*Article* **Analysis of the Influence of Parameter Condition on Whole Load Power Factor and Efficiency of Line Start Permanent Magnet Assisted Synchronous Reluctance Motor**

**Jin Wang, Yan Li \*, Shengnan Wu, Zhanyang Yu and Lihui Chen**

National Engineering Research Center for REPM Electrical Machines, Shenyang University of Technology, Shenyang 110178, China; wjcmf11@126.com (J.W.); imwushengnan@163.com (S.W.); ddzhanyang@sina.com (Z.Y.); clh143156@163.com (L.C.)

**\*** Correspondence: eeliyan@126.com

**Abstract:** Line start permanent magnet assisted synchronous reluctance motor (LSPMaSynRM) is an important high-efficiency and high-quality motor. Its parameter matching and operating characteristics are complex, with an increase in salient ratio resulting in a valley in the power factor curve. In this study, the formation principle of power factor curve valley was first deduced by the mathematical model of LSPMaSynRM. Then, the parameter matching principle of power factor curve valley was analyzed in detail. On this basis, the characteristics of load rate corresponding to the critical state of the power factor curve valley were obtained, and its influence on whole load efficiency was analyzed. The design principles for optimal efficiency in wide high-efficiency region and specific load point were obtained. Finally, a 5.5 kW LSPMaSynRM was designed and manufactured to verify the validity of the principle.

**Keywords:** line start permanent magnet assisted synchronous reluctance motor; power factor curve valley; efficiency; whole load region

### **1. Introduction**

High-efficiency, light-weight, and high-quality motor systems are the basic component of high-end equipment in the field of engineering. Permanent magnet synchronous motor (PMSM) has the advantages of high efficiency and high power density [1,2]. It is often used in high-quality motor but requires high cost. Without permanent magnet material, synchronous reluctance motor (SynRMs) relies on reluctance torque to drive the motor. Its cost is low, but its power factor and torque density are also low. Meanwhile, its current is large, and it is difficult for it to be efficient and light.

Permanent magnet assisted synchronous reluctance motor (PMaSynRM) is a special permanent magnet motor that combines the respective characteristics of PMSM and SRM [3]. Reasonable selection and design of rotor blades can improve the salient ratio and power factor of PMaSynRM to reduce the running current. At the same time, the performance of the motor can be optimized under certain constraints by optimizing the matching of permanent magnet torque [4,5].

Many studies have been carried out in this field, including the aspects of power factor and efficiency. In [6], the influence of the shapes of flux barriers and the number of "rotor virtual slots" was investigated based on the multiphysics model, which can achieve low vibration for PMSynRMs. In [7], in order to obtain maximum torque and minimum torque ripple in the design, optimal values of motor parameters were obtained by improving the rotor geometry of the three-phase PMaSynRM. In [8], the influence of permanent magnet flux linkage on power factor was analyzed, and it was proposed that the power factor could be raised to more than 0.8 when the permanent magnet flux linkage was more than 3 times the q-axis. In [9], a PMaSynRM prototype with four poles was designed

**Citation:** Wang, J.; Li, Y.; Wu, S.; Yu, Z.; Chen, L. Analysis of the Influence of Parameter Condition on Whole Load Power Factor and Efficiency of Line Start Permanent Magnet Assisted Synchronous Reluctance Motor. *Energies* **2022**, *15*, 3866. https://doi.org/10.3390/ en15113866

Academic Editor: Djaffar Ould-Abdeslam

Received: 10 April 2022 Accepted: 19 May 2022 Published: 24 May 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

by placing ferrite magnets inside the rotor of a SynRM, and experimental measurements were performed under various loading conditions. In [10], the optimizing rotor structure was found to improve the power factor, with the power factor of the prototype increasing from 0.879 to 0.918. In [11], the power factor was found to increase from 0.35 to 0.63 by adding AlNiCo on the basis of synchronous reluctance motor. In [12,13], the multilayer magnetic barrier structure was considered, and it was shown that the choice of the first permanent magnet thickness had a great influence on the power factor. In [14], the shape parameters were used to redefine the load rate so as to realize optimization of the load rate to meet the requirements of high efficiency. In [15], a simple structure was proposed with the topology composed of an internally inserted V-shape permanent magnet (IVPM) machine and a synchronous reluctance machine (SynRM). The main novelty was that the PMs in the rotor were diverted so that the reluctance component of the torque and the magnetic component of the torque reached their maximum values at the same load angle, which eventually led to a higher output torque for the same volume. In [16], a PM-assisted-SynRM design was proposed for high torque performance. Although it used the torque components to the fullest, it suffered from high torque ripple and relatively complex rotor geometry. In [17], time-stepping FEM and multiobjective genetic algorithm were used to optimize PMaSynRM, which improved the motor efficiency to more than 92% under rated working conditions and met the IE4 standards. In [18], a two-pole multibarrier ferrite-assisted SynRM for water pumps was designed, with the prototype having high power factor and efficiency. Many scholars have analyzed the operating characteristics of PMaSynRM, such as efficiency and power factor, but most of them have focused on specific rotor structure and geometric parameters. There have been few studies on the efficiency and power factor characteristics of PMaSynRM in the whole load range from the perspective of parameter matching.

In this study, the formation principle of power factor curve valley was deduced by the mathematical model of LSPMaSynRM. The deduction was not confined to any specific rotor structure so that the conclusion could be universal. On this basis, the matching of parameters and the characteristics of the corresponding load rate were analyzed in detail. Then, further analysis of the impact on whole load efficiency was carried out. Finally, a 5.5 kW LSPMaSynRM was designed and manufactured to verify the validity of the principle.

