On the Use of Multistage Stochastic Programming for the...

Department of Chemical and Biological Engineering

Illinois Institute of Technology

On the Use of Multistage Stochastic

Programming for the Design of Smart Grid

Coordinated Systems

Donald J. ChmielewskiOluwasanmi Adeodu and Jin Zhang

Department of Chemical and Biological EngineeringIllinois Institute of Technology

Minimally

Backed-off

Operating

Point

Different Controller

Tuning Values

Expected

Dynamic

Operating

Regions

Steady-State

Operating

Line

Optimal Steady-State

Operating Point

1



Motivation

Dispatch Capable

Generation Power Grid

Smart Grid Electric Power Network:

Demand

(Consumers)

Renewable

Generation

Responsive

Demand Energy Storage

Existing

Components

Expected

Future

Components

0 5 10 15 20

0

200

400

600

800

time (days)

Po

wer

Req

uir

ed f

rom

Dis

pa

tch

ab

le G

ener

ato

rs

(MW

)

Baseline

Baseline with Renewable Power

0 5 10 15 20

0

200

400

600

800

time (days)

Baseline with Renewable Power

Impact of Storage and DR

2



Building HVAC Systems

Analysis requires details

of operating policy

Multistage Stochastic

Programming (MSP)

framework

59 60 61 62-100

-50

0

50

100

150

200

250

Time (days)

kW

hr

/ d

ay

Heat from Room

Heat to Cooler

Heat to TES Unit

59 60 61 62-50

0

50

100

150

200

250

Time (days)

kW

hr

/ d

ay

Heat from Room

Heat to Cooler

Heat to TES Unit

3

Heat from

BuildingBuilding

Heat from

Environment

Power

Consumption Chiller

Heat to

TES

Thermal

Energy Storage

Heat to

Chiller



Integrated Gasification Combined Cycle

4



Dispatch Capable IGCC

- Respond to Market Prices - Increase Average Revenue

Electricity Price

Opportunity:

100 101 102 103 104 105 106 107 108 109

70

80

90

100

110

Time (days)

Ele

ctr

icit

y V

alu

e (

$/M

W h

r)

5



Simplified view of IGCC

Gasification Block(Includes ASU Distillation,

Gasifier and Acid Gas Removal)

Power Block(Includes Expansion Turbine,

Combustion Turbine, HRSG,

and Steam Turbine)

nASUns,A

nC

ncoal

ns,H2nH2

nG

H2 Storage(MH2)

Compressed

Air Storage(MA)

MACASU Main

Air Compressor

PGPC

PN

6



Dispatch of IGCC Power Generation

100 101 102 103 104 105 106 107 108 109

0

50

100

150

Va

lue (

$/M

W h

r)

100 101 102 103 104 105 106 107 108 109

0

500

1000

1500

Po

wer (

MW

)

100 101 102 103 104 105 106 107 108 109

0

500

1000

Time (days)

Ma

ss (

ton

nes)

Instantaneous

Average

Maximum

PG

Ce

MH2

7



Instantaneous and Average Revenue

100 101 102 103 104 105 106 107 108 109

0

50

100

150

Time (days)

Rev

en

ue (

$1

00

0/h

r)

Dispatch

No Dispatch

8



Dispatch Requires Equipment Upgrade





and Steam Turbine)

nASUns,A

nC

ncoal

ns,H2nH2

nG

H2 Storage(MH2)

Compressed

Air Storage(MA)

MACASU Main

Air Compressor

PGPC

PN

Net Present Value analysis is required

9



Presentation Outline

Motivation for Multistage Stochastic

Programming (MSP)

Review of MSP

Proposed Solution Method for MSP

Future Directions

10



Review of Stochastic Programming

Two-stage Stochastic Program:

bAxxQxcT

x s.t )(min

)(s.t )(min)( where hWyTxyqExQ T

y

x are here-and-now (equipment) variables

y are wait-and-see (operating) variables

are random (stochastic) variables

c and q() are capital and operating costs

h() is the disturbance

ym , m = 1 … M

m , m = 1 … M

11



Review of Stochastic Programming

Scenario Based Approximation:

