Economic MPC and Real-time Decision Making with Application to Large-Scale HVAC Energy Systems

Transcription

1 Economic MPC and Real-time Decision Making with Application to Large-Scale HVAC Energy Systems J.B. Rawlings N.R. Patel M.J. Wenzel R.D. Turney M.J. Risbeck C.T. Maravelias FOCAPO/CPC 2017 Tucson, Arizona January 8 12, 2017 FOCAPO/CPC / 34

2 Outline 1 Stanford Energy System Innovations (SESI) project 2 Expanding the Industrial Scope of Model Predictive Control Discrete decisions Is the control robust? Economic MPC with periodic constraint 3 Conclusions 4 Future Work FOCAPO/CPC / 34

3 The $485-million Stanford Energy System Innovations (SESI) project; replaced an aging 50-MW natural-gas-fired cogeneration plant with a new heat-recovery system to provide heating and cooling to the campus. FOCAPO/CPC / 34

4 A new 80-megavolt-ampere electrical substation brings electricity from the grid. Crews also converted 155 campus buildings from steam to hot-water distribution and installed a 22-mile-long network of new pipe. FOCAPO/CPC / 34

5 The star of the show: three heat-recovery chillers the largest in the U.S. that strip waste heat from 155 campus buildings. FOCAPO/CPC / 34

6 Johnson Controls developed the Central Energy Plant Optimization Model (CEPOM); the algorithm optimizes a 10-day forecast every 15 minutes, considering campus loads, weather patterns, price of electricity, available equipment and many other factors. FOCAPO/CPC / 34

7 Large-scale commercial application FOCAPO/CPC / 34

8 Control Decomposition Disturbance Forecasts, Electricity Pricing High Level Disturbance Estimate Cooling Load Aggregate System Curve Demand Profile Low-Level Airside Airside Subsystem 1 Airside Subsystem 2 Airside Subsystem n Low-Level Waterside Measurements Temperature Setpoints Measurements Equipment Setpoints Airside PID 1 Airside PID 2 Airside PID n Waterside PID FOCAPO/CPC / 34

9 The disturbance forecast: weather and electricity prices Amb.Temp. ( C) Elec. price ($/kwh) Time (days) FOCAPO/CPC / 34

10 High-level problem: Optimal production and average building temperatures (25 buildings) Cooling Load (MW) Demand Production Storage Unmet Avg. Temp. ( C) Time (days) FOCAPO/CPC / 34

11 Zone Temp. ( C) Low level airside: Optimal zone temperatures and setpoints (20 zones in each of 25 buildings) Temp. Setpoint ( C) Time (days) FOCAPO/CPC / 34

12 Low level waterside: Production and Gantt chart for central plant equipment Cooling Load (MW) Demand Production Storage Unmet Towers Pumps Chillers Time (days) FOCAPO/CPC / 34

13 Real-time computational requirements The optimizations were solved using Gurobi 6.0 via Matlab R2016b on a machine with 8GB RAM and 2.66GHz Intel Core 2 Quad Processor Q8400. The high-level problem took 35 seconds to solve. The low-level airside subproblems took about 15 seconds each to solve. The low-level waterside subproblem was given two minutes of computation time, after which the incumbent solution (with an optimality gap of 0.2%) was accepted. Since control executions occur every 15 minutes, this decomposition can easily be implemented online. Solution times can be further decreased by using a horizon shorter than one week (Risbeck, Maravelias, Rawlings, and Turney, 2016). FOCAPO/CPC / 34

14 SESI operation summary In operation since December 2015 (Wenzel, Turney, and Drees, 2016). The central plant was run in autonomous mode about 90% of the time (including time off-line for plant maintenance). Achieved 10% to 15% additional savings in operating costs compared to control by the best team of trained human operators (Stagner, 2016). This large-scale implementation demonstrates the significant potential benefits to applying model-based optimization to large HVAC systems FOCAPO/CPC / 34

