Controlling conventional generation to minimize forecast error cost

Transcription

1 Controlling conventional generation to minimize forecast error cost Ksenia Chernysh Heriot-Watt University 25 April 2013 Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

2 Introduction Energy supply scheme Renewable power is hard to predict Conventional power plants require considerable time to ramp up Imported power is expensive Demand must be always met Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

3 Introduction Energy supply scheme System operators predict net demand Net demand is power produced by wind farms subtracted from the total demand The power system acts autonomously to cover predicted net demand but is unable to deal with errors Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

4 Time Wind prediction data Introduction tsaerror Figure: Errors in MW in wind power prediction for 1,5 year period Suppose that power system covered predicted net demand and afterwards no other mananging policies were applied On the picture above: If error is negative there is no additional need to purchase energy If it is positive, there is a shortfall in the power supply, which must be covered It cannot be done immediately using conventional supply as it requires time to ramp up Therefore, expensive imported power must be bought Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

5 Introduction Natural questions What if we run some additional conventional power plants in order to save at shortfall/blockouts moments? Would it work? What types of policies for managing the amount of additional conventional supply are optimal? Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

6 Introduction Only large errors are important As smaller errors may be covered by local reserve only large errors are important The figure below is obtained from the previous one by removing shortfails below level 1000MW errormw Time Figure: Only errors, which are higher than 1000MW, for 1,5 year period Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

7 Mathematical model Simplifications of the previous picture We assume that errors occur at (possibly random) time instance T i This is designed to model only larger shortfalls in power generation since smaller errors are covered by local reserve Times between errors are deterministic and identical Times between errors are deteministic and periodic Stochastic assumptions: Pairs (ε n, T n T n 1 ) are iid or They form a Markov chain; They form a regenerative point process Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

8 Mathematical model Notations k(t) is a level of production of (additional) conventional power plants at time t U > 0 and B > 0 are ramp constraints: for every t, δ > 0 holds Bδ k(t + δ) k(t) Uδ {T i } i 0 with T 0 = 0 are (random) moments of time then we need to meet the level of the demand equal to ε i τ n = T n T n 1 Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

9 Notations Mathematical model c 1 and c 2 are unit costs for conventional and imported supply respectively c 1 < c 2 T For time period [0, T ] amount of c 1 0 k(t)dt is paid for maintaining conventional supply at level k(t) If error ε occurs at time t then amount c 2 (ε k(t)) + is paid immediately as demand must be met in any case Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

10 Classes of policies Mathematical model Definition Almost surely differentiable function k : R + R + belongs to class K if the inequality L k (y) U holds wp1 Furthermore, K + = {k +, where k K and k + = max(k, 0)} We consider only stochastic processes, which realizations almost surely belong to class K + Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

11 Classes of policies Mathematical model Definition A function k : R + R + belongs to class K if there exists t [0, T ] such that k is differentiable everywhere except point t and k (t) = B, for y [0, t), and k (t) = U, for y (t, + ] We use notation K + as earlier Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

12 Deterministic interarrival times Cost-functional for deterministic interarrival times Assume that {T i } i 0 = it with T > 0 are moments of time then we need to meet the level of the demand equal to ε i Definition Partial cost functional l i : Ti l i (k) = c 1 E k(y)dy + c 2 E(ε i k(t i )) + T i 1 (1) Infinite-time cost functional L: L(k) = lim sup n 1 T n [ n ] l i (k) i=1 (2) Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

13 Deterministic interarrival times Main result for deterministis interarrival times Theorem Suppose that L + B = 1; T i = it ; ε i are iid positive-valued random variables with its pdf F ; F is continuous Then there exists a function ˆk K + such that 1 L(ˆk) = min k L(k), 2 ˆk(iT ) = ˆk(0) = ā, 3 ā is a solution of c 2 F (a) = c 1 min ( T, a BU ), Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

14 Deterministic interarrival times How does optimal policy look like? The value at time it is defined by equation c 2 F (a) = c 1 min ( T, a BU ) a c 2 F (a) = c 1 BU : c 2 F (a) = c 1 T Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

15 Deterministic interarrival times The key idea of the proof Definition Suppose that p [0, 1] (q = 1 p) For k K + ST functional by the equation we define p-cost l i (p, k) = c 1 Ti Simple observation: T i 1 k(y)dy + c 2 [ pe(ξ k(ti 1 )) + + qe(ξ k(t i ))) +] L(k) = lim sup n (3) [ n ] 1 l i (p, k) (4) nt i=1 Lemma For ˆk as in theorem (1) holds l 1 (u, ˆk) = min k KST l 1 (u, k) and ˆk(T ) = ˆk(0) Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

16 Random interarrival times Case of Poisson process Suppose that interarrival times have exponential distrubition with common parameter λ Then (T i ) i 0 is a Poisson process with rate λ Definition Infinite-time cost functional L: L(k) = lim sup T [ c 1 E ( T )] 0 k(t)dt + c 2E τi Π E(ε i k(τ i )) + We may use the memoryless property of PP: If there is no shortfall at time t, then there is no new information about the next arrival at all Should we change the policy at this moment? The answer is no T (5) Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

17 Random interarrival times Case of Poisson process Theorem Consider that errors are independent and have the same probability distribution function F Consider a homogeneous Poisson process N(t) with rate λ Then a policy optimizing the cost-functional is a constant function k, where k = F (1 1 λ c ) 1 c 2 We can see that the optimal level is determined only by the value of the function F 1 ( ) at a single point Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

18 Random interarrival times Generalizations of Poisson case and work in progress The same conclusion holds in a more general setting, where ε i may have different distributions with the same threshold level k Now we are trying to generalize results for periodically varying increments For example τ i has distribution exp(λ) then i is odd and exp(µ) otherwise We have a conjecture that constant policy is not optimal Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19

19 Random interarrival times Thank you for your attention Questions? Ksenia Chernysh (Heriot-Watt University) Controlling conventional generation 25 April / 19