where:


Relationship (6) describes the maximum theoretical voltage drop value. Considering the measurement accuracy, both the statistical and dynamic errors, and the fact that the value of *dv* % is a certain average value, the criterion value of . *V*HVmax should be adopted as considerably smaller, as confirmed by the simulation study results reported further in this section.

In cases when d *Q*/d*V* > 0, a change (reduction) of the preset voltage value occurs in the proposed algorithm. The purpose of this is to reduce the uptake of reactive power. The preset voltage value should be as small as possible and, at the same time, should satisfy the following two conditions:


Bearing the above in mind, and also to estimate the risk of occurring a dramatic reactive power uptake in situations, where the coe fficient d *Q*/d*V* < 0, it would be advisable to determine the voltage susceptibility characteristic, *V* = f( *Q*), for each HV/MV substation. The above-mentioned characteristic can be determined through: simulation studies, a network experiment, or, preferably, by the analysis of the results of both investigation types. Regardless of the adopted method, a revision of the current state of the network should be made, including the parameters of network elements, voltage e ffects in vulnerable network nodes, caused by a change in transformer voltage ration, or, last but not least, the applied types and settings of the safeguards of induction machines in individual network nodes. The accuracy of the model of particular network elements, including safeguards, will determine the reliability of results obtained by means of simulation studies. In the case of a network experiment, a good knowledge of the network under examination will enable the experiment to be safely carried out and reliable results to be obtained. This e ffect will only be obtained, when a su fficiently low voltage level is successfully obtained, and no emergency trips of the drives occur due to carrying out the network experiment. As indicated by the characteristics shown in Figures 4 and 5, an extremum of the function occurs at voltages lower than the permissible long-lasting voltage; hence, it is essential that the network experiment is carried out at a possibly low voltage. A low voltage level in HV/MV substations increases the risk of emergency trips of asynchronous drives inside the network, caused by an excessive current increase or too low voltage. In practice, two types of safeguards are used for protecting asynchronous drives connected to an MV network. The primary type includes overcurrent protections with a time-dependent characteristic. A secondary, less often used safeguard are undervoltage protections—with typical settings being (0.7–0.8) *V*n.

#### *6.2. Methodology for dQ*/*dV Coe*ffi*cient Determination*

In order to determine the *V* = f( *Q*) characteristic and the value of *V*ext., a change of supply voltage should be induced. The simplest and most e ffective method of changing voltage in an HV/MV substation is by changing the voltage ratio of the power transformer.

The conditions for conducting the experiment can be defined as follows:


The proposed course of the test:


In a situation where a bank of capacitors is installed in the HV/MV substation. The tests should be carried out with the capacitors bank turned on and <sup>o</sup>ff, respectively. It should be borne in mind that in the conditions of normal operation, in the situation of lowered voltage, the capacitor bank shall be turned on. Its voltage susceptibility characteristic (negative value of the d*Q*/d*V* coefficient) may noticeably change the *Q*subst = f(*V*subst) curve. If turning the capacitor bank on–off during the experiment could cause the maximum voltage to be exceeded, the bank should stay turned <sup>o</sup>ff.

To emulate the course of the test to determine the *Q*subst = f(*V*subst) characteristic, simulation studies were carried out using a model of a 91-node real MV network. The model included 58 MV nodes (interconnected with 66 MV cable lines) and 33 MV/LV substations. A simplified diagram of the network is shown in Figure 9.

Two types of load models are used in each node. The constant admittance model and the dynamic asynchronous motor model were both taken from the library of the PowerFactory program. The motor power was selected to obtain the assumed percentage of asynchronous machines while retaining the reference power value in a node. As the effect of the voltage level at which measurements are made is significant, two variants of the initial supply voltage value were considered: the first, where *<sup>V</sup>*MV(t = 0) = 1 pu, and the second one, where *<sup>V</sup>*MV(t = 0) = 0.95 pu. The taps were changed until a voltage collapse was triggered. The obtained characteristics for a network with a 15-percent proportion of induction machines are shown in Figure 10.

