Congestion Management in a Smart Grid via Shadow Prices

Transcription

1 Congestion Management in a Smart Grid via Shadow Prices Benjamin Biegel, Palle Andersen, Jakob Stoustrup, Jan Bendtsen Systems of Systems October 23,

2 This presentation use of distributed Receding Horizon Control (RHC) to control flexible consumption 2

3 Interconnected flexible consumers p 2,1 C 7 C 2 C 1 p 1,2 p 1,1 f 4 C 6 f 5 p 2,2 f 1 f 6 p 3,3 C 8 f 7 f 3 f 2 C 4 p 3,1 C 3 p 2,3 p 3,2 C 5 3

4 Dynamics and constraints for flexible consumers flexible consumers belong to different balancing responsible parties (BRPs) consumers under BRP i have hourly energy consumptions p i = (p i,1,...,p i,mi ) R m i p i (k) = p i (k)+ p i (k) where p i R m i is the controllable consumption and p i R m i is the uncontrollable consumption. this gives a stored energy e i (k +1) = D i e i (k)+ p i (k) where D i R m i m i is diagonal with diagonal elements describing the proportional drain loss of each energy storage. 4

5 Model of the flexible consumers under BRP i. p i Flexible Consumers p i e max i p i e i (I D i)e i power and capacity constraints p min i p i (k) p max i 0 e i (k) e max i 5

6 Model of the flexible consumers under BRP i. p 2,1 C 7 C 2 C 1 p 1,2 C 3 p 1,1 f 4 f 7 f 3 C p 4 3,1 C 6 f 5 p 2,2 f 1 f 6 p 3,3 p 2,3 p 3,2 C 5 f 2 C 8 partial flow caused by BRP i t i R n L + t i (k) = R i p i (k) given by 6

7 Coupling line constraints total flows f = (f 1,...,f nl ) R n L + over the distribution lines f(k) = n B i=1 t i (k) where f j is the flow through line j. distribution line capacities are given by f max R n L +, thus f(k) f max. 7

8 Control objective for BRP i energy purchased by BRP i at spot market q spot,i (k) R + necessary (undesired) balancing energy q bal,i (k) R q bal,i (k) = 1 T p i (k) q spot,i (k) Energy [MWh] Time [h] Figure : Energy purchase q spot,i (k) (black) and energy consumption 1 T p i (k) (red) under BRP i. 8

9 Control objective for BRP i strategy is to minimize balancing energy convex cost function of the balancing energy l i (q bal,i (k)) : R R 9

10 Receding Horizon Control (RHC) approach look K steps into the future using consumption predictions convex optimization problem at time k minimize Φ(η(k)) = subject to n B i=1 Φ i (η i (k)) E i (k) E i, P i (k) P i i = 1,...,n B F(k) F max where the variables are η i (k) and the objective function is Φ i (η i (k)) = k+k 1 κ=k l i (q bal,i (κ)) Solving the optimization problem gives us p i (k),... p i (k +K 1); we apply first input p i (k) 10

11 Motivation for de-centralization all data must be centralized to solve the problem in practice this means, that all BRPs have to share all local information unlikely due to the competitive nature of the energy market 11

12 Separation by dual decomposition constraints are affine: we can separate by dual decomposition the partial Lagrangian of the centralized problem is L(η(k),Λ(k)) = Φ(η(k))+Λ T (k)(f(k) F max ) where Λ(k) R n LK + is the Lagrange multiplier, or shadow prices, associated with the inequality F(k) F max 12

13 Separation by dual decomposition the dual function is given by g (Λ(k)) = inf η(k) ( Φ(η(k))+Λ T (k)(f(k) F max ) a subgradient of the negative dual is given by S(k) ( g)(λ(k)) R n LK ) where S(k) = F(k) F max with F(k) being the solution to the optimization problem minimize subject to for i = 1,...n B. Φ(η(k))+Λ T (k)f(k) E i (k) E i, Pi (k) P i this optimization is completely separable between the n B BRPs, and can therefore be solved distributedly. 13

14 Separation by dual decomposition this optimization is completely separable between the n B BRPs, and can therefore be solved distributedly. the subproblem for each BRP becomes minimize Φ i (η i (k))+λ T (k)t i (k) subject to E i (k) E i, P i (k) P i 14

15 Algorithm The distributed problem is solved by the following algorithm: 1. Initialize dual variable Λ(k) := Λ 0 (k) 0, e.g. using Λ 0 (k) = 0 or Λ 0 (k) = Λ(k 1). 2. loop Optimize flows using the dual variable Λ(k) by locally solving problem Determine capacity margins S(k) based on the solutions T i (k) to the subproblems. Update dual variables Λ(k) := (Λ(k)+αk S(k)) Terminate when certain optimality gap is obtained or after a given number of iterations by publishing final price. 4. Increase k by one and go to step 1. 15

16 Market implementation of algorithm Loads Consumers BRPs DSO State, prediction State, prediction Initial prices Price iterations Activation Clearing Activation Economical settlement only between BRPs (not DSO). 16

17 Numerical example BRP 1 C 1 p1,1 f 1 f 2 p 1,2 C 2 BRP 2 p 2,1 C 3 Cost Φ(1) [-] Prices Λ(1) Iterations [-] 17

18 Conclusion different BRPs sharing the same distribution grid can obtain global optimum (within horizon )without sharing local information, by using the shadow prices at the distribution grid capacities. 18

19 Thank you 19