An Adjustable Robust Optimization Approach to Scheduling of Continuous Industrial Processes Providing Interruptible Load

Qi Zhang a, Michael F. Morari

b, Ignacio E. Grossmann a,

Arul Sundaramoorthy c, Jose M. Pinto

c

a Center for Advanced Process Decision-making (CAPD), Department of Chemical Engineering, Carnegie Mellon University

b Department of Chemical and Bioengineering, ETH Zurich

c Praxair, Inc., Business and Supply Chain Optimization R&D

Enterprise-wide Optimization Meeting Pittsburgh, September 2015

A power grid matches electricity supply and demand

Objective: Supply = Demand

Generation Transmission Consumption

Base Load Generation

Peak Load Generation

Renewables

Residential

Commercial

Industrial

Transmission Network

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Operating reserves ensure the reliability of the grid

Supply Demand

Supply Demand

Supply Demand

Shortage of supply compensated by generators with short ramp-up times

Referred to as operating reserve (spinning and non-spinning)

To provide spinning reserve, generators have to be already running

Expensive, requires underutilization of generation facilities

Supply-demand mismatch eliminated by reducing electricity consumption

Also referred to as interruptible load Can be regarded as spinning reserve Increases flexibility in the grid, reduces

the need for building new power plants

<

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Interruptible load provides new opportunities for power-intensive industries

Interruptible load is specified as the maximum possible reduction in electricity consumption that can be requested by the grid operator

Provision of interruptible load is encouraged by attractive financial incentives → great potential benefits for large industrial electricity consumers

Target power consumption

Minimum power consumption

Time

Power Consumption

Interruptible Load

Load reduction requested

Load reduction requested

Actual power consumption

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Challenge lies in the proper modeling of process flexibility and uncertainty

Target power consumption

Minimum power consumption

Time

Power Consumption

Interruptible Load

Load reduction requested

Load reduction requested

Need detailed scheduling model that incorporates uncertainty.

How flexible is the process? How much interruptible load potential does the plant really have?

How does a load reduction event impact the process? Can we still meet

all product demands?

Load reduction demand is not known in advance. How do we

consider this uncertainty?

Actual power consumption

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Plant is represented by different operating modes

Assume that plant can operate in different operating modes

For each operating mode, we need to know the range of possible production rates and the corresponding electricity consumption

Approximate the feasible operating region by a set of polyhedral regions in the product space

In each subregion, the electricity consumption is approximated by a linear function of the production rates

Surrogate model is created by using data from the real process or a detailed mathematical model1

3a

3b

2

P1

P2

1

1. Zhang et al. (2015). Optimization and Engineering.

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Use surrogate model in a scheduling framework that considers transitions between operating modes

Time horizon is discretized into time intervals of equal length

Length of time interval depends on process characteristics and electricity price

For every time interval, operating mode and production rates are determined

Constraints on mode transitions can be imposed1

Time [h] 0 1 2 -1 168 167

Off Startup On after 4 hours after at least

24 hours

after 2 hours

1. Mitra et al. (2012). Computers and Chemical Engineering.

Shutdown after at least 12 hours

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Providing interruptible load is associated with high uncertainty

Load reduction is requested in case of contingency → Not known in advance: When? How much? For how long?

To provide interruptible load, load reduction upon request has to be guaranteed

Still financially attractive because payment is made regardless how much load reduction has actually been requested

0 1 2 3 4 5 6 7 8 9

Interruptible load 𝐿𝐿

𝑡 0 1 2 3 4 5 6 7 8 9

Interruptible load 𝐿𝐿

𝑡

Worst case: maximum load reduction Best case: no load reduction

Assuming worst case is too conservative. Need a more realistic approach.

