Dual decomposition approach for dynamic spatial equilibrium models

Size: px

Start display at page:

Download "Dual decomposition approach for dynamic spatial equilibrium models"

Cordelia Todd
5 years ago
Views:

1 Dual decomposition approach for dynamic spatial equilibrium models E. Allevi (1), A. Gnudi (2), I.V. Konnov (3), G. Oggioni (1) (1) University of Brescia, Italy (2) University of Bergamo, Italy (3) Kazan Federal University, Russia Computational Management Science Conference (CMS2017) May 30-31, June 1, 2017 Bergamo

2 Our aim... We develop a dynamic equilibrium model for a system of auction markets joined by transmission lines subject to joint balance and capacity flows constraints. The model involves commodity storage within a given time period. We propose the single-level model to represent this dynamic system of markets, which are typical for energy systems. We construct a single-level variational inequality problem whose solution yields an equilibrium trajectory of this system. We apply a dual decomposition method to find its solution. The case study is based on the Italian day-ahead electricity market.

3 Main References Harker, P.T., editor. Spatial Price Equilibrium: Advances in Theory, Computation and Application. Springer-Verlag, Berlin, Nagurney, A. Network Economics: A Variational Inequality Approach. Kluwer, Dordrecht, Konnov, I.V. Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam, 2007a. Konnov, I.V. On variational inequalities for auction market problems. Optimization Letters, 1: , Konnov, I.V. Decomposition approaches for constrained spatial auction market problems. Networks and Spatial Economics, 9: , 2009.

4 Static spatial equilibrium problem Let us consider a system of K markets of a homogeneous commodity, which are joined in a network by transmission lines. Let I k and J k denote index sets of sellers and buyers at the k-th market, respectively. The i-th seller proposes his/her offer volume x i from the conditions: 0 α i x i β i +, i I k, The j-th buyer announces his/her bid volume y j from the conditions: 0 α j y j β j +, j J k. Together with offer/bid volumes, the sellers and buyers announce their prices g i, i I k and h j, j J k, respectively.

5 Auction market problem We define the offer/bid volumes vectors at the k-th market: x (k) = (x i ) i Ik and y (k) = (y j ) j Jk, The feasible sets of offer/bid volumes at the k-th market are: X (k) = [α i, β i ] and Y (k) = j, β j ]; i I k j J k [α Since prices depend on volumes, we have: g i = g i (x (k), y (k) ), i I k, and h j = h j (x (k), y (k) ), j J k. Assumption 1 (A1): The functions g i and h j are monotone and continuous.

6 Auction market problem (2) The k-th market problem then consists in finding vectors x(k) X (k), y(k) Y (k), and a number pk such that: g i (x(k), y (k) ) k i i, β i ), pk if xi = β i, pk = p if if xi = α i, x (α i I k, (1) and p h j (x(k), y (k) k if yj = α j, ) = pk if yj (α j, β j ), pk if yj = β j, j J k, (2) subject to the market balance condition yj u k = 0. j J k (3) where xi i I k Let u k denote the excess supply (excess demand) volume for the k-th market.

7 Dynamic spatial equilibrium problem Let us consider the k-th market model for the dynamic case. For simplicity, we consider a discrete time; i.e., we divide the time interval into subintervals t = 1, 2,..., T. Let: f kl,t be the flow from market k to market l at time interval t; v k,t be the volume of the commodity stored in the k-th market at the time intervals from (t 1) to t; p k,t be the price of the commodity at the k-th site at time interval t. In addition, the following capacity bounds hold: f kl,t [0, a kl ], a kl 0, k, l = 1,..., n, v k,t [0, b k ], b k 0, k,..., n, t = 1, 2,..., T.

8 Dynamic spatial equilibrium problem (2) If the vectors f = (f 11,1,..., f KK,T ) T, v = (v 1,1,..., v K,T ) T, p = (p 1,1,..., p K,T ) T are known, one can determine c kl,t : the transportation cost of a unit commodity from the k-th to the l-th market at the t-th time interval, r k,t : the cost of storing a unit commodity at the k-th market at the time intervals from (t 1)-th to t-th, u k,t : the excess supply (excess demand) of the k-th auction market at time interval t. Assumption 2 (A2): The functions c kl,t and r k,t are monotone and continuous.