MmhWyTx

bAxyqpxc

mm

M

m

m

T

mm

T

yx m ...1s.t min

1,

s.t min1

M

m

kmmm

T

mmy

k

T xThWyyqpxcm

Finite support of scenarios: m , m = 1 … M

Each with outcomes: qm = q(m) and hm = h(m)

Each with a probability: pm = p(m)

Corresponding wait-and-see variables: ym , m = 1 … M

Decomposition Methods Iterate over:

12



Multistage Stochastic Programming Variables indexed in time:

y(1), y(2), y(3), …, y(N) and (1), (2), (3), …, (N)

Non-anticipatory constraint requires past decisions cannot be changed:

ym11(1) = ym12(1) = … = ym33(1) and ymn1(2) = ymn2(2) = ymn3(2)

where pmnl = p(m(1), n(2), l(3)) is the joint probability of scenario mnl

MlMnMmhyWyWTx

hyWyWTx

hyWTxbAx

yqpyqpyqpxc

mnlmnlmnl

mnlmnlmnl

mnlmnl

M

m

M

n

M

l

mnl

T

lmnl

M

m

M

n

mnl

T

nmn

M

m

mnl

T

mm

T

...1,...1,...1)3()3()2(

)2()2()1(

)1()1(

s.t

)3()2()1(min

10

10

0

1 1 11 11

If horizon N = 3, then scenario approximation is:

Nested Decomposition

Solution Methods Required Other Solution

Methods Required 13



MSP Operating Policy Solution Methods*

Cost Function Approximations- Uses reserve constraints in place of non-anticipatory constraints

- Sub-optimal due conservatism of reserve constraints

Scenario Approximations - Computationally intensive (as discussed previously)

- Easily extends to equipment design

Policy and Value Function Approximations- Same as Approximate Dynamic Programming (curse of dimensionality)

- A bit difficult to extend to equipment design

Look-Ahead Policies - Same as Economic MPC.

- Closed-loop implementation is non-anticipatory

*Powell AI Magazine 2014

14



Economic MPC for IGCC





and Steam Turbine)

nASUns,A

nC

ncoal

ns,H2nH2

nG

H2 Storage(MH2)

Compressed

Air Storage(MA)

MACASU Main

Air Compressor

PGPC

PN

1

0)(

)()(minN

t

GetP

tPtCG

15



EMPC Simulation

16

0 1 2 3 4 5 6 7 8 9 10

50

100

150

Time (days)

Va

lue

of

Ele

ctri

city

($

/MW

)

Value of Electricity

0 1 2 3 4 5 6 7 8 9 100

500

1000

1500

Time (days)

Po

wer

Gen

era

ted

(M

W)

EMPC

0 1 2 3 4 5 6 7 8 9 100

500

1000

Time (days)

H2 i

n S

tora

ge

(to

nn

es)

EMPC



Design of Smart Grid Coordinated Systems





and Steam Turbine)

nASUns,A

nC

ncoal

ns,H2nH2

nG

H2 Storage(MH2)

Compressed

Air Storage(MA)

MACASU Main

Air Compressor

PGPC

PN

17



Presentation Outline

Motivation for Multistage Stochastic

Programming (MSP)

Review of MSP

Proposed Solution Method for MSP

Future Directions

18



A Solution Method

6.0max

210 HMcc

Monte Carlo

Simulation using

EMPC

Average

Operating Cost

Search over NPV

min { Capital Cost

+ Operating Costs }

Equipment

Size 1,0

Capital Cost =

where

NPV(Equip Size)

is non-convex

19



Local Minima in NPV

Equipment Variables

Ne

t P

rese

nt

Va

lue

20



NPV using a Surrogate Policy

Equipment Variables

Ne

t P

rese

nt

Va

lue

21



Initial Point for Monte Carlo Search

Equipment Variables

Ne

t P

rese

nt

Va

lue

22



Novel Two-Step Solution Procedure

Economic Linear

Optimal Control

(ELOC)

