# Understanding price functions to control domestic electric water heaters for demand response

A bilevel approach to adapt power consumption to availabilty

## Abstract

A well-known mechanism for demand response is sending price signals to customers a day ahead. Customers then postpone or advance their usage of electricity to minimize cost. Setting up price functions that adapt the customers’ load to availability is a big challenge. This paper investigates the feasibility of finding day-ahead price functions to induce a desired load profile of domestic electric water heaters (DEWHs) minimizing their electricity cost for demand response. Bilevel optimization is applied for a single DEWH using a simplified linear model and full knowledge. This leads to a solvable bilevel problem and allows understanding optimality of price functions and resulting heating profiles. It is shown that with the resulting price functions the DEWH may select many significantly different heating profiles leading to the same cost. Thus the price does not uniquely induce the desired heating profile. The acquired knowledge forms the basis for a procedure to create price functions for controlling the load profile of many DEWHs.

This is a preview of subscription content, log in to check access.

## Abbreviations

C :

Heat capacity of water in DEWH (J/K)

$$C_{\rho }$$ :

Specific heat capacity of water (J/kg K)

$$\mathbf {c}=(c_1,\ldots ,c_n)$$ :

Distribution of hot water consumption

$$c_{DEWH}$$ :

Total cost paid for heating DEWH ($) $$E_{total}$$ : Energy at least needed by DEWH (J) $$\varepsilon$$ : Minimal price for electricity ($/GJ)

$$\mathbf {\varepsilon } = (\varepsilon _1,\varepsilon _2,\ldots ,\varepsilon _n)$$ :

Minimal prices with $$\varepsilon _i = \varepsilon$$ ($/GJ) G : Thermal conductivity between tank of DEWH and environment (W/K) $$\gamma , \gamma (\varDelta t)$$ : Factor describing rate of cooling and heating, see (8) or (13) HWHW(t): (Constant/Function of) Power consumed by drawing hot water (W) $$\mathbf {HW} = (HW_1,\ldots ,$$ $$HW_n)$$ : Power consumed by drawing hot water in slot i (W) $$\mathbf {h}=(h_1,\ldots ,h_n)$$ : Rate of heater’s power used in slot i $$\hat{\mathbf {h}}=(\hat{h}_{1},\hat{h}_{2},\ldots ,\hat{h}_{n})$$ : Rate of heater’s power that should be used in slot i n : Number of time slots in the horizon PP(t): (Constant/Function of) Electrical power consumed by DEWH (W) $$\mathbf {P}=(P_1,\ldots ,P_n)$$ : Average electrical power consumed by DEWH in slot i $$\hat{P}(t)$$ : Power available for DEWH (W) $$\hat{\mathbf {P}}=(\hat{P}_{1},\ldots ,\hat{P}_{n})$$ : Power available for DEWH in slot i (W) $$\hat{\mathbf {P}}_{total}$$ : Power available for all devices (W) $$P_{heater}$$ : Nominal heating power of DEWH (W) p(t): Retail price for electricity ($/J)

$$\mathbf {p}=(p_1,\ldots ,p_n)$$ :

Retail price for electricity in slot i ($/J) $$\mathbf {p}_{ex}=(p_{ex,1},\ldots ,$$ $$p_{ex,n})$$ : Exchange price for electricity ($/J)

T(t):

Water temperature in DEWH ($$^\circ$$C)

$$\mathbf {T}=(T_1,\ldots ,T_n)$$ :

Water temperature in slot i ($$^\circ$$C)

$$T_0$$ :

Initial temperature in DEWH ($$^\circ$$C)

$$T_{cold}$$ :

Temperature of cold water ($$^\circ$$C)

$$T_{env}$$ :

Temperature of environment ($$^\circ$$C)

$$T_{min}$$ :

Minimum temperature in DEWH ($$^\circ$$C)

$$T_{max}$$ :

Maximum temperature in DEWH ($$^\circ$$C)

$$T_{\infty }$$ :

Limit the water temperature converges to at constant conditions, see (10) ($$^\circ$$C)

$$\mathbf {T}_{\infty }$$ :

Value of $$T_{\infty }$$ for each slot i ($$^\circ$$C)

$$\tau , \tau _{ref}, \tau _k, \tau _{price}$$ :

Constants describing rate of temperature change, see (9) (s)

t :

A point in time (s)

$$t_0$$ :

Start time (s)

$$\varDelta t$$ :

Length of each time slot (s)

V :

Volume of the tank of DEWH (l)

$$V_{HW}$$ :

Demand of water with $$T_{min}$$ per day (l)

## References

1. 1.

Chang TH, Alizadeh M, Scaglione A (2013) Real-time power balancing via decentralized coordinated home energy scheduling. IEEE Trans Smart Grid 4(3):1490–1504. doi:10.1109/TSG.2013.2250532

2. 2.

Dempe S (2002) Nonconvex optimization and its applications—foundations of bilevel programming. Kluwer Academic Publisher, Dordrecht

3. 3.

Department for Environment, Food and Rural Affairs (Defra) (2008) Measurement of domestic hot water consumption in dwellings. Tech. rep., Department for Environment, Food and Rural Affairs (Defra). https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/48188/3147-measure-domestic-hot-water-consump.pdf

4. 4.

Destatis (2017) Destatis—statistisches bundesamt. Website. Statistisches bundesamt, Wiesbaden, Germany. https://www.destatis.de/

5. 5.

DOE (2006) Benefits of demand response in electricity markets and recommendations for achieving them. Tech. rep., U. S. Department of Energy. https://emp.lbl.gov/sites/all/files/report-lbnl-1252d.pdf

6. 6.

