STOCHASTIC OPTIMIZATION AND CONTROL
FOR ENERGY MANAGEMENT
Nicolas Gast
Joint work with Jean-Yves Le Boudec,
Dan-Cristian Tomozei
March 2013
1
STOCHASTIC OPTIMIZATION AND CONTROL
Randomness due to:
• Volatility
• Forecast errors
Design of control policies
• Online algorithms
• Dynamic optimization
• Performance guarantees
FOR ENERGY MANAGEMENT
• Storage Management
• Energy scheduling
2
Production scheduling w. forecasts errors
Base load production scheduling
Deviations from forecast
Use storage to compensate
Social planner point of view
Quantify the benefit of storage
Obtain performance baseline
what could be achieved
no market aspects
load
renewables
renewables + storage
Compare two approaches
1.
Deterministic approach
Pump hydro, Cycle efficiency ≈ 80%
try to maintain storage level at ½ of its capacity using updated forecasts
2.
Stochastic approach
Use statistics of past errors.
3
Energy scheduling with delays
production (control)
Easily dispatchable
Used last-minute
Wind forecast
𝑓
𝑊𝑡 𝑡 + 𝑖 𝑖≤48ℎ
𝒇
𝑷𝒕 (𝒕 + 𝒏)
Demand forecast
𝑓
𝐷𝑡 𝑡 + 𝑖 𝑖≤48ℎ
Storage
𝑡+𝑛
𝑡
𝑡+1
…
𝑛 ≈ 6ℎ
time
Unmatched
demand
𝑫𝒆𝒎𝒂𝒏𝒅
Forecast uncertainties
Non-dispatchable
Renewables
Production
surplus
Reserve (Fast
ramping generators)
Losses
(e.g. wind curtailements)
«Large» system (national level)
Example of scheduling heuristics
Fixed offset policy
Needed
generation
Forecasted needed power
Offset
Already commited
generation
Max
?
Forecasted storage level
Max/2
Past
Schedule
Future
Time
Max
Fixed storage level policy
Storage
level
Forecasted
storage level
Max/2
Max
Forecasted storage level
Max/2
Time t
Time t+n
Time
5
Metric and performance (large storage)
Fixed storage
Energy may be wasted when
Storage is full
Unnecessary storage (cycling
efficiency < 100%)
Max
Target=50%
FO +200
FO +0
Fast ramping energy sources
(𝐶𝑂2 rich) is used when
storage is not enough to
compensate fluctuation
Numerical evaluation: data from the UK
(BMRA data archive https://www.elexonportal.co.uk/)
•
National data (wind prod & demand)
•
3 years
•
Corrected day ahead forecast: MAE = 19%
Forecasted storage level
Max/2
FO -200
Fixed offset
Max
Max/2
Questions:
• can we do better?
• How to compute optimal offset?
6
Fixed offset is optimal for large storage
Let
with
Theorem. If the forecast error is distributed as . Then:
1.
(l,g) is a lower bound: for any policy 𝜋, there exists 𝑢 such that:
2. FO is optimal for large storage: if the capacity is > 𝐵𝛿 , then
3.
The optimal offset is for u such that:
=
Target=80%
• Uses distribution of error
• Fixed reserve is Pareto-optimal
Target=50%
Lower bound
Optimal fixed offset
FO +50
FO
7
Scheduling Policies for Small Storage
Dynamic offset policy:
choose offset as a function of forecasted storage level
Stochastic optimal control (general idea)
Compute a value 𝑉 𝐵 of being at storage level B
𝑢 = arginf 𝔼[ cost 𝑢 + 𝑉 𝜙 𝑏, 𝑢 ]
𝑢 ∈offset
Expectation on
possible errors
Instant cost (losses or
fast-ramping energy)
Storage level at
next time-slot
Computation of V: depends on problem
Here: solution of a fixed point equation:
𝑉 𝑏 = 𝑔 + inf 𝔼[ cost 𝑢 + 𝑉 𝜙 𝑏, 𝑢 ]
𝑢 ∈offset
Approximate dynamic programming if state space is too large
Can be extended to more complicated state V(t,B,B’,…)
8
DO outperforms other heuristics
Large storage capacity
(=20h
of average production of wind energy)
Power = 30% of average wind power
Small storage capacity (=3h of
average production of wind energy)
Power = 30% of average wind power
Fixed storage level
Fixed storage level
Fixed Offset
Fixed offset
Dynamic offset
Dynamic
offset
Lower
bound
Fixed Offset & Dynamic offset
are optimal
DO is the best heuristic
Maintaining storage at fixed level: not optimal
There exist better heuristics
9
Conclusion
Maintain the storage at fixed level is not optimal
Statistics of forecast error are important
Our heuristics are close to optimal (and optimal for large capacity)
Social planner point of view
Provide a baseline
Guidelines to design a market structure?
Can be used to dimension storage
Other work:
Economic implications of storage
Does the market leads to a socially optimal use of storage? (partially yes)
Algorithmic aspects for charging EV (Distributed storage systems)
10
Questions ?
[1] Nicolas Gast, Dan-Christian Tomozei, Jean-Yves Le Boudec. Optimal Storage
Policies with Wind Forecast Uncertainties. Greenmetrics 2012, London, UK.
Related work:
[2] Nicolas Gast, Jean-Yves Le Boudec, Alexandre Proutière, Dan-Christian
Tomozei. Impact of Storage on the Efficiency and Prices in Real-Time
Electricity Markets. ACM E-Energy 2013
[3] Bejan, Gibbens, Kelly, Statistical Aspects of Storage Systems Modelling in
Energy Networks. 46th Annual Conference on Information Sciences and
Systems, 2012, Princeton University, USA.
[4] Cho, Meyn – Efficiency and marginal cost pricing in dynamic competitive
markets with friction, Theoretical Economics, 2010
11
Computing the optimal storage
Losses & Gaz used decreases
Develop two rules-of-thumb
to compute optimal storage
characteristics:
Optimal power C is when
P(forecast error >= C) < 1%.
optimal capacity B
Losses + Fast ramping generation
As capacity increases
As maximum power increases
Storage capacity (in AverageWindPower-hour)
Optimal capacity B, when
P(sum of errors over n slots >= B/2) < 1%.
12
Optimal storage and time horizon
Optimal storage power
Optimal storage capacity
We schedule the base production n hours in advance
13