These figures show the obtained voltage characteristics, where the bold black solid line represents the trajectory of voltage variation for the linear variation of MV voltage. Two characteristic points are highlighted in Figure 10a,b. They indicate the levels of voltages, at which the network experiment should be aborted. A further change in the voltage ratio of the power transformer would threaten with an emergency trip of the drives due to the exceeding of the rated current value of the machine (denoted as "Crit. I", assuming *I*max = 1 pu), or because of too low voltage (denoted as "Crit. V", assuming *V* ≤ 0.85 *V*n). Figure 10c shows voltage levels at individual load nodes, including the maximum number of the tap, which would not cause the activation of the undervoltage protection. Similarly, Figure 10d shows the drive load level, including the maximum number of the tap, which would not cause the activation of overcurrent protection.

Based on the obtained graphs of *V* and *Q*, points for plotting the static characteristics were determined (Figure 11).

**Figure 10.** Selected results for the case of voltage change in the HV/MV substation caused by the change of the transformer ratio ("Tap" corresponds to the tap number): (**a**) characteristic *V* = f(*Q*)—case *<sup>V</sup>*MV(t = 0) = 1; (**b**) characteristic *V* = f(*Q*)—case *<sup>V</sup>*MV(t = 0) = 0.95; (**c**) nodes voltage on distribution grid; (**d**) values of the load currents of induction machines.

**Figure 11.** Characteristic *V* = f(*Q*) approximation based on measurement points (numerical values correspond to the number of taps position): (**a**) case *<sup>V</sup>*MV(t = 0) = 1; (**b**) case *<sup>V</sup>*MV(t = 0) = 0.95.

The numerical values correspond to successive tap numbers, while the range of data (defined as the tap number) used for determining a specific curve is given in parentheses. The characteristics were obtained by approximation with a third-order polynomial (*i* = 3), which limits the number of required measurement points while maintaining the acceptable accuracy of determining the value of *V*ext. If the number of measurement points "*j*" is equal to the degree of the polynomial, "*j* = *i*", then one has an approximated function defined at "*i* + 1" points and an approximating polynomial of degree "*i*" is searched for. In this case, one deals with an interpolation, in which the mean-square error equals 0. It is obvious that measurement data obtained from tests on a real object can be burdened with large errors; therefore, in order to reduce them, a possibly large number of measurement points should be used. If a limited number of measurement points is available, it is preferable to adopt points of the lowest voltage. Having the above in mind and also comparing the characteristics shown in Figure 11a,b, a conclusion can be drawn that the test under discussion is best carried out in the conditions of the lowest possible voltage.

#### *6.3. Simulations Studies to Verify the E*ff*ectiveness of the Proposed Algorithm*

To verify the effectiveness of action of the proposed algorithm (Figure 8), a series of simulation studies were carried out using the network model described above (Figure 9). A disturbance to be modelled was the linear variation of electric power system voltage. Sample results are illustrated in Figures 12 and 13.

At the first step, the parametrization of the model had to be made. The following settings were adopted: tap changer switchover opening time, *<sup>t</sup>*op = 7 s; time lag associated with the decay of transient processes and the establishment of measurements values, *t*db = 1 s; independent lag delay value, τ = 180 s (which corresponds to the actual switch-off time at a deviation of Δ*V* = 0.2%·*V*n). Three variants of the transformer controller algorithm were considered, namely:


The driving force was a linear variation in supply voltage on the HV side. The rate and depth of the voltage drop were selected to, on the one hand, bring about a situation that would cause the risk of occurring a voltage collapse, and, on the other hand, to enable the analysis of voltage ratio control effects. In the first case, the HV side voltage drop rate was approx. 2.5 kV/min, which corresponds to the adopted limiting value of . *V*HVmax—it is about three times smaller than the one determined from Relationship (6). In the second case, the HV side voltage drop rate was at a level of 10 kV/min, corresponding to the value that occurred during the system failure describe in the study [46].

