**2.1 Electric water heater description**

The electric water heater is a multiple input single output (MISO) system. The controlled output water temperature will be called hot water temperature (*hwt(t)*). This variable depends of the cold water temperature (*cwt(t)*), water flow (*wf(t)*), power (*p(t)*) and of the electric water heater dynamics. The hot and cold water temperature difference is called delta water temperature (*Δt(t)*).

The electric water heater is physically composed by an electric resistance, a permutation chamber and several sensors used for control and security of the system as shown on figure 2.

Operating range of the *hwt(t)* is from 20 to 50ºC. Operating range of the *cwt(t)* is from 5 to 25ºC. Operating range of the *wf(t)* is from 0,5 to 2,5 litters / minute. Operating range of the *p(t)* is from 0 to 100% of the available power.

Fig. 2. Schematic of the electric water heater: sensors and actuator.

The applied energy in to the heating resistance is controlled using 100 alternated voltage cycles (one second). In each sample, the applied number of cycles is proportional to the delivery energy to the heating element.

Figure 3 shows one photo of the electric water heater and the micro-controller board.

Adaptable PID Versus Smith Predictive Control Applied to an Electric Water Heater System 149

1 1 ( ) - -

*Qe s <sup>M</sup> Wf <sup>s</sup> <sup>s</sup> Wf M*

*t s WfCe WfCe M s td s td e e*

Passing to the discrete domain, with a sampling period of h=1 second and with discrete time

, the final discrete transfer function is illustrated in equation 3.

 <sup>1</sup> ( 1) () 1 ( ( )) *Wf Wf tk e tk M M e Qe k d k WfCe*

*d k*( ) that became from the transformation of energy 2

*d k*( ) that became from the water flow *wf(k)* that circulates in the

 

 

4 ( ) 1,75 /min ( ) 5 1,00 /min ( ) 1,75 /min 6 ( ) 1,00 /min

Considering now the possibility of changes in the water flow, in the discrete domain

2

*e Q M e k d k*

( ( ))

*wf k d k*

<sup>1</sup> <sup>1</sup> ( ( )) ( ( ))

Observing the real data of the system, the absorbed energy *Qe(t)* is a linear static function *f(.)*

*Qe k d k f p k d k* ( ( )) ( ( ))

2

*e f M p k d k*

( ( ))

<sup>1</sup> <sup>1</sup> ( ( )) ( ( ))

*wf k d k*

*to wf k l d k to l wf k l to wf k l*

*d k* , the final transfer function is given in equation 5.

2

( ( ))

*wf k d k tk e M t k*

proportional to the applied electric power *p(t)* as expressed in equation 6.

( 1) ( )

Finally, the discrete global transfer function is given by equation 7.

2

( ( ))

*wf k d k tk e M t k*

 *dk d k d k* () () () 

( 1) ( )

*Wf*

(2)

(3)

*dk s* () 3

(4)

(5)

(7)

*d k*( ) and is the

is given in equation 4, where 1

(6)

( ) <sup>1</sup>

delay ( ) ( ) int( ) 1 *td t d k*

is the fixed part of

permutation chamber.

*Wf=wf(k)* and ( ) <sup>2</sup> 

variable part of

*h*

The real discrete time delay 1 2

2

*wf k d k Ce*

2

*wf k d k Ce*

Fig. 3. Photo of the electric water heater and the micro-controller board.