15 Literature review for multiple problem elements Buildings Stochastic MPC for buildings Oldewurtel, Parisio, Jones, Gyalistras, Gwerder, Stauch, Lehmann, and Morari (2012) Ma, Matuško, and Borrelli (2015) Scheduling for central plants with TES Kapoor, Powell, Cole, Kim, and Edgar (2013) Scheduling/control for TES and buildings Mayer, Killian, and Kozek (2015) Touretzky and Baldea (2016) Discrete Actuators Mixed logical dynamical/piecewise affine systems Bemporad and Morari (1999) Lazar, Heemels, Weiland, and Bemporad (2006) Switched Systems El-Farra and Christofides (2003) Quantization as a disturbance Quevedo, Goodwin, and De Doná (2004) Kobayshi and Hiraishi (2013) Economic MPC Stability Diehl, Amrit, and Rawlings (2011) Ellis, Durand, and Christofides (2014) Grüne and Stieler (2014) Average Performance Angeli, Amrit, and Rawlings (2012) Müller, Angeli, and Allgöwer (2014) Periodic Systems Linear Nonlinear Böhm, Raff, Reble, and Allgöwer (2009) Limon, Alamo, de la Peña, Zeilinger, Jones, and Pereira (2012) Huang, Harinath, and Biegler (2011) Zanon, Gros, and Diehl (2013) Falugi and Mayne (2013) FOCAPO/CPC / 34

16 MPC: Supporting theory for this class of problems Problem 1 stabilizing a steady state X N X f κ f ( ) x(0) Stability Assumption: V f (f (x, κ f (x))) V f (x) l(x, κ f (x)) FOCAPO/CPC / 34

17 Nominal Stability Result stabilizing a steady state Denoting the (possibly multivalued) control law as κ N ( ), closed-loop system is x + F (x) := {f (x, u) u κ N (x)} (1) Theorem 1 (Exponential stability of (sub)optimal MPC) The origin of the closed-loop system (1) is exponentially stable on (arbitrarily large) compact subsets of the feasible set X N. (Pannocchia, Rawlings, and Wright, 2011) Feasible set X N is the set of x that can reach X f within N steps while meeting constraints. FOCAPO/CPC / 34

18 Problem 1 Extension to discrete actuators So the first question of interest is how much effort is required to extend all of the existing MPC theory to handle discrete actuators. The answer, surprisingly, is none. Consider the main assumption about the input feasible set (Rao and Rawlings, 1999) The set U is compact and contains the origin. FOCAPO/CPC / 34

19 Problem 1 Extension to discrete actuators So the first question of interest is how much effort is required to extend all of the existing MPC theory to handle discrete actuators. The answer, surprisingly, is none. Consider the main assumption about the input feasible set (Rao and Rawlings, 1999) The set U is compact and contains the origin. Notice that U is not required to contain the origin in its interior as is common in much of the early MPC literature. Therefore, to treat discrete actuators, we simply change the set U to enforce discreteness in some subset of the actuators. FOCAPO/CPC / 34

20 Continuous and mixed continuous-discrete actuators u 2 u 2 u 1 u 1 (a) (b) Figure 1: Typical input constraint sets U for (a) continuous actuators and (b) mixed continuous-discrete actuators; the origin ( ) is the equilibrium of interest. FOCAPO/CPC / 34

21 Problem 2 Inherent robustness of (sub)optimal MPC Real systems are affected by disturbances that cause the nominal model to no longer hold Process errors d lead to model error x + = f (x, u) + d Measurement errors e corrupt x estimate to x + e FOCAPO/CPC / 34

22 Problem 2 Inherent robustness of (sub)optimal MPC Real systems are affected by disturbances that cause the nominal model to no longer hold Process errors d lead to model error x + = f (x, u) + d Measurement errors e corrupt x estimate to x + e With disturbances, the closed-loop system becomes x + F de (x) := {f (x, u) + d u κ N (x + e)} (2) Theorem 2 (Robust exponential stability of (sub)optimal MPC) The origin of the perturbed closed-loop system (2) is robustly exponentially stable on (arbitrarily large) compact subsets of the feasible set X N. (Pannocchia et al., 2011) FOCAPO/CPC / 34

23 Problem 2 Extension to discrete actuators Once again, the extension to discrete actuators is immediate The set U need not be convex, connected, etc. it need only contain the origin FOCAPO/CPC / 34

24 Problem 2 Extension to discrete actuators Once again, the extension to discrete actuators is immediate The set U need not be convex, connected, etc. it need only contain the origin However, design choices become more striking with discrete actuators: Theory precludes a large change in control action at the setpoint System must be locally stabilizable using only unsaturated actuators Discrete actuators are always saturated FOCAPO/CPC / 34

25 Problem 2 Extension to discrete actuators Once again, the extension to discrete actuators is immediate The set U need not be convex, connected, etc. it need only contain the origin However, design choices become more striking with discrete actuators: Theory precludes a large change in control action at the setpoint System must be locally stabilizable using only unsaturated actuators Discrete actuators are always saturated Single (set)point stabilization may no longer be an appropriate goal FOCAPO/CPC / 34

