**3. Results**

From the results of the simulation studies, it can be concluded that there is a need to design an intermediate protection characteristic situated between Extremely Inverse and Invertim. In relation to the courses of starting currents from Figure 6, depicting a situation where the stator voltage of the motor is in range (0.8 ÷ 1) Un, the parameters α and *C* of Equation (1) were identified in order to designate a characteristic of an overcurrent protection with features intermediate between protections with Standard Inverse characteristics and those with Invertim characteristics. It was assumed in a simplistic manner that the winding temperature in induction devices at starting time up to 20 s will not exceed the permissible temperature for a given class of insulation. In such a situation, the investigated characteristic should run above the characteristics of starting currents for the assumed voltage range, and its shape should be as similar as possible to the Extremely Inverse characteristic. The values of designated parameters of Equation (1) describing the investigated time-current characteristic are α = 2.3 and *C* = 178, respectively. The course of the characteristic with designated parameters α and *C* is marked with the number "6" in Figure 7. As can be seen in that figure, the protection with the characteristic marked as "6" allows for the starting of the auxiliary ventube fan motor in the stator voltage range (0.8 ÷ 1)·Un. At the same time, the protection with this characteristic ensures safety from the effects of a failed starting process, so, in the case where U = 0.7·Un. It must be noted that while this protection is not as effective as devices with the characteristics of Standard Inverse, Very Inverse and Extremely Inverse, it is better than protection devices with Long Inverse and Invertim characteristics. The detailed methodology is described below.

**Figure 7.** Proposed time-current characteristic of an overcurrent protection for an induction motor powering the auxiliary jet fan against the time-current characteristics of overcurrent protections with characteristics in accordance with References [4,16] and time courses of RMS values of starting currents of this motor for various stator voltages.

We attempted to determine a new time-current characteristic of the overload protection, as none of the standard characteristics of the overload protection given in Reference [4] take into account the specific supply and operation of mining machines with extended start-up times. The following assumptions were made to identify the Equation (1) parameter values:


The identification of the Equation (1) parameter values was carried out for the data points specified in the characteristics of Figure 8 that fulfill the above assumptions. Data point values for currents above 3.2 *In* are arbitrary so that the third assumption is met.

In order to calculate the values of the Equation (1) α and *C* parameter values, the Equation was transformed into the following form (2):

$$\hat{I}(k) = I\_{\text{ll}} \left( \frac{\text{C}}{T(k)} + 1 \right)^{\frac{1}{\alpha}} \tag{2}$$

where

*T*(*k*)—data points of the response time values,

ˆ *I*(*k*)—calculated values of currents for given times *T*(*k*) and values of coefficients α and *C*.

*k* = 1, 2, ... , *N*,

*N*—number of data points.

**Figure 8.** Data points selected for the determination of a new time-current characteristic on the background of simulated current RMS waveforms and Very Inverse characteristics. 1—Very Inverse characteristics, 2—selected data points.

Subsequently, iterative computer calculations were carried out to determine from the Equation (2) the values of the *I*(*k*) currents with given values of time *T*(*k*) for the parameters α and *C*, which were selected from the ranges 2.0, 3.0 and 150, 190, respectively. The identification task was therefore reduced to determine such values of parameters α and *C* that would minimize the accepted criterion in the form of the mean square error (MSE). This condition can be written as follows (3):

$$\mathbb{P}(a,\mathbb{C}) = \min\{MSE\} = \min\left\{ \frac{1}{N} \sum\_{k=1}^{N} \left( I(k) - \hat{I}(k) \right)^{2} \right\} \tag{3}$$

where:

*I*(*k*)—data points of current values (from Figure 8) for the corresponding *T*(*k*) time values.

The dependence of MSE value surface on the parameters α and *C* in the whole analyzed area is shown in Figure 9.

In order to verify the results obtained from the identification of the parameters of Equation (1) described above, a second method was used to determine these constants. Equation (1) can be transformed into the following form:

$$y = \left(\frac{\mathbb{C}}{T(k)} + 1\right) = \alpha \ln\left(\frac{I(k)}{I\_n}\right) = \alpha \cdot x \tag{4}$$

This formula represents a linear function. In such a situation, the α coefficient at a given *C* (from the range as in Figure 9) can be determined either by linear regression, or by the method of least squares. The value of Criterion (8) is then calculated for the obtained data. Using Equation (4) significantly reduces the calculation time.

**Figure 9.** Mean square error (MSE) value surface dependence on α and *C* parameter values in the whole analyzed area for the studied motor case

If Equation (4) transforms due to α to the form:

$$\alpha(k) = \frac{\ln\left(\frac{C}{T(k)} + 1\right)}{\ln\left(\frac{I(k)}{I\_w}\right)}\tag{5}$$

then its value for a given *C* should be constant for each point *k*. In fact, there are small differences for different points, so it can be determined as an arithmetic mean for a given value of *C*.

$$\alpha = \frac{1}{N} \sum\_{k=1}^{N} \frac{\ln\left(\frac{C}{T(k)} + 1\right)}{\ln\left(\frac{I(k)}{I\_n}\right)}\tag{6}$$

The performed identification calculations using the Formula (6) and Criterion (7) for *C* in the range as shown in Figure 9 gave identical results as the method described above—using Criterion (3). The dependence of the criterion on the *C* parameter for different α values obtained in this situation is shown in Figure 10.

$$\mathcal{C} = \min \{ MSE \} = \min\_{\mathcal{C}} \left\{ \frac{1}{N} \sum\_{k=1}^{N} \left( I(k) - \hat{I}(k) \right)^{2} \right\} \tag{7}$$

Minimum MSE values were obtained for α = 2.3 and *C* = 178, which is exactly the same as the method based on Formula (2) and Criterion (3). The values of Criteria (3) and (7) are 123 in both cases.

The protection device with the proposed "6" characteristics exhibits features which are intermediate between protections with Very Inverse characteristics and those with Invertim characteristics, while the shape of this characteristic is similar to Extremely Inverse characteristics, which is consistent with the previous assumptions. In more complex operating conditions (e.g., consecutive starts) and using a more complicated model description, characteristics design can be made using more complicated and much more time-consuming evolutionary computations—like Particle Swarm Optimization, as in Reference [22].

**Figure 10.** MSE values obtained when Formula (7) is used to identify α and C parameters.