9 Dynamic spatial equilibrium problem (3) Let us consider the k-th market at the t-th time interval. Each parameter above, including offer/bid volumes, their bounds, auction prices, and export values, now depends on time interval t. We can define the offer/bid vectors: x (k,t) = (x i,t ) i Ik and y (k,t) = (y j,t ) j Jk, and the feasible sets of offer/bid volumes: X (k,t) = i I k [α i,t, β i,t ] and Y (k,t) = Let u k,t denote the export volume. [α j,t, β j,t ]; j J k Together with offer/bid volumes, the sellers and buyers announce their prices g i,t = g i,t (x (k,t), y (k,t) ), i I k, h j,t = h j,t (x (k,t), y (k,t) ), j J k. Assumption 1 (A1) also applies to g i and h j.

10 Dynamic auction market problem At the t-th time interval, the k-th market problem consists in finding vectors x(k,t) X (k,t), y(k,t) Y (k), and a number pk,t such that: g i,t (x(k,t), y (k,t) ) pk,t if xi,t = α i,t, = pk,t if xi,t (α i,t, β i,t ), i I k, (4) pk,t if xi,t = β i,t, and p h j,t (x(k,t), y (k,t) k,t if yj,t = α j,t, ) = pk,t if yj,t (α j,t j,t ), pk,t if yj,t = β j,t j J k, (5) subject to the market balance condition xi,t yj,t k,t = 0; i I k j J k (6)

11 Equilibrium conditions We first write the volume balance condition at each node: n n fkl,t flk,t v k,t + v k,t+1 u k,t = 0, l=1 l=1 k = 1,..., K ; t = 1,..., T ; (7) We define equilibrium conditions for costs and stored volumes as the following VI: Find vk,t [0, b k ] such that [ ] r k,t (vk,t ) + p k,t 1 p k,t (v k,t vk,t ) 0 v k,t [0, b k ], (8) k = 1,..., K ; t = 1,..., T ; We define equilibrium conditions for network flows as the following VI: Find fkl,t [0, a kl ] such that [ ] c kl,t (fkl,t ) + p k,t p l,t (f kl,t fkl,t ) 0 f kl,t [0, a kl ], k, l = 1,..., K ; t = 1,..., T. (9)

12 Equilibrium conditions (2) A trajectory {(x(k,t), y (k,t), f kl,t, v k,t, p k,t )}, k, l = 1,..., K, t = 1,..., T, is called equilibrium if it satisfies conditions (4) (9). We reduce the problem of finding an equilibrium trajectory to a suitable VI. Define the trajectory vectors x = (x (k,t) ) k=1,...,k ;t=1,...,t, y = (y (k,t) ) k=1,...,k ;t=1,...,t, v = (v k,t ) k=1,...,k ;t=1,...,t, p = (p k,t ) k=1,...,k ;t=1,...,t, f = (f kl,t ) k,l=1,...,k ;t=1,...,t ; and the corresponding feasible capacity sets T K T K X = X (k), Y = Y (k), t=1 k=1 t=1 k=1 V = T K [0, b k ], F = T K K [0, a kl ]. t=1 k=1 t=1 k=1 l=1

13 Equilibrium conditions (3) We then define the general feasible set: ( n f kl,t n ) f lk,t v k,t + v k,t+1 l=1 (x, y, v, f ) ( l=1 W = X Y V F x i,t ) y j = 0, i I k j J k,t k = 1,..., K ; t = 1,..., T. (10) We define the problem of finding elements (x, y, v, f ) W such that T K g i,t (x(k,t), y (k,t) )(x i,t xi,t ) h j,t (x(k,t), y (k,t) )(y j,t yj,t ) t=1 k=1 i Ik j J { k T K K K } + c kl,t (fkl,t )(f kl,t fkl,t ) + r k,t (vk,t )(v k,t vk,t ) 0 t=1 k=1 l=1 k=1 (x, y, v, f ) W. (11)

14 Equilibrium conditions (4) Theorem (i) Let (x, y, v, f ) be a solution of VI (10) (11). Then there exist a vector p and numbers uk,t, k = 1,..., n, t = 1,..., T, such that relations (4) (6) for k = 1,..., n, t = 1,..., T, and (7) (9) are fulfilled. (ii) If (x, y, v, f ) X Y V F, a vector p and numbers uk,t, k = 1,..., n, t = 1,..., T, satisfy relations (4) (6) for k = 1,..., n, t = 1,..., T, and (7) (9), then (x, y, v, f ) is a solution of VI (10) (11). This theorem implies that we can solve VI (10) (11) instead of the systems (4)-(6), (7) (9).

