2024 AIChE Annual Meeting

(116a) Handling Model Mismatch in Economic Model Predictive Control for Heat Pump Water Heaters

Authors

dela Rosa, L. - Presenter, California State University Long Beach
Mande, C., University of California, Davis
Ellis, M., University of California, Davis
Water heating accounts for a considerable portion of energy consumption (18%) and greenhouse gas emissions (15%) in U.S. residential buildings [1,2]. As renewable energy sources continue to penetrate the U.S. electric grid, transitioning to electric heat pump water heaters (HPWHs) is key to decarbonize buildings. HPWHs employ a vapor-compression heat pump (HP) cycle, making them more energy efficient than electric resistance water heaters. However, most HPWHs are hybrid devices, consisting of both a HP and backup electric resistance elements. This combination is necessary due to the lower heating capacity of conventional HPs compared to resistive elements, ensuring rapid recovery during high hot water use [3]. Yet, the rise in HPWH installations increases grid demand [4]. Economic model predictive control (MPC) is ideal for automating load flexibility in HPWHs, as it can explicitly account for utility rate signals and hot water use forecasts to preheat during low-demand periods while maintaining comfort.

In the context of HPWHs, the MPC can utilize a tank thermal dynamic model to predict tank temperature evolution, facilitating the design of a preheating strategy aimed at maintaining the tank temperatures within the desired comfort range. Several works have developed and validated PDEs and multi-node ODEs for modeling tank thermal stratification (e.g., [5]-[7]). However, these models are often nonlinear to account for the buoyancy effect caused by temperature inversion. Moreover, HPWHs typically use a HP with a constant-speed compressor and resistance heating with constant resistance. Thus, the MPC decisions for HPWHs are binary. Utilizing a nonlinear predictive model alongside binary decisions in the MPC results in a mixed-integer nonlinear problem (MINLP) which is not suitable for real-time control. Alternatively, several studies use a one-node, linear model of a HPWH to convert the MPC problem into an MILP, predicting the average tank temperature while neglecting thermal stratification (e.g., [8]-[10]). In contrast, Jin et al. employs a 2-node model of a HPWH in the MPC, yielding higher HP utilization and lower resistance element use compared to a 12-node model. This is due to underestimation of the lower tank temperature and overestimation of the upper tank temperature in the 2-node model [11]. Despite the prevalent use of one-node, linear HPWH models in the MPC literature, there is a lack of understanding on the implications of this modeling simplification on MPC performance, particularly when considering the use of back-up resistive heating. For example, Buechler et al. use a one-node model for a two-element resistive water heater in the MPC, enabling only the activation of the lower resistance element despite the potential preference of using the upper element to heat water drawn from the top of the tank [12].

In this work, we examine the impact of model mismatch on MPC for HPWHs equipped with two backup resistive elements. Specifically, a 20-node nonlinear model is used to simulate the actual HPWH, while a one-node, linear HPWH model is used as the prediction model in the MPC. This study highlights adverse effects that can arise from such modeling discrepancies, such as overheating and unnecessary tank heating attributed to the use of resistance heating. To mitigate these issues, we introduce constraints on resistance heating elements within the MPC to reduce scalding risks and minimize additional energy costs. We then apply this MPC approach to the simulated HPWH to evaluate its effectiveness in alleviating the adverse effects stemming from model mismatch.

References

[1] “2020 residential energy consumption survey,” U.S. Energy Information Administration, 2023. [Online]. Available: https://www.eia.gov/consumption/residential/data/2020/c&e/pdf/ce3.1.pdf

[2] J. Leung, “Decarbonizing U.S. buildings,” Center for Climate and Energy Solutions, Tech. Rep., 2018.

[3] C. Shapiro and S. Puttagunta, “Field performance of heat pump water heaters in the northeast,” U.S. Department of Energy, Tech. Rep., Feb 2016.

[4] A. Brooks, “Thermal load shifting with heat pump water heaters: Technology brief,” Association for Energy Affordability, Tech. Rep., 2021.

[5] A. L. Nash, A. Badithela, and N. Jain, “Dynamic modeling of a sensible thermal energy storage tank with an immersed coil heat exchanger under three operation modes,” Applied Energy, vol. 195, pp. 877–889, 2017.

[6] K. M. Powell and T. F. Edgar, “An adaptive-grid model for dynamic simulation of thermocline thermal energy storage systems,” Energy Conversion and Management, vol. 76, pp. 865–873, 2013.

[7] Z. Xu, R. Diao, S. Lu, J. Lian, and Y. Zhang, “Modeling of electric water heaters for demand response: A baseline PDE model,” IEEE Transactions on Smart Grid, vol. 5, pp. 2203–2210, Sept. 2014.

[8] E. M. Wanjiru, S. M. Sichilalu, and X. Xia, “Model predictive control of heat pump water heater-instantaneous shower powered by integrated renewable-grid energy systems,” Applied Energy, vol. 204, pp. 1333–1346, 2017.

[9] L. dela Rosa, C. Mande, and M. J. Ellis, “Supervisory multi-objective economic model predictive control for heat pump water heaters for cost and carbon optimization,” in ASHRAE Transactions, Atlanta, GA, USA, Feb. 2023, pp. 517–525.

[10] L. dela Rosa, C. Mande, H. Richardson, and M. J. Ellis, “Integrating greenhouse gas emissions into model predictive control of heat pump water heaters,” in Proceedings of the American Control Conference, San Diego, CA, USA, Jun 2023, pp. 4044–4050.

[11] X. Jin, J. Maguire, and D. Christensen, “Model predictive control of heat pump water heaters for energy efficiency,” in Proceedings of the 18th ACEEE Summer Study on Energy Efficiency in Buildings, Pacific Grove, CA, USA, Aug. 2014, pp. 133–145.

[12] E. Buechler, A. Goldin, and R. Rajagopal, “Improving the Load Flexibility of Stratified Electric Water Heaters: Design and Experimental Validation of MPC Strategies,” IEEE Transactions on Smart Grid, 2024.