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Slide 1
SMART-Invest: A stochastic, dynamic policy model for optimizing investment in wind, solar and storage
Federal Enegy Regulatory CommissionWorkshop on Increasing Market and Planning Efficiency
Through Software
June 21-23, 2015
Warren B. PowellJavad Khazaei
PENSA LaboratoryPrinceton University
http://energysystems.princeton.edu
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Energy from wind
1 year
Wind power from all PJM wind farms
Jan Feb March April May June July Aug Sept Oct Nov Dec
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Energy from wind
30 days
Wind from all PJM wind farms
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Solar energy
Princeton solar array
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Solar energy
PSE&G solar farms
Sept Oct Nov Dec Jan Feb March April May June July Aug
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Solar energy
Solar from a single solar farm
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Solar energy
Within-day sample trajectories
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99.9 percent from renewables!
Fossil Backup
BatteryStorage
Wind &Solar
20 GW
750 GWhr battery!
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SMART-ISO
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A spectrum of modelsSMART-Invest» Aggregate model of
PJM energy markets and planning process.
» Models entire year in hourly increments.
» No grid model.» But does explicitly
plan day-ahead steam generation, and hour-ahead gas turbines.
» Optimized robust policies using reserve optimization.
» Requires that all loads be covered by wind, solar, storage and fossil.
SMART-ISO» Detailed model of
unit commitment, grid constraints, planning process
» Robust policy using optimized reserves.
» Performs week-long simulations in 5-minute increments.
» 2-3 hours for a single week-long simulation.
Budischak et al» Model is 6 lines of
code:» Store energy if
wind+solar> load» Withdraw if
wind+solar<load» Optimize (total
enumeration) investment in wind, solar and storage.
» Assume grid is available at all times for instantaneous backup.
» Enumerates “28 billion configurations”
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SMART-Invest
Features:» Find the optimal mix of wind, solar and storage, in the
presence of two types of fossil generation:• Slow (steam) generation, which is planned 24 hours in advance• Fast (turbine) generation, which is planned 1 hour in advance• Real-time ramping of all fossils within ramping limits
» Simulates entire year in hourly increments, to capture all forms of variability (except subhourly)
» Minimizes investment and operating costs, possibly including SRECs and carbon tax.
» Able to directly specify the cost of fossil generation (anticipating dramatic reduction in fossils).
» Properly accounts for the marginal cost of each unit of investment.
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The value of storage
The marginal value of storage» On the margin, value of storage can be expensive!
Ener
gy in
stor
age
Time
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8760
,1
min ( ) ( , ( | ))Invi
inv inv opr opr invt t t tx i I
tC x C S X S x
SMART-Invest
The investment problem:
Investment cost in wind, solarand storage.
Capital investment cost inwind, solar and storage
( )oprtX S is the operating policy.
Operating costs of fossil generators,energy losses from storage, misc. operating costs of renewables.
Wind
Solar
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SMART-Invest
Operational planning
» Meet demand while minimizing operating costs» Observe day-ahead notification requirements for
generators» Includes reserve constraints to manage uncertainty» Meet aggregate ramping constraints (but does not
schedule individual generators)
24 hour notification of steam
1 hour notification of gas
Real-time storage and ramping decisions (in hourly increments)
36 hour planning horizon using forecasts of wind and solar
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1) Policy function approximations (PFAs)» Lookup tables, rules, parametric functions
2) Cost function approximation (CFAs)»
3) Policies based on value function approximations (VFAs)»
4) Lookahead policies» Deterministic lookahead/rolling horizon proc./model predictive control
» Stochastic lookahead /stochastic program/Monte Carlo tree search
» “Robust optimization”
Designing robust operating policies
( )( | ) arg min ( , | )
t t
CFAt t tx
X S C S x
X
,' ',..., ' 1
( ) arg min ( , ) ( , )tt t t H
TLA Dt t tt tt tt ttx x t t
X S C S x C S x
( ) arg min ( , ) ( , )t
VFA x xt t x t t t t t tX S C S x V S S x
'' '
' 1( ) arg min ( , ) ( ) ( ( ), ( ))
t
TLA S t tt t tt tt tt tt
t tX S C S x p C S x
xtt , xt ,t1,..., xt ,tT
,' ',..., ( ) ' 1
( ) arg min max ( , ) ( ( ), ( ))tt t t H t
TLA ROt t tt tt tt ttx x w W t t
X S C S x C S w x w
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Stochastic optimization modelsThe objective function
Given a system model (transition function)
We call this the base model.