15 Dual decomposition method Under Assumptions A1 and A2, we can derive the equivalence to the (convex) optimization problem: where T n min µ (x,y,v,f ) X Y V F i,t (x i,t ) η j,t (y j,t ) + (12) t=1 k=1 i Ik j Jk [ T n ] n n + σ kl,t (f kl,t ) + θ k,t (v k,t ) subject to t=1 k=1 l=1 (x, y, f, v) W µ i,t : [α i,t, β i,t ] R, i I k ; t = 1,..., T η i,t : [α i,t, β i,t ] R, σ kl,t : [0, a kl ] R, θ kl,t : [0, b k ] R, i I k ; t = 1,..., T k=1 k, l = 1,..., K ; t = 1,..., T k = 1,..., K ; t = 1,..., T are convex differentiable functions such that: µ i,t (x i,t ) = g i,t (x i,t ), η j,t (y j,t ) = h j,t (y j,t ), σ kl,t (f kl,t ) = c kl,t (f kl,t ), θ k,t (v k,t ) = r k,t (v k,t ).

16 Dual decomposition method (2) where maximize Ψ(p) (13) subject to p R n T K Ψ(p) = min µ (x,y,v,f ) X Y V F i,t (x i,t ) η j,t (y j,t ) + (14) t=1 k=1 i Ik j Jk [ T K ] K K + σ kl,t (f kl,t ) + θ k,t (v k,t ) + t=1 k=1 l=1 k=1 ( T K n ) K + p k,t f kl,t f lk,t v k,t + v k,t+1 x i,t y j,t t=1 k=1 l=1 l=1 i Ik j Jk If the functions µ i,t, η j,t, σ kl,t, and θ k,t are non strictly convex, it is possible to replace this problem with a sequence of regularized problems, so that, at the s-iteration, we solve a dual regularized problem.

17 Dual decomposition method (3) where maximize Ψ s(p) (15) subject to p R n T K Ψ s(p) = min ( µ (x,y,v,f ) X Y V F i,t (x i,t ) + 0.5λ s(x i,t x s 1 i,t ) 2) + (16) t=1 k=1 i Ik ( η j,t (y j,t ) 0.5λ s(y j,t y s 1 j ) 2) + j Jk [ T K ] K ( + σ kl,t (f kl,t ) + 0.5λ s(f kl,t f s 1 kl,t ) 2) K ( + θ k,t (v k,t ) + 0.5λ s(v k,t v s 1 k,t ) 2) + t=1 k=1 l=1 k=1 ( T K K ) K + p k,t f kl,t f lk,t v k,t + v k,t+1 x i,t y j,t = t=1 k=1 l=1 l=1 i Ik j Jk

18 T K = ( ) min µ i,t (x i,t ) + 0.5λ s(x i,t x s 1 t=1 k=1 i Ik x i,t [α i,β i,t ) 2 p k,t x i,t + (17) ] i max j Jk y j,t [α j [ T K K + t=1 k=1 l=1 K + min v k=1 k,t [0,b k ],β j ] ( ) η j,t (y j,t ) 0.5λ s(y j,t y s 1 j,t ) 2 p k,t y j,t min f kl,t [0,a kl ] + ( σ kl,t (f kl,t ) + 0.5λ s(f kl,t f s 1 kl,t ) 2 + (p k,t p l,t )f kl,t ) + ] ( ) θ k,t (v k,t ) + 0.5λ s(v k,t v s 1 k,t ) 2 + (p k,t 1 p k,t )v k,t

