Mixed-integer optimization methods for online scheduling in large-scale HVAC systems

Abstract

Due to time-varying utility prices, peak demand charges, and variable-efficiency equipment, optimal operation of heating ventilation, and air conditioning systems in campuses or large buildings is nontrivial. Given forecasts of ambient conditions and utility prices, system energy requirements can be reduced by optimizing heating/cooling load within buildings and then choosing the best combination of large chillers, boilers, etc., to meet that load while accounting for switching constraints and equipment performance. With the presence of energy storage, utility costs can be further reduced by temporally shifting production, which adds an additional layer of complexity. Furthermore, due to changes in market and weather conditions, it is necessary to revise a given schedule regularly as updated information is received, which means the problem must be tractable in real time (e.g., solvable within 15 min). In this paper, we present a mixed-integer linear programming model for this problem along with reformulations, decomposition approaches, and approximation strategies to improve tractability. Simulations are presented to illustrate the effectiveness of these methods. By removing symmetry from identical equipment, decomposing the problem into subproblems, and approximating longer-timescale behavior, large instances can be solved in real time to within 1% of the true optimal solution.

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References

  1. 1.

    Powell, K.M., Cole, W.J., Ekarika, U.F., Edgar, T.F.: Optimal chiller loading in a district cooling system with thermal energy storage. Energy 50, 445–453 (2013)

    Article  Google Scholar 

  2. 2.

    Albadi, M.H., El-Saadany, E.F.: Demand response in electricity markets: an overview. In: 2007 IEEE Power Engineering Society General Meeting, IEEE, pp. 1–5 (2007)

  3. 3.

    Touretzky, C.R., Baldea, M.: Integrating scheduling and control for economic MPC of buildings with energy storage. J. Process Control 24(8), 1292–1300 (2014)

    Article  Google Scholar 

  4. 4.

    Henze, G.P.: Energy and cost minimal control of active and passive building thermal storage inventory. J. Sol. Energy Eng. 127(3), 343–351 (2005)

    Article  Google Scholar 

  5. 5.

    Henze, G.P., Felsmann, C., Knabe, G.: Evaluation of optimal control for active and passive building thermal storage. Int. J. Therm. Sci. 43(2), 173–183 (2004)

    Article  Google Scholar 

  6. 6.

    Henze, G.P., Biffar, B., Kohn, D., Becker, M.P.: Optimal design and operation of a thermal storage system for a chilled water plant serving pharmaceutical buildings. Energy Build. 40(6), 1004–1019 (2008)

    Article  Google Scholar 

  7. 7.

    Rawlings, J., Patel, N., Risbeck, M., Maravelias, C., Wenzel, M., Turney, R.: Economic MPC and real-time decision making with application to large-scale hvac energy systems. Comput. Chem. Eng. 114, 89–98 (2017)

    Article  Google Scholar 

  8. 8.

    Ma, J., Qin, J., Salsbury, T., Xu, P.: Demand reduction in building energy systems based on economic model predictive control. Chem. Eng. Sci. 67(1), 92–100 (2012)

    Article  Google Scholar 

  9. 9.

    Oldewurtel, F., Parisio, A., Jones, C.N., Gyalistras, D., Gwerder, M., Stauch, V., Lehmann, B., Morari, M.: Use of model predictive control and weather forecasts for energy efficient building climate control. Energy Build. 45, 15–27 (2012)

    Article  Google Scholar 

  10. 10.

    Ma, Y., Matuško, J., Borrelli, F.: Stochastic model predictive control for building HVAC systems: complexity and conservatism. IEEE Trans. Control Syst. Technol. 23(1), 101–116 (2015)

    Article  Google Scholar 

  11. 11.

    Ma, Y., Borrelli, F., Hencey, B., Coffey, B., Bengea, S.C., Haves, P.: Model predictive control for the operation of building cooling systems. IEEE Control Syst. Technol. 20(3), 796–803 (2012)

    Article  Google Scholar 

  12. 12.

    Touretzky, C.R., Baldea, M.: A hierarchical scheduling and control strategy for thermal energy storage systems. Energy Build. 110, 94–107 (2016)

    Article  Google Scholar 

  13. 13.

