Power systems and Queueing theory: Storage and Electric Vehicles - PowerPoint PPT Presentation

Power systems and Queueing theory: Storage and Electric Vehicles (Joint work with Lisa Flatley, Richard Gibbens, Stan Zachary, Seva Shneer) James Cruise Maxwell Institute for Mathematical Sciences Edinburgh and Heriot-Watt Universities December 14, 2017

Plan Lecture 1: Storage and Arbitrage Lecture 2: Storage, Buffering and Competition Lecture 3: Stability and Electric Vehicles

Today: 1 Arbitrage Continued Reminders Competition 2 Extending to Buffering Stochastic Dynamic Programming Problem and Reformulation Price Taker Example Price Maker General Theory 3 Directions of Research Multiple Stores and Resource Pooling Demand Side Response Stochastic Prices

Reminder: Cost function At any ( discrete ) time t , C ( x ) t buy P P x sell C t ( x ) = cost of increasing level of store by x (positive or negative) Assume convex (reasonable). This may model market impact efficiency of store rate constraints

Reminder: Problem Let S t = level of store at time t , 0 ≤ t ≤ T . S 0 = S ∗ S T = S ∗ Policy S = ( S 0 , . . . , S T ), 0 (fixed), T (fixed). Define also x t ( S ) = S t − S t − 1 ( energy “bought” by store at time t – positive or negative) Problem: minimise cost T � C t ( x t ( S )) t =1 subject to S 0 = S ∗ S T = S ∗ 0 , T and 0 ≤ S t ≤ E , 1 ≤ t ≤ T − 1 .

N stores in competition We assume the stores are sufficiently large as to have market impact : the activity of each store negatively affects profits which can be made by the others. Model costs at each time t as being derived from a price function p t ( · ), where p t ( x ) is the price per unit traded when the total amount traded is x . Suppose that each store i buys x it at time t . Then store i incurs a total cost over time of T n � � � � x it p t x jt . (*) t =1 j =1 What happens now depends of the RULES OF THE GAME (the mechanism by which the market is cleared ).

Bertrand competition Stores bid “prices” . Any single unconstrained store able to offer lowest price corners entire market. If overcapacity then typically little or no profits to be made.

Bertrand competition Stores bid “prices” . Any single unconstrained store able to offer lowest price corners entire market. If overcapacity then typically little or no profits to be made. CAPITAL COSTS CANNOT BE RECOVERED .

Cournot competition Stores bid “quantities” . Each store optimises its own profit given the activities over time of all the other stores . Result. There exists at least one Nash equilibrium .

Cournot competition Stores bid “quantities” . Each store optimises its own profit given the activities over time of all the other stores . Result. There exists at least one Nash equilibrium . Given some set of strategies x = ( x 1 , . . . , x n ) of all the Proof. stores over all time 1 , . . . , T , suppose that each store i (simultaneously) updates its entire strategy from x i to x ′ i given the activities x j of all the remaining stores j � = i . This defines a mapping ( x 1 , . . . , x n ) → ( x ′ 1 , . . . , x ′ n ) which is continuous and defined on a compact convex set. Hence by the Brouwer fixed point theorem, the above mapping has a fixed point ( x 1 , . . . , x n ). This (by definition) is a Nash equilibrium.

Behaviour of stores At a Nash equilibrium each store tend to “overtrade” (compared to an optimal cooperative solution): it thereby increases its revenue , the excess costs (in terms of price impact) being borne by the other stores .

Example: Competition example 10.0 7.5 Total store level Stores 1 5.0 2 3 2.5 0.0 01/01 02/01 03/01 04/01 05/01 06/01 07/01 08/01 09/01 10/01 11/01 12/01 13/01 14/01 15/01 70 Market clearing price Stores 60 1 2 50 3 40 30 01/01 02/01 03/01 04/01 05/01 06/01 07/01 08/01 09/01 10/01 11/01 12/01 13/01 14/01 15/01

Linearised price functions Assume that, for each t , p t ( x ) = a t + b t x (a reasonable first approximation). Result. There is a unique Nash equilibrium , given by the minimiser ( x 1 , . . . , x n ) of the quadratic function � n T � n n � 2 �� � x it + 1 � � � � x 2 a t 2 b t it + x it (**) t =1 i =1 i =1 i =1 subject to the given rate and capacity constraints on each store. Proof. For each i , minimisation in x i of the above function is equivalent to minimisation of the earlier function (*).

Unconstrained stores in competition Assume linear prices and n stores subject to neither capacity nor rate constraints , and each of which has the same starting and finishing level. Result. At the (unique and necessarily symmetric) Nash equilibrium , the quantity traded per store is proportional to 1 / ( n + 1) and the profit per store is proportional to 1 / ( n + 1) 2 . This follows easily from the observed symmetry of the Proof. solution and the use of strong Lagrangian theory to minimise the function (**) subject to the constraint � n i =1 x it = 0 for all i .