0
min , ( )T
tt t t
tE C S X S
Decision function (policy)State variableCost function
Finding the best policy
Expectation over allrandom outcomes
1 1, , ( )Mt t t tS S S x W
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Robust policies
Lookahead model:» Objective function
» Reserve constraint:
» Other constraints:• Ramping• Capacity constraints• Conservation of flow in storage• ….
Tunable policy parameter
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Robust policies
Policy search – Optimizing reserve parameter
Low carbon tax, increased usage of slow fossil, requires higher reserve margin ~19 percent
High carbon tax, shift from slow to fast fossil, requires minimal reserve margin ~1 pct
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SMART-Invest
Operational planning model
» Model plans using rolling 36 hour horizon» Steam plants are locked in 24 hours in advance» Gas turbines are decided 1 hour in advance
Slow fossil running units1 24 36
1 24 36Slow fossil running units
1 24 36Slow fossil running units
1 24 36Slow fossil running units
1 24 36Slow fossil running units1 24 36
Slow fossil running units1 24 36
Slow fossil running units1 24 36
Slow fossil running units
…
…
…1 2 3 4 5 8760
1 year
Lock in steam generation decisions 24 hours in advanceThe tentative plan is discarded
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ParallelizationThe simulation» It was very important to model the entire year so that we capture
all forms of variability, including seasonal.» This means solving 8,760 linear programs to compute a single
simulation.» Solution: We divided the problem into 52 weeks, where we would
simulate 8 days, using a 1-day warmup:
Week 1
Week 2
Week 3
Week 4
Week 52
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Parallelization
CPU times with and without parallelization
Objective functions with and without parallelization
» Errors in optimal solution introduced by parallelization were .02, .10 and 1.2 percent, respectively.
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SMART-Invest
Stage 1:Investment
Hour 1 Hour 2 Hour 3 Hr 8758 Hr 8759 Hr 8760…
Wind CapacitySolar Capacity
Battery CapacityFossil Capacity
Wind
Solar
Find search direction
Update investments
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The search algorithm
Gradient-based search» Let
» Compute numerical derivatives:
» Update investment solution
» We then use an adaptive stepsize rule that finds the largest stepsize that produces an improvement (but we limit how small it may be).
8760
1( ) ( ) ( , ( | ))inv inv inv opr opr inv
t t t tt
F x C x C S X S x
( ) ( )( ) inv invinvi i i
invi i
F x F xF xx
( )
( )( )
inv
invwind
invx
inv
invsolar
F xx
F xF xx
1 1( , )n n n nn xx x F x W
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The search algorithm
Robustness of stochastic search algorithm
Different Starting Points
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The search algorithm
Empirical performance of algorithm» Starting the algorithm from different starting points
appears to reliably find the optimal (determined using a full grid search).
» Algorithm tended to require < 15 iterations:• Each iteration required 4-5 simulations to compute the
complete gradient• Required 1-8 evaluations to find the best stepsize• Worst case number of function evaluations is 15 x (8+5) = 195.• Budischak paper required “28 billion” for full enumeration
(used super computer)
» Run times• Aggregated supply stack: ~100 minutes• Full supply stack: ~100 hours
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Policy studies
Base cost parameters» Data taken from eia.gov
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Policy studies
Renewables as a function of cost of fossil fuels
Solar
Wind
$/MWhr cost of fossil fuels
100
80
60
40
20
0Perc
ent f
rom
rene
wab
le Total renewables
0 20 50 60 70 80 90 100 150 300 400 500 1000 3000
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Policy studies
Sensitivity to wind costs
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Policy studies
Sensitivity to CO2 tax.
Slow fossil Fast fossil
Nuclear
WindSolar
“Other” fast fossil
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SMART-Invest
http://energysystems.princeton.edu