    Kapoor, K., Powell, K.M., Cole, W.J., Kim, J.S., Edgar, T.F.: Improved large-scale process cooling operation through energy optimization. Processes 1(3), 312–329 (2013)

    Article  Google Scholar 

  14. 14.

    Risbeck, M.J., Maravelias, C.T., Rawlings, J.B., Turney, R.D.: A mixed-integer linear programming model for real-time cost optimization of building heating, ventilation, and air conditioning equipment. Energy Build. 142, 220–235 (2017)

    Article  Google Scholar 

  15. 15.

    Maravelias, C.T.: General framework and modeling approach classification for chemical production scheduling. AIChE J. 58(6), 1812–1828 (2012)

    Article  Google Scholar 

  16. 16.

    Harjunkoski, I., Maravelias, C.T., Bongers, P., Castro, P.M., Engell, S., Grossmann, I.E., Hooker, J., Méndez, C., Sand, G., Wassick, J.: Scope for industrial applications of production scheduling models and solution methods. Comput. Chem. Eng. 62, 161–193 (2014)

    Article  Google Scholar 

  17. 17.

    Kondili, E., Pantelides, C., Sargent, R.: A general algorithm for short-term scheduling of batch operations–I MILP formulation. Comput. Chem. Eng. 17(2), 211–227 (1993)

    Article  Google Scholar 

  18. 18.

    Pantelides, C.C.: Unified frameworks for optimal process planning and scheduling. In: Proceedings on the Second Conference on Foundations of Computer Aided Operations, pp. 253–274 (1994)

  19. 19.

    Méndez, C.A., Cerdá, J., Grossmann, I.E., Harjunkoski, I., Fahl, M.: State-of-the-art review of optimization methods for short-term scheduling of batch processes. Comput. Chem. Eng. 30(6–7), 913–946 (2006)

    Article  Google Scholar 

  20. 20.

    Velez, S., Maravelias, C.T.: Reformulations and branching methods for mixed-integer programming chemical production scheduling models. Ind. Eng. Chem. Res. 52(10), 3832–3841 (2013)

    Article  Google Scholar 

  21. 21.

    Vin, J.P., Ierapetritou, M.G.: A new approach for efficient rescheduling of multiproduct batch plants. Ind. Eng. Chem. Res. 39(11), 4228–4238 (2000)

    Article  Google Scholar 

  22. 22.

    Mendez, C.A., Cerdá, J.: An milp framework for batch reactive scheduling with limited discrete resources. Comput. Chem. Eng. 28(6–7), 1059–1068 (2004)

    Article  Google Scholar 

  23. 23.

    Touretzky, C.R., Harjunkoski, I., Baldea, M.: Dynamic models and fault diagnosis-based triggers for closed-loop scheduling. AIChE J. 63(6), 1959–1973 (2017)

    Article  Google Scholar 

  24. 24.

    Rawlings, J.B., Mayne, D.Q., Diehl, M.M.: Model Predictive Control: Theory, Computation and Design. Nob Hill Publishing, Madison (2017)

    Google Scholar 

  25. 25.

    Gupta, D., Maravelias, C.T.: On deterministic online scheduling: major considerations, paradoxes and remedies. Comput. Chem. Eng. 94, 312–330 (2016)

    Article  Google Scholar 

  26. 26.

    Gupta, D., Maravelias, C.T., Wassick, J.M.: From rescheduling to online scheduling. Chem. Eng. Res. Des. 116, 83–97 (2016)

    Article  Google Scholar 

  27. 27.

    Lee, T.-S., Liao, K.-Y., Lu, W.-C.: Evaluation of the suitability of empirically-based models for predicting energy performance of centrifugal water chillers with variable chilled water flow. Appl. Energy 93, 583–595 (2012)

    Article  Google Scholar 

  28. 28.

    Li, Z., Ierapetritou, M.G.: Process scheduling under uncertainty: review and challenges. Comput. Chem. Eng. 32(4–5), 715–727 (2008)

    Article  Google Scholar 

  29. 29.