as surrogate policy

Global Search over

approximate NPV

Initial

Search

Point

Monte Carlo

Simulation using

EMPC

Search over NPVmin { Capital Cost

+ Operating Costs }

Equipment

Size

Average

Operating

Cost

23



Minimally

Backed-off

Operating

Point

Expected

Dynamic

Operating

Regions

Steady-State

Operating

Line


Operating Point

Minimally

Backed-off

Operating

Point

Different Controller

Tuning Values

Expected

Dynamic

Operating

Regions

Steady-State

Operating

Line


Operating Point

Economic Linear Optimal Control (ELOC)

iELOCi xLu

24



Economic Linear Optimal Control ELOC for Minimal Operating Cost

0)(

)(

0)(

)(

))(sqrt(

..

)(min

minmax

cos.

,,,,,,

XBYAX

BYAXGGX

XYDXD

YDXD

diag

qqqq

mDsDqpGmBsAsts

qg

T

T

w

T

ux

uxz

zz

zz

ux

top

YXqms

zz

Branch and

Bound with

SDP solver

25



ELOC Simulation

26

0 1 2 3 4 5 6 7 8 9 10

-2000

0

2000

4000

Time (days)

Po

wer

Gen

era

ted

(M

W)

EMPC ELOC

0 1 2 3 4 5 6 7 8 9 10

50

100

150

Time (days)

Va

lue

of

Ele

ctri

city

($

/MW

)


0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

2000

Time (days)

H2 i

n S

tora

ge

(to

nn

es)

EMPC ELOC



Primal Problem

(SDP solver)

Master Problem

(BARON)

Master Problem

Primal Problem

Economic Linear Optimal Control ELOC Based Design (Global Solution)

0)(

)(

0)(

)(

))((

..

)(min

minmax

cos.

,,,,,,

XBYAX

BYAXGGX

XYDXD

YDXD

diagsqrt

qqqq

mDsDqpGmBsAsts

qg

T

T

w

T

ux

uxz

zz

zz

ux

top

YXqms

zz

),()(min maxmin

cos.cos.

,

,,,,,,,

minmax

qqgqg tcaptop

qq

YXqms

zz

Generalized Benders

Decomposition

27



Example of ELOC Based Design

6.0

10

6.0max

2

2

12

2

0

new

G

G

G

G

H

H

H

H

Pcc

Mcc

Capital Cost =

1,02 Hwhere





and Steam Turbine)

nASUns,A

nC

ncoal

ns,H2nH2

nG

H2 Storage(MH2)

Compressed

Air Storage(MA)

MACASU Main

Air Compressor

PGPC

PN

28

1,0G

max

22

max

20 HHH MM

and maxmax0 GGG PP



Discontinuous Non-convex Capital Costs

29

0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

Hydrogen Storage Unit Size - MH2

max (tonnes )

Ca

pit

al

Co

st (

mil

lio

n $

)

0 100 200 300 400 500 600 7000

50

100

150

200

250

New Power Block Size - PG

new (MW)

Ca

pit

al

Co

st (

mil

lio

n $

)



Novel Two-Step Solution Procedure

Economic Linear

Optimal Control

(ELOC)

as surrogate policy

Global Search over

approximate NPV

Initial

Search

Point

Monte Carlo

Simulation using

EMPC


+ Operating Costs }

Equipment

Size

Average

Operating

Cost

30



Genera

ted P

ow

er

(MW

)

H2 in Storage (tonnes)

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

2000

Solution to the Design Problem

31

Genera

ted P

ow

er

(MW

)


0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

2000

Genera

ted P

ow

er

(MW

)


0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

2000

Negative NPV



EMPC with Different Equipment Sizes

32

0 1 2 3 4 5 6 7 8 9 10

50

100

150

Time (days)

Va

lue

of

Ele

ctri

city

($

/MW

)


0 1 2 3 4 5 6 7 8 9 100

1000

2000

Time (days)

H2 i

n S

tora

ge

(to

nn

es)

0 1 2 3 4 5 6 7 8 9 100

1000

2000

Time (days)

Po

wer

Gen

era

ted

(M

W)

ELOC Search Gradient Search



0

1000

2000

3000 0500

10001500

2000

-100

0

100

200

300

400

500

Generated Power (MW)H2 in Storage (tonnes)

NN

PV

($

10

6)

Global Solution?

