Stochastic control principles
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1 Stochastic control principles Sven Thomsen, Henrik Aa. Nielsen A/S Lyngsø Allé 3 Hørsholm FlexPower, WP4 workshop, RISØ-DTU Sven C. Thomsen & Henrik Aa. Nielsen 1
2 Agenda The house problem revisited Issues with receding horizon solution The electric car problem Conclusions Optimal policy for simplified (dice) example Sven C. Thomsen & Henrik Aa. Nielsen 2
3 House optimization revisited Receding horizon methodology With a quadratic penalty on temperature deviation + the natural price penalty the following forecasts are required Conditional expectation of consumer consumption (or the outdoor temperature) Conditional expectation of the price At any given time t the optimal open-loop trajectory is calculated feedback is indirect by recalculating the trajectory at every sample time. Sven C. Thomsen & Henrik Aa. Nielsen 3
4 House optimization con t This approach is not globally optimal in a math. sence but could very well be the most practical solution MPC does not explicitly go for a control policy feedback is not explicitly accounted for Not an issue for linear systems with additive gaussian uncertainty However, the electricity price is multiplicative Accounting for the joint (correlation in time) power price distribution by building some heuristic on top of the problem might be interesting Sven C. Thomsen & Henrik Aa. Nielsen 4
5 Solving the for an optimal policy The natural way to find the optimal policy - and hence the controller that gives the global optimum is by using stochastic dynamic programming Principle of optimality (Bellmans equation) V t x t =min u U E [ L x t,u t, w t Immediate cost ] V N x N =E [ x N,w N Terminal V t1 x t1 cost.to.go Combining the system equation and Bellmans equation the problem can be solved brute force - discretized state space & terminates in finite time ] Sven C. Thomsen & Henrik Aa. Nielsen 5
6 Stochastic programming con t Pitfalls using brute force Curse of dimensionality Not generally possible to separate predictions and control Example of a nice solution Linear system, gaussian uncertainty, quadratic cost (See e.g. Intro. to stochastic control theory Åstrøm) Estimation problem and control problem decoupled Kalman filter (stochastic optimal state estimator) LQ controller (deterministic optimal controller) Sven C. Thomsen & Henrik Aa. Nielsen 6
7 Electric car problem - the dice example by Bjarke (Actua) Electricity price at time t is given by outcome of random variable P t (fair dice with 6 sides) We need to charge the car within the horizon N. The car has c 0 units when it start. Full charge is c L units The car can be charged one unit (u=1) or zero units (u=0) in a given sample time. V 0 c 0 =min u0, u 1,..,u N 1 E [ J = c t1 =c t u t c t 0,1,2,..., c L, u t 0,1 N 1 t=0 u t P t c L c N 2 ] Sven C. Thomsen & Henrik Aa. Nielsen 7
8 Example con t Closed form solution for α= (i.e.fully charged at t=n) uc=1if pr cl c, N 1 R c L c 1, N 1 6 R i, j = 1 6 k =1 minkr i 1, j 1, R i, j 1 R 1,1 =3.5 Numerical example 4 hours in plug Needs 2 hours Sven C. Thomsen & Henrik Aa. Nielsen 8
9 Example con t The dice example clearly illustrates that more is needed than the conditional expectation. The dice example is however not realistic. Uniform distribution between price limits & no time correlation Implications of using receding horizon solution wanish as we move further to the right Sven C. Thomsen & Henrik Aa. Nielsen 9
10 Example con t If time allows Sven C. Thomsen & Henrik Aa. Nielsen 10
11 Conclusion The electric car example shows the need for including the price distribution in the optimization problem. The stochastic programming problem is tractable with the simplified (dice) setup. Accounting for correlated prices might call for heuristics. The house heating example is not globally optimal in a math. sence. However, given the complications of solving for a policy and no finite terminal time it is arguably the most practical approach. Dice example shows worst case - real life situation has more structure. Sven C. Thomsen & Henrik Aa. Nielsen 11
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