    Li, Z., Floudas, C.A.: A comparative theoretical and computational study on robust counterpart optimization: III improving the quality of robust solutions. Ind. Eng. Chem. Res. 53(33), 13112–13124 (2014)

    Article  Google Scholar 

  30. 30.

    Shi, H., You, F.: A computational framework and solution algorithms for two-stage adaptive robust scheduling of batch manufacturing processes under uncertainty. AIChE J. 62(3), 687–703 (2016)

    Article  Google Scholar 

  31. 31.

    Lappas, N.H., Gounaris, C.E.: Multi-stage adjustable robust optimization for process scheduling under uncertainty. AIChE J. 62(5), 1646–1667 (2016)

    Article  Google Scholar 

  32. 32.

    Du, J., Park, J., Harjunkoski, I., Baldea, M.: A time scale-bridging approach for integrating production scheduling and process control. Comput. Chem. Eng. 79, 59–69 (2015)

    Article  Google Scholar 

  33. 33.

    Nie, Y., Biegler, L.T., Villa, C.M., Wassick, J.M.: Discrete time formulation for the integration of scheduling and dynamic optimization. Ind. Eng. Chem. Res. 54(16), 4303–4315 (2015)

    Article  Google Scholar 

  34. 34.

    Feng, J.D., Chuang, F., Borrelli, F., Bauman, F.: Model predictive control of radiant slab systems with evaporative cooling sources. Energy Build. 87, 199–210 (2015)

    Article  Google Scholar 

  35. 35.

    Mendoza-Serrano, D.I., Chmielewski, D.J.: HVAC control using infinite-horizon economic MPC. In: IEEE 51st Annual Conference on Decision and Control (CDC), pp. 6963–6968 (2012)

  36. 36.

    Vielma, J.P., Ahmed, S., Nemhauser, G.: Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper. Res. 58(2), 303–315 (2010)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison (2009)

    Google Scholar 

  38. 38.

    Wolsey, L.A.: Integer Programming. Wiley, New York (1998)

    MATH  Google Scholar 

  39. 39.

    Patel, N.N.R., Risbeck, M.J., Rawlings, J.B., Wenzel, M.M.J., Turney, R.D.: Distributed economic model predictive control for large-scale building temperature regulation. In: American Control Conference, Boston, MA, pp. 895–900 (2016)

  40. 40.

    Zavala, V.M., Constantinescu, E.M., Krause, T., Anitescu, M.: On-line economic optimization of energy systems using weather forecast information. J. Process Control 19(10), 1725–1736 (2009)

    Article  Google Scholar 

  41. 41.

    ElBsat, M.N., Wenzel, M.J.: Load and electricity rates prediction for building wide optimization applications. In: 4th International High Performance Buildings Conference at Purdue, West Lafayette, IN (2016)

Download references

Acknowledgements

Funding, equipment models, and sample data provided by Johnson Controls, Inc. Additional funding provided by the National Science Foundation (Grant #CTS-1603768).

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Correspondence to Christos T. Maravelias.

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Appendices

Appendix A: Notation

A.1 Sets

\(i \in \mathbf{I}\) :

Temperature zones

\(j \in \mathbf{J}\) :

Type of generators

\(k \in \mathbf{K}\) :

Resources/utilities

\(t \in \mathbf{T}\) :

Time periods

\(m \in \mathbf{M}\) :

Interpolation regions for equipment models

\(n \in \mathbf{N}\) :

Interpolation points for equipment models

\(\mathbf{M}_{j} \subseteq \mathbf{M}\) :

Interpolation regions for equipment j

\(\mathbf{N}_{jm} \subseteq \mathbf{N}\) :

Interpolation points contained in region m for equipment j

A.2 Parameters

The following parameters are given based on available forecasts or desired system performance.