33

Negative NPV



Global Solution?

34

0 500 1000 1500 2000 25000

500

1000

1500

2000

-100

0

100

200

300

400

500


Generated Power (MW)

NN

PV

($

10

6)

Negative NPV



Global Solution?

35

0 500 1000 1500 2000 2500-100

-50

0

50

100

150

200

250

300

350

NN

PV

($

1

06)


0 500 1000 1500 2000-80

-60

-40

-20

0

20

40

60

80

100

NN

PV

($

1

06)


Negative NPV

MWh/$202eC



Change in Electricity Price Variance?

36

Negative NPV

MWh/$152eC

0500

10001500

20002500 0

500

1000

1500

20000

50

100

150

200

250

300

350

400

450



NN

PV

($

10

6)



0 500 1000 1500 20000

20

40

60

80

100

120

NN

PV

($

1

06)


0 500 1000 1500 2000 25000

50

100

150

200

250

300

NN

PV

($

1

06)


Change in Electricity Price Variance?

37

Negative NPV

MWh/$152eC



Acknowledgements

Former Students:

David Mendoza (PhD, 2013)

Benjamin Omell (PhD, 2013)

Ming-Wei Yang (PhD, 2010)

Jui-Kun (Michael) Peng (PhD, 2004)

Amit Manthanwar (MS, 2003)

Funding:

National Science Foundation (CBET – 1511925)

Wanger Institute for Sustainable Engineering Research (IIT)

38



Conclusions

Identified Multistage Stochastic Programming as the appropriate

framework for the design of smart grid coordinated systems

Scenario based solution procedure seems intractable

EMPC seems reasonable as an operating policy

Proposed a novel two-step solution procedure

Global search using ELOC as a surrogate policy

Followed by gradient search using EMPC

Illustrated that multiple local minima exist

39



Proposed Solution Method

Economic Linear

Optimal Control

(ELOC)

as surrogate policy

Global Search over

approximate NPV

Initial

Search

Point

Monte Carlo

Simulation using

EMPC


+ Operating Costs }

Equipment

Size

Average

Operating

Cost

Constrained ELOC

40



iii

ikikik

iNi

T

iNi

Ni

ik

ik

T

ikik

T

ikux

xx

BuAxx

tsPxxRuuQxxikik

|

|||1

||

1

||||,

..)(min||

iLQRi xLu

Predictive Form of ELOC

iii

ikikik

iNiELOC

T

iNi

Ni

ik

ikELOC

T

ikikELOC

T

ikux

xx

BuAxx

tsxPxuRuxQxikik

|

|||1

||

1

||||,

..)(min||

iELOCi xLu

* see Chmielewski & Manthanwar (2004) for details

Linear Quadratic Regulator

41



Predictive Form of ELOC Constrained ELOC

max

|

min

|||

zzz

uDxDz

ik

ikuikxik

iii

ikikik

iNiELOC

T

iNi

Ni

ik

ikELOC

T

ikikELOC

T

ikux

xx

BuAxx

tsxPxuRuxQxikik

|

|||1

||

1

||||,

..)(min||

42



Constrained ELOC Simulation

EMPC Horizon 24 hours

Constrained ELOC Horizon 3 hours

43

0 1 2 3 4 5 6 7 8 9 10

-2000

0

2000

4000

Time (days)

Po

wer

Gen

era

ted

(M

W)

EMPC ELOC Constrained ELOC

0 1 2 3 4 5 6 7 8 9 10-1000

0

1000

2000

Time (days)

H2 i

n S

tora

ge

(to

nn

es)

EMPC ELOC Constrained ELOC

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