Unconstrained stores in competition Assume linear prices and n stores subject to neither capacity nor rate constraints , and each of which has the same starting and finishing level. Result. At the (unique and necessarily symmetric) Nash equilibrium , the quantity traded per store is proportional to 1 / ( n + 1) and the profit per store is proportional to 1 / ( n + 1) 2 . This follows easily from the observed symmetry of the Proof. solution and the use of strong Lagrangian theory to minimise the function (**) subject to the constraint � n i =1 x it = 0 for all i . Consequence. In comparison to the optimal cooperative solution, the n unconstrained stores in Cournot competition overtrade by a factor 2 n / ( n + 1) make a total profit proportional to 4 n / ( n + 1) 2 .

1 Arbitrage Continued Reminders Competition 2 Extending to Buffering Stochastic Dynamic Programming Problem and Reformulation Price Taker Example Price Maker General Theory 3 Directions of Research Multiple Stores and Resource Pooling Demand Side Response Stochastic Prices

Buffer shocks Major role of storage is to provide very fast response: TV pick-up Buffering renewable generation (forecast errors) Generation failure Leads to cost dependent on storage level. Random fluctuations in storage due to unexpected demands.

Stochastic Dynamic Programme Decision at time t . As before: Current store level s t − 1 Cost functions C t Purchase amount x t Additionally: Random cost D ( s t − 1 + x t ) Random storage level at next time period, s t .

Problem (Diagram)

Solution Strategy Instead of writing down full SDP consider non-standard version: V t − 1 ( s t − 1 ) = min [ C t ( x t ) + A t ( s t − 1 + x t ) + V t ( s t − 1 + x t )] , x t ∈ X t s t − 1 + x t ∈∩ [0 , E t ] (1) C t costs as before, A t ( s t − 1 + x t ) expected cost of the shock (see later) V t the expected future cost under optimal strategy. Recast as a deterministic optimisation which we solve with using Lagrangian methods.

Price taker example Consider special case of cost function: c ( b ) � x , if x ≥ 0 t C t ( x ) = (2) c ( s ) x , if x < 0 . t Then: Proposition Suppose that, for each t, we have c ( b ) = c ( s ) = c t say; define t t s t = argmin s ∈ [0 , E t ] [ c t s + A t ( s ) + V t ( s )] . ˆ (3) Then, for each t and for each s t − 1 , we have ˆ x t ( s t − 1 ) = ˆ s t − s t − 1 provided only that this quantity belongs to the set X t .

Price taker example If the store is not total efficient we need A t is convex. Define: s ( b ) = argmin s ∈ [0 , E t ] [ c ( b ) s + A t ( s ) + V t ( s )] (4) t t and similarly define s ( s ) = argmin s ∈ [0 , E t ] [ c ( s ) s + A t ( s ) + V t ( s )] . (5) t t Proposition Suppose that the cost functions C t are as given by (2) and that the functions A t are convex. Then the optimal policy is given by: for each t and given s t − 1 , take min( s ( b ) if s t − 1 < s ( b )  − s t − 1 , P It ) , t t   if s ( b ) ≤ s t − 1 ≤ s ( s ) x t = 0 , (6) t t max( s ( s ) if s t − 1 > s ( s )  − s t − 1 , − P Ot ) .  t t

Relation to previous work Interested in buffering against wind forecast errors, minimising excess conventional generation. Bejan, Kelly, Gibbens, ”Statistical aspects of storage systems modelling in energy networks” Gast,Tomozei,Le Boudec, ”Optimal storage policies with wind forecast uncertainties”

Function A t Reminder: A t ( s t − 1 + x t ) average cost of the shock Made of two parts: 1 Cost of due to the shock, e.g. energy unserved 2 Cost due to random fluctuation in store level.

Estimating A t ¯ A t ( s t − 1 + x t ) the expected additional cost to immediately returning the level of the store to its planned level s t − 1 + x t by the end of time period. The cost of the energy which will be purchased to rectify the situation as well as penalty costs. Here ¯ A t is readily determinable, since it does not depend on how the store is controlled outside the time period t .

Estimating A t Then ¯ A t is a good approximation of A t if one of the following is true: Linear cost functions, C t ( x ) = c t x . since the optimal level for the store is unchanged. Shocks are rare but expensive. since the major contribution to A t is the cost due to the shock not the readjustment. Approximation can be improved by allowing longer time periods for the coupling. In many applications the value of A t may need to be determined by observation.