19 Case study: the Italian electricity market Auction model on the Italian electricity market: this market is subdivided into fifteen zones (i, j = 1,..., 15) that are classified into three main groups: 1 Foreign countries represented by France (i = 1), Switzerland (i = 2), Austria (i = 3), Slovenia (i = 4), and Greece (i = 15) to which Italy is connected; 2 Geographical zones represented by North (i = 5), Central North (i = 6), Central South (i = 7), Sardinia (i = 8), South (i = 9), and Sicily (i = 10); 3 Virtual zones represented by Rossano (i = 11), Foggia (i = 12), Brindisi (i = 13), and Priolo (i = 14) defining areas with limited power production. FRANCE SWITZERLAND AUSTRIA SLOVENIA NORTH CENTRAL NORTH SARDINIA CENTRAL SOUTH SOUTH ROSSANO FOGGIA SICILY BRINDISI PRIOLO GREECE

20 Case study: the Italian electricity market (2) Time interval: a day partitioned into 24 hours (t = 1,..., 24). The day is divided into daytime (t = 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) and nighttime (t = 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 24). Flows from site i to site j at time intervals from (1 t)-th to t-th is limited by transfer capacity. The transfer capacity data are provided by Terna 1 and by ENTSO-E 2. Nine different technologies: wind, photovoltaic, run-of-river, geothermal, nuclear, coal, CCGT, other gas and oil based power plants. Each of these technologies is characterized by variable costs based on fuel and CO 2 emission costs. 1 See 2 See

21 Case study: the EU-ETS The European Union Emissions Trading System (EU-ETS) is a cap and trade system that limits and imposes a price on CO 2 emissions generated by specific production installations. All network zones, except for Switzerland (i = 2), are directly involved in the EU-ETS. Zone i = 2, corresponding to the Switzerland, is not involved in the EU-ETS, but it applies an own ETS, denoted as Swiss ETS. In our numerical experiments, we assume a CO 2 price of 5 e/ton.

22 Numerical experiments The traders bid function g i,t is defined by the following affine function: g i,t = a g i,t + bg i,t x i,t Where the parameters a g i,t and bg i,t are estimated using reference values for generation and variable costs; We consider 4 electricity companies participating to the different auction markets. The electricity buyers bid function h j,t is defined by the following affine function: h j,t = a h j,t bh j,t y j,t where the reference demand and price parameters a h j,t and bh j,t are estimated using data taken from the Italian Market Operator website; We consider 1 consumer group per auction market.

23 Results Variable x i,t ranges between 0 (α i,t) and the maximum available capacity in each market (β i,t). We consider two possible intervals for variable y j,t : Case 1 α j,t = 0 and β j,t = peak load in market k; Case 2 α j,t = average demand in market k and β j,t = peak load in market k increased by 20%; 20 Case 1 #itera3ons 20 Case 2 #itera3ons Seconds (execu3on 3me) Seconds (execu3on 3me) lambda=0.02 lambda=2 lambda=3 lambda=0.02 lambda=2 lambda=3 lambda=4 lambda=5 lambda=100 lambda=4 lambda=5 lambda=100

24 Final remarks and conclusions We conduct a sensitivity analysis on parameter λ s; The analysis shows that the proposed methods is quite efficient (low number of iterations and low computation time); We are testing this approach on the gas market.

25 Thank you for your attention!

26 Reference marginal cost Reference generation Table 8. Reference marginal production costs and reference generation level in zone i

27 a g i b g i Table 9. Values of parameters a g i and b g i of the offer price function g i

28 MWh Table 10. Reference demand per time segment t and zone i

29 e/mwh Reference price Table 11. Reference price per time segment t

30 MWh Table 12. Values of parameters α i,t corresponding to the reference offer values submitted by market i in time t to the Italian day-ahead market

31 MWh Table 13. Values of parameters β i,t corresponding to the total available capacity in market i and in time t

32 MWh Table 14. Values of parameters α j,t corresponding to reference bid demand values submitted by market i and time t to the Italian day-ahead market

33 MWh Table 15. Values of parameters β j,t corresponding to the average power demand in market i and time t

Energy System Modelling Summer Semester 2018, Lecture 7

Energy System Modelling Summer Semester 2018, Lecture 7 Dr. Tom Brown, tom.brown@kit.edu, https://nworbmot.org/ Karlsruhe Institute of Technology (KIT), Institute for Automation and Applied Informatics