MPC-based interval number optimization for electric water heater scheduling in uncertain environments

doi:10.1007/s11708-019-0644-9

Front. Energy

2021, Vol. 15

Issue (1) : 186-200 https://doi.org/10.1007/s11708-019-0644-9

RESEARCH ARTICLE

MPC-based interval number optimization for electric water heater scheduling in uncertain environments

Jidong WANG¹, Chenghao LI¹, Peng LI¹, Yanbo CHE¹(

), Yue ZHOU², Yinqi LI³

¹. Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China
². School of Engineering, Cardiff University, Cardiff CF24 3AA, UK
³. State Grid Chengdu Power Supply Company, Chengdu 610041, China

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Abstract

In this paper, interval number optimization and model predictive control are proposed to handle the uncertain-but-bounded parameters in electric water heater load scheduling. First of all, interval numbers are used to describe uncertain parameters including hot water demand, ambient temperature, and real-time price of electricity. Moreover, the traditional thermal dynamic model of electric water heater is transformed into an interval number model, based on which, the day-ahead load scheduling problem with uncertain parameters is formulated, and solved by interval number optimization. Different tolerance degrees for constraint violation and temperature preferences are also discussed for giving consumers more choices. Furthermore, the model predictive control which incorporates both forecasts and newly updated information is utilized to make and execute electric water heater load schedules on a rolling basis throughout the day. Simulation results demonstrate that interval number optimization either in day-ahead optimization or model predictive control format is robust to the uncertain hot water demand, ambient temperature, and real-time price of electricity, enabling customers to flexibly adjust electric water heater control strategy.

Keywords electric water heater load scheduling interval number optimization model predictive control uncertainty

Corresponding Author(s): Yanbo CHE

Online First Date: 04 September 2019 Issue Date: 19 March 2021

Cite this article:

Jidong WANG,Chenghao LI,Peng LI, et al. MPC-based interval number optimization for electric water heater scheduling in uncertain environments[J]. Front. Energy, 2021, 15(1): 186-200.

URL:

https://academic.hep.com.cn/fie/EN/10.1007/s11708-019-0644-9
https://academic.hep.com.cn/fie/EN/Y2021/V15/I1/186

Fig.1 Thermal characteristic curve of EWH.

Fig.2 Time frame of receding horizon optimization.

Fig.3 Flowchart of the optimization process.

Fig.4 Interval curve of electricity charge.

Fig.5 Interval curve of hot water demand.

Fig.6 Interval curve of ambient temperature.

Tab.1 Parameters of EWH

Fig.7 All possible actual water temperatures when ignoring uncertain information.

Fig.8 All possible actual water temperatures in zero tolerance degree scheme.

Schemes	σ₁	σ₂	Cost/$
Schemes	σ₁	σ₂	p^c	p^w	$p ‾$
Scheme 1	0	0	2.0205	0.2020	1.4695
Scheme 2	0.2	0.2	1.9114	0.1911	1.3901
Scheme 3	0.1	0.3	1.9328	0.1933	1.4056
Scheme 4	0.3	0.1	1.8383	0.1838	1.3369
Scheme 5	0.4	0.4	1.8439	0.1844	1.3410
Scheme 6	0.3	0.5	1.9170	0.1917	1.3942
Scheme 7	0.5	0.3	1.8304	0.1830	1.3312

Tab.2 Cost of different schemes

Fig.9 Hot water temperature in different conservative schemes.

Fig.10 Violation degrees in different conservative schemes.

Fig.11 Actual water temperature under an unexpected situation.

Fig.12 Actual water temperature after receding horizon optimization.

Fig.13 Actual water temperature after MPC-based interval number optimization.

Fig.14 Actual water temperature in the receding horizon optimization.

Fig.15 Electricity cost under different optimization time domain Ts in Scheme 1.

θ_n	Hot water temperature at time t_n
θ_e_,_n	Ambient temperature
Q	Capacity of electric water heater
R	Thermal resistance of electric water heater
C	Thermal capacitance of electric water heater
$X n$	Power states of electric water heater (ON/OFF)
t_n/h	Time at nth step
d_n	Hot water demand at time $t n$
M	Mass of water in full tank
A^I, B^I	Interval number
a^up, a^down	Upper bound and low bound of interval number A^I
B^up, B^down	Upper bound and low bound of interval number B^I
a^c, a^w	Middle point and radius of interval number
$θ n + 1 I$ , $θ n I$	Interval of hot water temperature at time t_n₊₁ and t_n
$d n I$	Interval of hot water demand at time t_n₊₁ and t_n
$θ e, n I$	Interval of ambient temperature at time t_n₊₁ and t_n
$p n I$	Interval of real-time pricing
$θ WH I$	Comfort zone of electric water heater
$θ WH up$ , $θ WH down$	Upper bound and low bound of temperature range
P_EWH	Rated power of electric water heater
N	Number of all time steps over scheduling horizon
f^c(X_n)	Middle point of electricity bill band
f^w(X_n)	Radius of electricity bill band
β	Weight coefficient
δ, φ	Normalizing factors of the middle point and radius
σ	Tolerance degree for constraint violation
α	Penalty factor
r( )	Random number in the interval [0.1]
w	The inertia weight
c₁, c₂	Weighted coefficients of the global and personal best solutions

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