\(\phi _{kt}\):

Forecast secondary resource demand

\(\rho _{kt}\):

Forecast resource prices

\(\rho ^{\text {max}}_{k}\):

Demand charge

\(\psi _{kt}\):

Exogenous demand included in peak calculation

\(\beta _{kt}\):

Penalty for unmet secondary demand

\(\varTheta ^-_{it}, \varTheta ^+_{it}\):

Lower and upper temperature comfort bounds for zones

\(\varPi ^{\text {max}}_{k}\):

Previous maximum resource demand

\(\mu _j\):

Number of units of each type available for use

\(\zeta _{jknt}\):

Interpolation points for piecewise-linear models

\(\sigma _k\):

Fractional retention of stored resource

\(\delta ^+_j, \delta ^-_j\):

Minimum on/off dwell time for generators

\(\varPi _k\):

Upper bound for resource purchase

\(\varUpsilon _k\):

Bound for single-period storage charge/discharge

\(\varSigma _k\):

Storage capacity

\(\alpha _{ii'}, \omega _{ii'k}\):

Dynamic coefficients in airside model

\(\theta _{it}\):

Time-varying temperature disturbance for zones

\(\chi ^+_{it}, \chi ^-_{it}\):

Penalties for comfort violation in zones

\(\varGamma _{ik}\):

Upper bound on resource flow to zones

\(\xi _{jnt}\):

Part-load coefficients for piecewise-linear models

\(\lambda _{ikt}\):

Dual multiplier for resource supply to airside

A.3 Decision variables

The following decision variables are primarily for the waterside subsystem.

\(U_{jt} \in \{0,\cdots ,\mu _j\}\):

Integer number of generators on during a given time

\(U^+_{jt}, U^-_{jt}\):

Number of generators switched on/off at the beginning of a time period

\(Q_{jkt}\):

Production of resources in equipment

\(P_{kt} \in {[}0,\varPi _k{]}\):

Amount of purchased resources

\(P^{\text {max}}_{k} \in [\varPi ^{\text {max}}_{k},\infty )\):

Maximum single-period purchase of resource k

\(Y_{kt} \in {[}-\varUpsilon _k,\varUpsilon _k{]}\):

Charge (\(<0\)) or discharge (\(>0\)) of stored resources

\(S_{kt} \in {[}0,\varSigma _k{]}\):

Stored inventory at the end of a time period

\(B_{kt} \in {[}0,\phi _{kt}{]}\):

Unmet secondary demand

\(V_{jmt} \in \{0,\cdots ,\mu _j\}\):

Number of units operating in each interpolation region

\(Z_{jmnt} \in {[}0,\mu _j{]}\):

Weighting of interpolation points in each region

\(X_{lt}\):

States of the airside dynamic model

\(T_{it}\):

Zone temperature of zone at end of a time period

\(G_{ikt} \in {[}0,\varGamma _{ik}{]}\):

Primary demand of resources by zones

\(T^+_{it}, T^-_{it}\):

Positive and negative comfort violation for zones

Appendix B: Formulation comparison

The following table shows performance results for different formulations and decomposition strategies applied to various instance. The first four columns give the number of temperature zones I, generators J, resources K, and time points T for the instance. The remaining four columns give the performance the following four formulations:

  • The original Symmetric formulation

  • The Symmetry-Free reformulation from Sect. 3.1

  • The Airside/Waterside decomposition based on Lagrangian relaxation

  • The Hierarchical decomposition using aggregate system curves.

Each entry shows the objective function and estimated optimality gap, along with the solution time required to obtain those values. Note that in each row, the objective function is normalized to the best value at 0% with other objectives given as a percentage increase over that value. Gaps are also given as a percentage of the best solution. For problems without an airside model, the two decomposition strategies are unnecessary (and thus listed as N/A for those rows).

