an adjustable robust optimization approach to scheduling...
TRANSCRIPT
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
4
8
12
16
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
4
12
16
20
24
28
32
36
40
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
0
40
80
120
160
0
4
8
12
16
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
16
20
24
28
32
36
40
44
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
0
40
80
120
160
0
4
8
12
16
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
16
20
24
28
32
36
40
44
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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