.pdf · Table:Notation and Parameter...

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