The analysis of the simulation study results (Figures 12 and 13) shows that the proposed authors' solution brings about expected positive results. It should also be noted that the correction (reduction) of the preset voltage value, when the coefficient d*Q*/d*V* > 0, is an advantageous solution. Because of this operation, a reduction in reactive power demand by the HV/MV substations occurs, which is faster than resulting from the voltage decrease rate, and the time of operation with lower power increases (Figures 12b and 13b). The time increase is associated with the increase in the transformer control range due to the opposite directions of tap change in individual operation phases. In the first phase, the reduction of the preset voltage value is followed by a tap change towards negative tap numbers; in the second phase, when d*Q*/d*V* < 0, the taps are switched over in the direction opposite to the initial direction (Figures 12d and 13d). It should be noted that, although the power changes in the network under examination are not big, assuming that the proposed solution would be implemented in the

majority of transformer controllers in HV/MV substations, the total value of reduced power can be significant from the point of view of stable network operation. An undesirable effect resulting from the reduction of the HV/MV substation's preset voltage value is a voltage drop in load nodes (Figures 12e and 13e—the voltage variations are determined in the node of the lowest voltage).

**Figure 12.** Verification of the effectiveness of the proposed algorithm (voltage drop on the HV side—2.5 kV/min): (**a**) voltage on MV bus of the HV/MV substation; (**b**) reactive power of the transformer; (**c**) characteristics *V*MV = f(*Q*MV); (**d**) tap position; (**e**) voltage of loads; (**f**) IRMS of the asynchronous motor (expressed as a percentage of the rated current).

**Figure 13.** Verification of the effectiveness of the proposed algorithm (voltage drop on the HV side—10 kV/min): (**a**) voltage on MV bus of the HV/MV substation; (**b**) reactive power of the transformer; (**c**) characteristics *V*MV = f(*Q*MV); (**d**) tap position; (**e**) voltage of loads; (**f**) IRMS of the asynchronous motor (expressed as a percentage of the rated current).

In the network under analysis, the attained values do not exceed the threshold values typical of undervoltage protection settings. A consequence of voltage reduction in load nodes is an increase in the induction machines' current. However, the attained current magnitudes will not cause a fast shutdown of the equipment. The applied overcurrent protections make use of time-dependent characteristics with a large trip delay set value, in the order of several minutes. In the case under analysis, the maximum overload value did not exceed 2% for the case illustrated in Figure 12f and 10%, as shown in Figure 13f.

With the correct settings of the protection devices, the possible risk of disconnection of the devices connected to the MV grid is small. Please note that the issues discussed in this paper concern the regulation of power transformers supplying the distribution network in conditions of voltage collapse. That's why the incidental disconnection of single devices, in a situation of saving the network from a catastrophic failure, ceases to be relevant.

#### *6.4. Tests on the Physical Model*

To make an additional verification of the operational effectiveness and to explore the possibility of implementing the proposed solution in a real facility, tests were carried out in the laboratory using a physical model to simulate the operation of the HV/MV substation. A simplified equivalent circuit diagram is shown in Figure 14. The object of control was the transformer of a functional unit, LOAD1. The disturbance was a change in the voltage *V*1 (Bus 4) caused by a linear variation in the reactive power uptake of receivers LOAD2 and LOAD3. The test system was power supplied from a 15 kV network of the Gda ´nsk University of Technology (GUT) via a power transformer of *S*nT = 630 kVA.

**Figure 14.** Simplified network diagram used in laboratory tests.

Three algorithm variants, identical to the ones used in the simulation studies, were implemented in the transformer controller. Three cases of average initial voltage decrease rate were emulated, namely:


In view of the fact that the transformed tested was equipped with a power electronic tap changer, tests were carried out to demonstrate its regulation capabilities, as well as to show the versatility of the proposed solution. In variant D, an assumption was made that switchovers were done without delay in all of the algorithms examined. The preliminary tests showed that a stable tap changing process was attained at a rate of switching between adjacent taps in the range of (300–400) ms.

**Figure 15.** Verification of the effectiveness of the proposed algorithm, tests on the physical model—variant A: (**a**) voltage *V*1; (**b**) voltage *V*2; (**c**) characteristics *V*1 = f(*Q*1); (**d**) characteristics *V*2 = f(*Q*2); (**e**) tap position; (**f**) reactive power *Q*2.

