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

A bilevel approach to adapt power consumption to availabilty


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.

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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)


(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


(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)


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)


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)


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Lübkert, T., Venzke, M., Vo, N. et al. Understanding price functions to control domestic electric water heaters for demand response. Comput Sci Res Dev 33, 81–92 (2018). https://doi.org/10.1007/s00450-017-0349-4

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  • Smart grid
  • Bilevel optimization
  • Demand response
  • Pricing schemes
  • Water heater