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Define uncertainty set of appropriate size

Apply robust optimization approach to guarantee feasibility

In practice, load reduction is only requested a few times in months1

Apply the following “budget” uncertainty set2 (limits the number of time periods in which maximum load reduction can be requested):

𝐼𝐿 𝐿𝐿

Γ

1. EnerNOC (www.enernoc.com/our-resources/brochures-faq) 2. Zhang et al. (2015). AIChE Journal.

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Apply adjustable robust optimization approach to incorporate recourse decisions

Decisions have to depend on the realization of the uncertainty

“Traditional” robust optimization: No recourse, only “here-and-now” decisions

Adjustable robust optimization1: Recourse decision variables are specified as functions of the uncertain parameters

For tractability reasons, restrict to affine functions:

1. Ben-Tal et al. (2004). Mathematical Programming.

actual production

planned production

uncertain load reduction

decision coefficient

multistage linear decision rule

𝑝𝑡 and 𝑞𝑡𝑡 are decision variables 𝜁 defines the extent of recourse

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Proposed model is applied to a real-world industrial case study provided by Praxair

MILP Model:

Benchmark Case: • Cryogenic air separation plant

• Products: liquid oxygen (LO2), liquid nitrogen (LN2)

• 90% plant utilization

minimize electricity cost + product purchase cost - interruptible load sales

subject to surrogate process model mass balances energy balances mode transition constraints initial conditions terminal constraints

for all possible realizations of the uncertainty, i.e. ∀ 𝒘 ∈ 𝑾(𝑰𝑰)

max( )

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If no interruptible load is provided, the solution suggests operating the plant such that high-price periods are avoided

0

40

80

120

160

0

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8

12

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0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

Pric

e [$

/MW

h]

Elec

tric

ity C

onsu

mpt

ion

Time [h]

Target Electricity Consumption Electricity Price

-10

-8

-6

-4

-2

0

2

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0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

LO2

In a

nd O

ut F

low

s

LO2

Inve

ntor

y Le

vel

Time [h] LO2 Production LO2 Purchase LO2 Demand LO2 Inventory

Final inventory level reaches the minimum

(= initial inventory)

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Providing interruptible load reduces the total operating cost, even with minimum extent of recourse (𝜻 = 𝟎)

Inventory buffer built to ensure

feasibility

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0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

Pric

e [$

/MW

h]

Elec

tric

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onsu

mpt

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Time [h]

Provided Interruptible Load Target Electricity Consumption Electricity Price Interruptible Load Price

-10

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-6

-4

-2

0

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0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

LO2

In a

nd O

ut F

low

s

LO2

Inve

ntor

y Le

vel

Time [h] Target LO2 Production Target LO2 Purchase LO2 Demand Production Recourse Purchase Recourse Target LO2 Inventory

1.2% cost savings

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Cost savings increase by 50% if greater extent of recourse is considered (𝜻 = 𝟐𝟐)

1.8% cost savings

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120

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0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

Pric

e [$

/MW

h]

Elec

tric

ity C

onsu

mpt

ion

Time [h] Provided Interruptible Load Target Electricity Consumption Electricity Price Interruptible Load Price

-10

-8

-6

-4

-2

0

2

4

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20

24

28

32

36

40

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0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

LO2

In a

nd O

ut F

low

s

LO2

Inve

ntor

y Le

vel

Time [h] Target LO2 Production Target LO2 Purchase LO2 Demand Production Recourse Purchase Recourse Target LO2 Inventory

No inventory buffer required

50% increase

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Cost savings depend on the level of plant utilization

Lower plant utilization implies higher process flexibility, which allows more effective load shifting

Higher plant utilization implies higher target production levels, which allow more interruptible load to be provided

0

0.5

1

1.5

2

2.5

50 55 60 65 70 75 80 85 90 95 100

Cos

t Sav

ings

Level of Plant Utilization [%]

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Highest cost savings are achieved at a plant utilization of 95%.

Conclusions

Developed robust MILP scheduling model for continuous power-intensive plants that can provide interruptible load

Proposed adjustable robust optimization approach accounts for uncertainty in load reduction demand and considers recourse actions in the form of linear decision rules

Proposed model has been applied to an industrial case study provided by Praxair

Results show the financial benefit of providing interruptible load, and demonstrate the effect of the considered extent of recourse

Further insight: Largest cost savings are achieved at a high, yet not maximum level of plant utilization

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