**Figure 16.** Verification of the effectiveness of the proposed algorithm, tests on the physical model—variant B: (**a**) voltage *V*1; (**b**) voltage *V*2; (**c**) characteristics *V*1 = f(*Q*1); (**d**) characteristics *V*2 = f(*Q*2); (**e**) tap position; (**f**) reactive power *Q*2.

(**c**) (**d**) 

**Figure 17.** Verification of the effectiveness of the proposed algorithm, tests on the physical model—variant C: (**a**) voltage *V*1; (**b**) voltage *V*2; (**c**) characteristics *V*1 = f(*Q*1); (**d**) characteristics *V*2 = f(*Q*2); (**e**) tap position; (**f**) reactive power *Q*2.

**Figure 18.** Verification of the effectiveness of the proposed algorithm, tests on the physical model—variant D: (**a**) voltage *V*1; (**b**) voltage *V*2; (**c**) characteristics *V*1 = f(*Q*1); (**d**) characteristics *V*2 = f(*Q*2); (**e**) tap position; (**f**) reactive power *Q*2.

The positive effect of the application of the proposed solution is best visible in the situation, where the voltage decrease rate is significantly smaller than the rate of the voltage ratio regulation process (Figure 15). With a typical HV/MV substation transformer voltage regulation algorithm (here called TCS—Transformer Control System), a stable preset value on the secondary transformer side is maintained (Figure 15b, the voltage *V*2). From the point of view of the quality of the supply of consumers, this is a desirable action. Unfortunately, in a situation of power deficit in the system, maintaining this control criterion results in a decrease in the limit of voltage stability on the primary transformer side (Figure 15c). Much more advantageous effects are achieved by using the proposed solution (Figure 8). The activation of one of the algorithm's elements results in a change in the preset voltage value, which causes an abrupt increase in deviation, resulting in a change in voltage ratio towards voltage reduction. Because, in the initial phase of the disturbance under examination, the d*Q*/d*V* coefficient is positive, a fast reduction of reactive power flowing through the transformer follows (Figure 15f). As a consequence, an increase in the reserve of voltage stability in the supply network is obtained (Figure 15c).

The reduction of the reactive power input also results in a decreased drop of the voltage *V*1 (Figure 15a). An added positive effect of the preset value adjustment is a change in tap switching direction (Figure 15e). As a result, the present regulation capabilities increase, which results in a longer time of maintaining reduced power. So, positive effects are observed for cases in which the voltage decrease rate is lower than the limiting value described by Relationship (6).

The confirmation is provided also by graphs shown in Figure 16. In a situation, where the voltage decrease rate exceeds the limiting value (Figure 17), a positive effect of reducing the preset value to the value of *V*ref = *U*ext. + ε is only observed in the first voltage decrease phase, when *V* ≤ *V*ext (Figure 17c). Due to the high voltage decrease rate, the voltage drop is not compensated for by the change of the voltage ratio, which leads to a dramatic increase in reactive power uptake by the HV/MV substation. Because, as a result of changing the critical preset voltage value, the critical voltage value has been attained sooner, the voltage collapse will also come up sooner (Figure 17a,b).

The emulated change rate of the supply voltage *V*1 in variant D was, more or less, 30 times greater compared to the remaining variants. As the regulation process rate, resulting from the use of the power electronic tap changer, was almost 20 times greater, the results obtained by applying the proposed solution are equally positive as those obtained from the system with the emulated power electronic tap changer (with the respectively lower voltage decrease rate). Just like for variants A and B, an increase in the voltage stability reserve (Figure 18c), a reduction in the decrease rate of the supply voltage *V*1 (Figure 18a), and an increase in the present regulation range (Figure 18f) were obtained.

The effects of the operation of the "New (ϑfreeze)" algorithm taken for reference should be regarded as moderate for each of the A-D cases under consideration. With the exception of the case, where the voltage decrease rate exceeds the transformer's regulation capacities, the performance of the reference algorithm is always poorer than the proposed author's algorithm and, at the same time, better than the algorithm currently in use. As the effects of the operation of the reference algorithm are noticeably lower compared to the author's algorithm, the solution shown in Figure 8 should be taken as recommended.
