## 2 Materials and Methods
### The Proposed Algorithm
The flow of the power through the buffers is shown in Figure 4.  
Figure 3: Temperature changes occurring in steady state, with an input signal of 10.1% duty cycle and a period of 35 s.  
The following notation is used here:  
\(T_{s}\)**--a sampling period of 200 ms used in the system under test;  
\(t_{i}=iT_{s}\)**--the \(i\)-th discrete time instant;  
\(n\)**--number of heaters;  
\(u_{j}(i)\in\{0,1\}\)**--the output of the controller of the heater \(j\) at a discrete time instant \(t_{i}\);  
\(z_{j}(i)\in\{0,1\}\)**--state of the relay activating heater \(j\) at a discrete time instant \(t_{i}\);  
\(e\)**--quantum of energy consumed by the switched on heater during the sampling period;  
\(e_{sp}\)**--buffered energy level set-point expressed as the number of quanta \(e\);  
\(k_{j}(i)\)**--number of units in the heater buffer \(j\) at discrete time instant \(t_{i}\);  
\(m(i)\)**--number of heaters switched on at a discrete time instant \(t_{i}\);  
\(P(i)\)**--power source load at a discrete time instant \(t_{i}\). The equation for \(k_{j}\) at time \(t_{i}\) is as follows:  
\[k_{j}(i)=k_{j}(i-1)+u_{j}(i)-z_{j}(i-1). \tag{1}\]  
In each time sample, the number \(l\) of buffers in which the level of stored energy is greater than the assumed set level \(e_{sp}\) is determined. The number of heaters that should be switched on indicates the level to which the system should move in a smooth way. By operating on the number of machines being switched on instead of controlling them separately, this difficult multidimensional problem is reduced to a one-dimensional system, where the number of heaters to be switched on is determined by averaging the number of heaters waiting to be switched on. Therefore, a simple Proportional-Integral (PI) controller has been used, in which the input is a one-step prediction error of the number of machines to be switched on, and the output is the number of machines that will be switched on at a given time \(t_{i}\). The applied controller with an integral element smoothly follows the changes by averaging the error signal \(e_{r}\), thus not causing peak loads on the power source. The PI controller equation is described by the following formula:  
\[m(i)=K\Bigg{(}e_{r}(i)+\frac{T_{s}}{T_{i}}\sum_{k=0}^{i}e_{r}(i)\Bigg{)}, \tag{2}\]  
where  
Figure 4: The flow of the power through the buffers. \(e_{r}(i)=l(i)-m(i-1)\) is a one step ahead prediction error;  
\(l(i)\)--the number of buffers for which the level of stored energy is greater than \(\varepsilon_{sp}\);  
\(K\)--controller gain;  
\(T_{i}\)--integration time. The presented controller can be interpreted as a low-pass filter of changes in the input signal [43], which is the number of switched-on machines, causing a constant component to appear at its output corresponding to the value of the average power used to heat the machines. This component follows the average load of the electrical network as it changes due to the disturbances affecting the system under study. The PI controller used in the system operates as a low-pass filter for the power signal that should be used to maintain the filling of the virtual energy buffers at a constant level. These buffers are connected between the output of each of the furnace temperature controllers and the input of the actuator. The filling of these is related to the amount of energy going into them from the controllers and its release in the form of heater activation. Filling the virtual buffers at a constant level means that energy is continuously transferred from the regulators to the heaters. Maintaining the average filling of the buffers at a constant level allows the heater activations to be shifted in time. In order to achieve temperature stabilisation at a set level, the dynamics of the filter built on the PI controller should be chosen to match the dynamics of the object, which is the furnace of the creep test machine. The developed algorithm specifies the number of heaters to be switched on at time \(t_{i}\) but does not specify exactly which heaters should be switched on.