26 Feasible Sets MPC is stabilizing on X N 30 Continuous Actuator 30 Discrete Actuator T T T 1 X f X 1 X 2 X 3 X 4 X 5 X 6 FOCAPO/CPC / 34

27 Feasible Sets MPC is stabilizing on X N but X N may not be what you expect 30 Continuous Actuator 30 Discrete Actuator T T T 1 X f X 1 X 2 X 3 X 4 X 5 X 6 FOCAPO/CPC / 34

28 Problem 3 Economic MPC with periodic constraints Let the sequences (x p, u p ) denote a given T -periodic solution to a periodic nonlinear system x p (i + 1) = f (x p (i), u p (i), i) x p (i + T ) = x p (i), u p (i + T ) = u p (i) In the third problem, we assume a periodic solution is available, but change the controller s goal from stabilization of the periodic solution (tracking) to optimization of economic performance. The periodic solution then serves as a useful end constraint for the economic optimization problem. The stage cost l(x, u) is free to be chosen as an economic profit function and has no connection to distance from (x p, u p ) as in the tracking case. FOCAPO/CPC / 34

29 Setup for economic MPC with periodic constraint x p (t + N) x p (t + 1) x p (t 1) x p (t) (x, t) X N (t) Figure 2: The periodic solution x p(t) as end constraint for economic MPC problem for a system with initial condition (x, t) and N = 2. FOCAPO/CPC / 34

30 Mixed-Integer Programming Mixed-integer optimization has the following general form: min x R n f (x) s.t. g(x) 0 x i I, i I {1,..., n} Objective function f, constraints g, and integer variables I FOCAPO/CPC / 34

31 Mixed-Integer Programming Mixed-integer optimization has the following general form: min x R n f (x) s.t. g(x) 0 x i I, i I {1,..., n} Objective function f, constraints g, and integer variables I Special case is mixed-integer quadratic program (MIQP) f is convex quadratic g is linear (i.e., affine) FOCAPO/CPC / 34

32 Mixed-Integer Programming Mixed-integer optimization has the following general form: min x R n f (x) s.t. g(x) 0 x i I, i I {1,..., n} Objective function f, constraints g, and integer variables I Special case is mixed-integer quadratic program (MIQP) f is convex quadratic g is linear (i.e., affine) MIQP software (e.g., Gurobi, SCIP, CPLEX) can find solutions much more efficiently than a brute force search. Relax integrality constraints and solve Branch on fractional variables to enforce integrality and repeat FOCAPO/CPC / 34

33 Computational Burden Exhaustive branching can take an extremely long time. Luckily, powerful solvers have a number of other techniques. Pre-solve to remove unnecessary variables or equality constraints Derive valid inequalities to improve relaxation bound Employ strong branching or other methods to choose the best fractional variable to branch on Retain basis information so re-solving a branch is quick FOCAPO/CPC / 34

34 Computational Burden Exhaustive branching can take an extremely long time. Luckily, powerful solvers have a number of other techniques. Pre-solve to remove unnecessary variables or equality constraints Derive valid inequalities to improve relaxation bound Employ strong branching or other methods to choose the best fractional variable to branch on Retain basis information so re-solving a branch is quick However, MIQP is attractive for a number of reasons. Discrete variables can model discrete (e.g., on/off) decisions A conservative estimate of optimality gap is available Less difficult than general nonlinear global optimization (with or without discrete variables) FOCAPO/CPC / 34

35 Example: Simplified Building Cooling q amb Ambient Zone, T T amb dt dt = k(t T amb) + q amb + q ch q tank ds dt = σs + q tank vq min q vq max q tank q ch, v {0, 1, 2} Chiller 1 Chiller 2 q ch q tank Tank, s x := (T, s) u := (q ch, q tank, v) d := (T amb, q amb ) Temperature must be maintained within preset bounds. Each chiller can be on or off. When on, chillers have minimum and maximum capacity. FOCAPO/CPC / 34

36 Parameters Stage Costs l econ (x, u, t) := ρ(t)q(t) 4 3 ρ l track (x, u, t) := x(t) x p (t) 2 Q T amb 23 Horizon N = 24 + u(t) u p (t) 2 R Periodic ρ, T amb, q amb Weights Q = R = I q amb Time FOCAPO/CPC / 34