IJKTSymmetricSymmetry-freeAirside/watersideHierarchical
01011480.04% (0.71% Gap)0.00% (0.00% Gap)N/AN/A
10.01 min0.40 min
010111200.21% (1.02% Gap)0.00% (0.01% Gap)N/AN/A
10.01 min10.00 min
010111680.39% (1.13% Gap)0.00% (0.03% Gap)N/AN/A
10.01 min10.00 min
02011480.03% (1.00% Gap)0.00% (0.00% Gap)N/AN/A
33.59 min3.40 min
020111200.53% (1.73% Gap)0.00% (0.18% Gap)N/AN/A
10.01 min10.00 min
020111681.19% (2.48% Gap)0.00% (0.23% Gap)N/AN/A
10.02 min10.00 min
03011480.41% (1.26% Gap)0.00% (0.00% Gap)N/AN/A
10.01 min0.82 min
03011120No Solution 0.00%(0.06% Gap)N/AN/A
10.01 min10.00 min
03011168No Solution 0.00%(0.06% Gap)N/AN/A
10.01 min10.01 min
05011480.09% (1.62% Gap)0.00% (0.00% Gap)N/AN/A
10.01 min3.30 min
05011120No Solution 0.00%(0.12% Gap)N/AN/A
10.02 min10.00 min
05011168No Solution 0.00%(0.18% Gap)N/AN/A
10.02 min10.00 min
010011480.27% (2.40% Gap)0.00% (0.00% Gap)N/AN/A
10.01 min3.18 min
010011120No Solution 0.00%(0.01% Gap)N/AN/A
10.02 min10.00 min
010011168No Solution 0.00%(0.02% Gap)N/AN/A
10.03 min10.00 min
IJKTSymmetricSymmetry-freeAirside/watersideHierarchical
4104480.09% (1.01% Gap)0.01% (0.07% Gap)0.00% (0.53% Gap)0.00% (1.28% Gap)
10.00 min10.00 min5.25 min5.28 min
41041200.06% (0.76% Gap)0.02% (0.13% Gap)0.00% (0.22% Gap)0.01% (2.46% Gap)
10.00 min10.00 min8.18 min5.31 min
41041680.32% (1.06% Gap)0.03% (0.17% Gap)0.00% (0.27% Gap)0.03% (2.70% Gap)
10.01 min10.00 min10.00 min6.61 min
20104480.26% (1.21% Gap)0.00% (0.02% Gap)0.00% (0.49% Gap)0.00% (1.74% Gap)
10.00 min10.00 min4.61 min4.05 min
201041200.95% (1.69% Gap)0.02% (0.26% Gap)0.00% (0.25% Gap)0.01% (3.06% Gap)
10.00 min10.00 min7.86 min5.39 min
201041680.88% (1.64% Gap)0.15% (0.50% Gap)0.00% (0.28% Gap)0.05% (5.13% Gap)
10.01 min10.01 min10.02 min6.14 min
50104480.31% (1.27% Gap)0.06% (0.20% Gap)0.00% (0.50% Gap)0.00% (1.75% Gap)
10.00 min10.00 min3.74 min5.16 min
50104120No Solution1.77% (0.00% Gap)0.00% (0.25% Gap)0.00% (1.53% Gap)
10.01 min6.67 min7.80 min10.12 min
50104168No Solution1.00% (1.44% Gap)0.00% (0.27% Gap)0.12% (8.33% Gap)
10.13 min10.03 min10.08 min8.08 min
100104480.38% (1.35% Gap)0.34% (1.02% Gap)0.00% (0.50% Gap)0.00% (1.75% Gap)
20.01 min20.03 min10.39 min5.33 min
100104120No Solution1.08% (1.71% Gap)0.00% (0.24% Gap)0.01% (1.50% Gap)
20.04 min20.03 min12.59 min14.89 min
10010416829.25% (29.99% Gap)No Solution0.02% (0.28% Gap)0.00% (2.46% Gap)
20.05 min20.04 min15.94 min14.54 min
25010448No SolutionNo Solution0.00% (0.51% Gap)0.00% (1.75% Gap)
30.06 min30.18 min15.92 min16.52 min
250104120No SolutionNo Solution0.01% (0.24% Gap)0.00% (2.23% Gap)
30.14 min30.14 min18.54 min19.53 min
250104168No SolutionNo Solution0.00% (0.27% Gap)0.05% (2.30% Gap)
30.24 min30.27 min21.17 min34.51 min

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Risbeck, M.J., Maravelias, C.T., Rawlings, J.B. et al. Mixed-integer optimization methods for online scheduling in large-scale HVAC systems. Optim Lett 14, 889–924 (2020). https://doi.org/10.1007/s11590-018-01383-9

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Keywords

  • Large-scale HVAC systems
  • Online optimization
  • Closed-loop scheduling