Kepplinger P, Huber G, Petrasch J (2015) Autonomous optimal control for demand side management with resistive domestic hot water heaters using linear optimization. Energy Build 100:50–55. doi:10.1016/j.enbuild.2014.12.016

7. 7.

Kondoh J, Aki H, Yamaguchi H, Murata A, Ishii I (2005) Future consumed power estimation of time deferrable loads for frequency regulation. In: CIRED 2005—18th international conference and exhibition on electricity distribution, pp 1–4. doi:10.1049/cp:20051215

8. 8.

Konstantin P (2013) Praxisbuch energiewirtschaft—energieumwandlung, -transport und -beschaffung im liberalisierten markt, 3rd edn. Springer, Berlin

9. 9.

Löfberg J (2004) Yalmip: a toolbox for modeling and optimization in matlab. In: Proceedings of the CACSD conference, Taipei, Taiwan

10. 10.

Löfberg J (2016) YALMIP. https://yalmip.github.io/

11. 11.

Lübkert T, Venzke M, Turau V (2016) Impacts of domestic electric water heater parameters on demand response: a simulative analysis of physical and control parameter impacts. Comput Sci Res Dev. doi:10.1007/s00450-016-0321-8

12. 12.

MathWorks: Matlab (2016). https://de.mathworks.com/products/matlab.html

13. 13.

Mayne D, Rawlings J, Rao C, Scokaert P (2000) Constrained model predictive control: stability and optimality. Automatica 36(6):789–814. doi:10.1016/S0005-1098(99)00214-9

14. 14.

Meng FL, Zeng XJ (2013) A Stackelberg game-theoretic approach to optimal real-time pricing for the smart grid. Soft Comput 17(12):2365–2380. doi:10.1007/s00500-013-1092-9

15. 15.

Nehrir M, Jia R, Pierre D, Hammerstrom D (2007) Power management of aggregate electric water heater loads by voltage control. In: IEEE power engineering society general meeting, 2007, pp 1–6

16. 16.

Paull L, Li H, Chang L (2010) A novel domestic electric water heater model for a multi-objective demand side management program. Electr Power Syst Res 80(12):1446–1451

17. 17.

Safdarian A, Fotuhi-Firuzabad M, Lehtonen M (2014) A distributed algorithm for managing residential demand response in smart grids. IEEE Trans Ind Inf 10(4):2385–2393. doi:10.1109/TII.2014.2316639

18. 18.

Safdarian A, Fotuhi-Firuzabad M, Lehtonen M (2016) Optimal residential load management in smart grids: a decentralized framework. IEEE Trans Smart Grid 7(4):1836–1845. doi:10.1109/TSG.2015.2459753

19. 19.

SaniTec Produkthandel GmbH: Fach-Information Warmwassergeräte (2014)

20. 20.

Shaad M, Momeni A, Diduch CP, Kaye M, Chang L (2012) Parameter identification of thermal models for domestic electric water heaters in a direct load control program. IEEE, pp 1–5

21. 21.

Shah JJ, Nielsen MC, Shaffer TS, Fittro RL (2016) Cost-optimal consumption-aware electric water heating via thermal storage under time-of-use pricing. IEEE Trans Smart Grid 7(2):592–599. doi:10.1109/TSG.2015.2483502

22. 22.

SPOT E (2017) Epex spot se: welcome. Website. EPEX SPOT SE, Paris. http://www.epexspot.com/

23. 23.

Stryi-Hipp G, et al (2007) Grosol—studie zu großen solarwärmeanlagen. Survey. Bundesverband-Solarwirtschaft e.V, Berlin, Germany. http://www.erneuerbare-energien.de/fileadmin/ee-import/files/pdfs/allgemein/application/pdf/studie_grosol.pdf

24. 24.

Sundström O, Binding C, Gantenbein D, Berner D, Rumsch WC (2012) Aggregating the flexibility provided by domestic hot-water boilers to offer tertiary regulation power in switzerland. In: 2012 3rd IEEE PES innovative smart grid technologies Europe (ISGT Europe), pp 1–7. doi:10.1109/ISGTEurope.2012.6465643

25. 25.

Vardakas JS, Zorba N, Verikoukis CV (2015) A survey on demand response programs in smart grids: pricing methods and optimization algorithms. IEEE Commun Surv Tutor 17(1):152–178. doi:10.1109/COMST.2014.2341586

26. 26.

Venzke M, Turau V (2016) Simulative evaluation of demand response approaches for waterbeds. In: Proceedings of the 2016 IEEE international energy conference (ENERGYCON)

27. 27.

Von Stackelberg H (1934) Marktform und gleichgewicht. Springer, Berlin

28. 28.

von Stackelberg H (2011) Market structure and equilibrium. Springer, Berlin. doi:10.1007/978-3-642-12586-7

29. 29.

Wei W, Liu F, Mei S (2015) Energy pricing and dispatch for smart grid retailers under demand response and market price uncertainty. IEEE Trans Smart Grid 6(3):1364–1374. doi:10.1109/TSG.2014.2376522

30. 30.

Xu Z, Diao R, Lu S, Lian J, Zhang Y (2014) Modeling of electric water heaters for demand response: a baseline PDE model. IEEE Trans Smart Grid 5(5):2203–2210. doi:10.1109/TSG.2014.2317149

31. 31.

Zugno M, Morales JM, Pinson P, Madsen H (2013) A bilevel model for electricity retailers’ participation in a demand response market environment. Energy Econ 36:182–197. doi:10.1016/j.eneco.2012.12.010

## Author information

Authors

### Corresponding author

Correspondence to Tobias Lübkert.

## Rights and permissions

Reprints and Permissions