37 Optimal Periodic Solution T q ch Economic Cost: s Time Periodic solution can be used for tracking or end constraint. Precooling reduces cooling during peak price hours. Bounds on q are determined by the (integral) value of v. FOCAPO/CPC / 34

38 Tracking MPC T q ch Economic Cost: s Time Tracking MPC converges to the periodic reference. Initial condition T (0) = 2, s(0) = 0. Stage cost penalizes changes in T, s, q ch, q tank, and v. FOCAPO/CPC / 34

39 Economic MPC T q ch Economic Cost: s Time Using the economic objective, cost is reduced by 3.3% Controller aggressively pursues lower cost Deviation in u is not penalized FOCAPO/CPC / 34

40 Conclusions MPC is well suited to high-level operational goals like energy or cost minimization. FOCAPO/CPC / 34

41 Conclusions MPC is well suited to high-level operational goals like energy or cost minimization. MPC of large-scale HVAC energy systems offers significant economic benefit over current operations. FOCAPO/CPC / 34

42 Conclusions MPC is well suited to high-level operational goals like energy or cost minimization. MPC of large-scale HVAC energy systems offers significant economic benefit over current operations. Hierarchical decomposition is the key design step. Several candidates remain viable for this step. FOCAPO/CPC / 34

43 Conclusions MPC is well suited to high-level operational goals like energy or cost minimization. MPC of large-scale HVAC energy systems offers significant economic benefit over current operations. Hierarchical decomposition is the key design step. Several candidates remain viable for this step. Supporting MPC theory: discrete actuators and general nonlinear systems Nominal stability of optimal and suboptimal MPC with a steady-state operating point (Rawlings and Risbeck, 2016) Inherent robustness of suboptimal MPC with mixed continuous/discrete actuators (Allan et al., 2016) Asymptotic stability of the periodic tracking problem with terminal region Asymptotic stability of economic MPC with periodic end constraint FOCAPO/CPC / 34

44 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) FOCAPO/CPC / 34

45 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? FOCAPO/CPC / 34

46 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? fama, nullus fortuna! FOCAPO/CPC / 34

47 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? fama, nullus fortuna! Hierarchical decomposition: quantify the complexity/performance tradeoff FOCAPO/CPC / 34

48 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? fama, nullus fortuna! Hierarchical decomposition: quantify the complexity/performance tradeoff Large energy systems as players in the real-time electricity market FOCAPO/CPC / 34

49 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? fama, nullus fortuna! Hierarchical decomposition: quantify the complexity/performance tradeoff Large energy systems as players in the real-time electricity market Combined scheduling and control FOCAPO/CPC / 34

50 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? fama, nullus fortuna! Hierarchical decomposition: quantify the complexity/performance tradeoff Large energy systems as players in the real-time electricity market Combined scheduling and control More large-scale applications. Move the needle on total US energy consumption FOCAPO/CPC / 34

51 Future Work Release test problem based on Stanford SESI system for the research community to benchmark new control system designs (other decompositions, robust MPC, stochastic MPC, etc.) Software! We all need it. We shouldn t all be duplicating it. Free source model recommended for research community. Why? fama, nullus fortuna! Hierarchical decomposition: quantify the complexity/performance tradeoff Large energy systems as players in the real-time electricity market Combined scheduling and control More large-scale applications. Move the needle on total US energy consumption Process industries: anything planned besides DMC(n)? FOCAPO/CPC / 34

52 Acknowledgments Thanks to Johnson Controls, Inc. for sample data, equipment models, and research funding. The authors gratefully acknowledge the financial support of the NSF through grant #CTS FOCAPO/CPC / 34

53 References I D. A. Allan, M. J. Risbeck, and J. B. Rawlings. Stability and robustness of model predictive control with discrete actuators. In American Control Conference, pages 32 37, Boston, MA, July 6 8, D. Angeli, R. Amrit, and J. B. Rawlings. On average performance and stability of economic model predictive control. IEEE Trans. Auto. Cont., 57(7): , C. Böhm, T. Raff, M. Reble, and F. Allgöwer. LMI-based model predictive control for linear discrete-time periodic systems. In L. Magni, D. Raimondo, and F. Allgöwer, editors, Nonlinear Model Predictive Control - Towards New Challenging Applications, pages Springer Berlin / Heidelberg, A. Bemporad and M. Morari. Control of systems integrating logic, dynamics, and constraints. Automatica, 35: , FOCAPO/CPC / 7

