advertisement

Convex Hull Stochastic Dynamic Programming Applied to Electric Vehicle Charging and Frequency Regulation Bids Jonathan Donadee and Marija IliΔ jdonadee@andrew.cmu.edu, milic@ece.cmu.edu Electric Energy Systems Group, Electrical and Computer Engineering Solution Algorithm o avoids using more integer variables in backwards recursion ο§ Optimal Value Function – Vh(Eh) o Def: Expected cost of making optimal decisions from stage h through the end of stage H given that the state of charge is Eh at the start of hour h 1. Discretize the feasible state space at the final decision time, Ebatt(H), into N points rate to follow control signal οΆ Existing Deterministic Optimization Methods Ignore Regulation Signal Effect on State of Charge ο§ Assume integrated signal energy is zero over one hour, or use expected value o Bad Assumption π ππ» πΈππππ π,π» = minππ,π» ,π΅π,π» πΌπ πΆπ» πΈππππ π,π» , ππ,π» , π΅π,π» , π π» + ππ»+1 πΈππππ π,π»+1 20 3. 15 Approximate ππ» β with ποΏ½π» β , a piecewise-linear function on the convex hull of the points ππ» πΈππππ π,π» 10 Approximation of Vh by Convex Hull Solving for V5(16.8) Using 30 Sample Regulation Signals 0.35 0.25 Solution 0.3 0 -1 -0.5 0 0.5 Optimal Value Function Cost ($) 5 Convex Hull 1 0.25 Hourly Sum of Normalized Regulation Energy Cost ($) ο§ EVs providing regulation First to model pro-rated contract penalties 25 Counts οΆ A Stochastic Method is Needed to Consider the Distribution of the Regulation Signal οΆ Contributions: ο§ First tractable stochastic model of 2. For i= 1, …, N solve Histogram of PJM Regulation Signal over 1 Week Providing Regulation by Charging Only 0.2 0.15 0.2 0.15 ποΏ½6 0.1 0.05 Energy at Pavg 17 18 19 20 21 22 23 24 22 20 Energy Bounds (from Regulation Bid) 18 Energy at Pavg 16 14 Actual Energy State 16 18 20 22 1 24 Energy (kWh) For i = 1, …, N solve πβ πΈππππ π,β = minππ,β ,π΅π,β πΌπ πΆβ πΈππππ π,β , ππ,β , π΅π,β , π βπ + πβ+1 πΈππππ π,β+1 οΆ Must solve for optimal average charge rate and regulation bids 24 12 For h= H-1, H-2, …, 1 π½π is a two stage deterministic equivalent stochastic MILP with 30 sample π regulation signals, π π» A. πͺπ -Stage h costs include i. Energy purchase costs and regulation revenues ii. Adjustments to energy purchase cost caused by following the regulation signal or reaching Emax iii. Pro-rated penalty for violating regulation contract for each sample B. Regulation Signal 1 26 Energy Sample Paths Hour 5 Initial SOC (kWh) ο§ Simulation Results Penaltyπ = Q ππ βt B ∗ ππ οΏ½ πΌπ‘ t 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 Time 16 o Inconvenience cost can be given by EV owner ($/hr late) Battery might reach capacity before regulation contract expires, breaking contract: o pro-rated penalty is a function of time, not energy o Binary indicator variables (π°π ) indicate if contract was violated in each sub-hourly timestep ο§ οΆ Problem Size: 1593 variables, 390 binaries, 3,784 equations ο§ Solved 133 times in algorithm οΆ Solver: GAMS w/ XPRESS on Intel 6 core 3.2Ghz cpu οΆ Results ο§ total simulation time ~3min ο§ ~75% of pts on the convex hull Energy Bounds (from Regulation Bid) 0 ο§ Pavg- average charge rate ο§ B – Regulation Bid (identical up and down) ο§ B ≤ Pavg and B ≤ Pmax - Pavg οΆ State of charge takes a random walk ο§ Battery might not be charged by desired time: PJM DA prices (12/1/2011), Plug in at midnight, Unplug at 7am 21 point state space discretization Hour 6 0.05 οΆ οΆ Setup ο§ Data:1 vehicle , Pmax=8kW, E0=12 kWh, Emax=24kWh, 0 0.1 3. Simulation Normalized 5 Minute Integral of Regulation Signal οΆ Convex Hull Stochastic Dynamic Programming ο§ Use a piecewise-linear, function to approximate the optimal value function οΆ Smart Charging of Electric Vehicles allows demand side participation in Energy and Ancillary Services Markets ο§ Intermittent renewable generation requires more ancillary services ο§ EV owners want to minimize charging costs ο§ EVs can provide Secondary Frequency Regulation by adjusting EV charge Battery State of Charge (kWh) Background and Contributions π½π―+π - cost function of the battery state of charge at scheduled vehicle unplug time A. Includes an Inconvenience penalty for the remaining time to reach Emax by charging at Pmax 2 3 4 5 6 Simulation Timestep (hr) 7 4 4.5 6 5.5 5 Simulation Timestep (hr) 6.5 7 8 Optimal Decisions Hour 1 2 3 4 5 6 7 B 0 0 0 4 4 4 0 Pavg 0 0 0 4 4 4 0 Conclusions and Future Work οΆ Conclusions ο§ Driver inconvenience is almost always avoided with penalty of $20/hr o Pavg is biased to finish early in last hour Regulation contract is almost always broken in the last hour, sometimes earlier Convex Hull is a good approximation of optimal value function If Pavg>4 kWh, then Pavg+B=8, else B=Pavg ο§ ο§ ο§ οΆ Future Work ο§ Understand convexity properties of the value function ο§ Uncertain energy and regulation prices, adds more states ο§ Compare with other methodologies ο§ Multiple vehicle bid aggregation ο§ Apply method to other storage devices (Stationary Batteries, Flywheels) T