54 References II M. Diehl, R. Amrit, and J. B. Rawlings. A Lyapunov function for economic optimizing model predictive control. IEEE Trans. Auto. Cont., 56(3): , N. H. El-Farra and P. D. Christofides. Coordinating feedback and switching for control of hybrid nonlinear processes. AIChE J., 49(8): , M. Ellis, H. Durand, and P. D. Christofides. A tutorial review of economic model predictive control methods. J. Proc. Cont., 24(8): , P. Falugi and D. Q. Mayne. Tracking a periodic reference using nonlinear model predictive control. In 52nd IEEE Conference on Decision and Control, pages , doi: /CDC L. Grüne and M. Stieler. Asymptotic stability and transient optimality of economic mpc without terminal conditions. J. Proc. Cont., 24(8): , FOCAPO/CPC / 7

55 References III R. Huang, E. Harinath, and L. T. Biegler. Lyapunov stability of economically oriented NMPC for cyclic processes. J. Proc. Cont., 21: , K. Kapoor, K. M. Powell, W. J. Cole, J. S. Kim, and T. F. Edgar. Improved large-scale process cooling operation through energy optimization. Processes, 1(3): , K. Kobayshi and K. Hiraishi. Computational techniques for model predictive control of large-scale systems with continuous-valued and discrete-valued inputs. J. Appl. Math., 2013:9, M. Lazar, W. Heemels, S. Weiland, and A. Bemporad. Stabilizing model predictive control of hybrid systems. IEEE Trans. Auto. Cont., 51(11): , FOCAPO/CPC / 7

56 References IV D. Limon, T. Alamo, D. de la Peña, M. N. Zeilinger, C. Jones, and M. Pereira. MPC for tracking periodic reference signals. In Proceedings of the IFAC Conference on Nonlinear Model Predictive Control, number EPFL-CONF , M. A. Müller, D. Angeli, and F. Allgöwer. Transient average constraints in economic model predictive control. Automatica, 50(11): , Y. Ma, J. Matuško, and F. Borrelli. Stochastic model predictive control for building HVAC systems: Complexity and conservatism. IEEE Trans. Cont. Sys. Tech., 23(1): , B. Mayer, M. Killian, and M. Kozek. Management of hybrid energy supply systems in buildings using mixed-integer model predictive control. Energ. Convers. Manage., 98: , FOCAPO/CPC / 7

57 References V F. Oldewurtel, A. Parisio, C. N. Jones, D. Gyalistras, M. Gwerder, V. Stauch, B. Lehmann, and M. Morari. Use of model predictive control and weather forecasts for energy efficient building climate control. Energ. Buildings, 45:15 27, G. Pannocchia, J. B. Rawlings, and S. J. Wright. Conditions under which suboptimal nonlinear MPC is inherently robust. Sys. Cont. Let., 60: , N. R. Patel, J. B. Rawlings, M. J. Wenzel, and R. D. Turney. Design and application of distributed economic model predictive control for large-scale building temperature regulation. In 4th International High Performance Buildings Conference at Purdue, West Lafayette, IN, July 11 14, D. E. Quevedo, G. C. Goodwin, and J. A. De Doná. Finite constraint set receding horizon quadratic control. Int. J. Robust and Nonlinear Control, 14(4): , FOCAPO/CPC / 7

58 References VI C. V. Rao and J. B. Rawlings. Steady states and constraints in model predictive control. AIChE J., 45(6): , J. B. Rawlings and M. J. Risbeck. Model predictive control with discrete actuators: Theory and application. Accepted to Automatica, M. J. Risbeck, C. T. Maravelias, J. B. Rawlings, and R. D. Turney. Closed-loop scheduling for cost minimization in HVAC central plants. In 4th International High Performance Buildings Conference at Purdue, West Lafayette, IN, July J. Stagner. Enterprise optimization solution (EOS) cost savings vs. manual plant dispatching. Report on Central Energy Facility, Stanford Energy System Innovations, C. R. Touretzky and M. Baldea. A hierarchical scheduling and control strategy for thermal energy storage systems. Energ. Buildings, 110: , FOCAPO/CPC / 7

59 References VII M. J. Wenzel, R. D. Turney, and K. H. Drees. Autonomous optimization and control for central plants with energy storage. In 4th International High Performance Buildings Conference at Purdue, West Lafayette, IN, M. Zanon, S. Gros, and M. Diehl. A lyapunov function for periodic economic optimizing model predictive control. In Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, pages , FOCAPO/CPC / 7