### **2. Analysis of the Principle of Power Factor Curve Valley of PMaSynRM**

Power factor is the rate of active power to apparent power, which is essentially the phase relationship between the voltage and current in a specific operating state of the motor determined by parameters under the determination of torque. According to the mathematical model of PMaSynRM, the torque equation can be obtained as follows:

$$T\_{\varepsilon} = \frac{mpE\_0 lI}{\omega X\_{\rm d}} \sin \theta + \frac{mplI^2}{2\omega} (\frac{1}{X\_{\rm q}} - \frac{1}{X\_{\rm d}}) \sin 2\theta \tag{1}$$

where *T*<sup>e</sup> is electromagnetic torque, *m* is the number of phases, *p* is poles, *E*<sup>0</sup> is no-load back electromotive force (EMF), *θ* is the angle between voltage and no-load back EMF, *U* is voltage, *ω* is angular frequency, *X*<sup>d</sup> is d-axis reactance, and *X*<sup>q</sup> is q-axis reactance. The vector diagram was shown in Figure 1.

The torque equation can also be expressed as follows:

$$T\_{\varepsilon} = p[\psi\_{\text{f}}i\_{\text{s}}\sin\beta + \frac{1}{2}(L\_{\text{d}} - L\_{\text{q}})i\_{\text{s}}^2 \sin 2\beta] \tag{2}$$

where *ψ*<sup>f</sup> is permanent magnet flux linkage, *β* is the angle between current and permanent magnet flux linkage, *i*<sup>s</sup> stator current, *L*<sup>d</sup> is d-axis inductance, and *L*<sup>q</sup> is q-axis inductance.

**Figure 1.** The vector diagram.

In sine steady state, the torque equation can be changed as follows:

$$T\_{\varepsilon} = mp[\frac{E\_0}{\omega}I\_{\rm s}\sin\beta + \frac{1}{2}(L\_{\rm d} - L\_{\rm q})I\_{\rm s}^2\sin 2\beta] \tag{3}$$

Ignoring resistance, the d-axis and q-axis current are as follows:

$$I\_{\rm d} = \frac{E\_0 - \mathcal{U}\cos\theta}{X\_{\rm d}}\tag{4}$$

$$I\_{\mathbf{q}} = \frac{\mathcal{U}\sin\theta}{X\_{\mathbf{q}}} \tag{5}$$

The stator current can be expressed as follows:

$$I\_{\rm s} = \sqrt{\left(\frac{E\_0 - \mathcal{U}\cos\theta}{X\_{\rm d}}\right)^2 + \left(\frac{\mathcal{U}\sin\theta}{X\_{\rm q}}\right)^2} \tag{6}$$

Putting it into torque Equation (3), the torque equation can be expressed as follows:

$$\begin{split} T\_{\mathbf{c}} &= \frac{mpE\_0}{\omega} \sqrt{\left(\frac{E\_0 - \mathbf{U}\cos\theta}{X\_{\mathbf{d}}}\right)^2 + \left(\frac{\mathbf{U}\sin\theta}{X\_{\mathbf{q}}}\right)^2} \sin\beta\\ &+ \frac{mp}{2\omega} (\mathbf{X\_{\mathbf{d}}} - \mathbf{X\_{\mathbf{q}}}) \left[\left(\frac{E\_0 - \mathbf{U}\cos\theta}{X\_{\mathbf{d}}}\right)^2 + \left(\frac{\mathbf{U}\sin\theta}{X\_{\mathbf{q}}}\right)^2\right] \sin 2\beta \end{split} \tag{7}$$

Torque is a function of *θ* and *β*. Equation (1) shows that the shape of the torque curve depends on *E*0, *U*, *X*d, and *X*q, and the torque increases as *θ* increases. Equation (7) shows that the shape of the torque curve depends on *E*0, *U*, *X*d, *X*q, and *θ*. For a manufactured motor, *E*0, *X*d, and *X*<sup>q</sup> are constants, and *U* can also be regarded as a fixed value. As the torque increases, there is a one-to-one correspondence between *θ* and *β*. This corresponding relationship depends on the motor parameters *E*0, *U*, *X*d, and *X*q, which can be attributed to two parameters, namely *λ* = *E*0/*U*, which indirectly reflects the amount of permanent magnets, and the salient ratio *ρ* = *X*q*/X*d. Different parameters have different corresponding relationships between *θ* and *β*, which ultimately reflect different power factor curve states.

The relationship between *θ* and *β* can be obtained by simultaneous Equations (1) and (7). Because the equation is very complicated, it is impossible to obtain the analytical expression of the relationship. One solution of *θ* corresponding to two *β* can be obtained by numerical methods. According to the running state of the motor, the true solution and false solution can be judged as shown in Figure 2.

The introduction of the square term of *θ* in the process of deriving Equation (7) results in two *β* solutions. According to the voltage and torque equations, the voltage circle and the torque curve are obtained in the current plane of the d–q axis, and the true solution is obtained according to the intersection point, as shown in Figure 3.

In this way, a torque curve with *θ* and *β* as independent variables can be obtained. The state of the curve depends on parameters *λ* and *ρ*. Under certain conditions, the two curves will have special states, as shown in Figure 4.

**Figure 2.** *β* corresponding solution at *θ* = 60.

**Figure 3.** The torque curve in plane of the d-q axis with per unit value of *i*d-*i*q.

**Figure 4.** The *θ*-*T*e and *β*-*T*e curves with per unit value.

In order to observe the relationship of the curves more clearly, the *β*-*T*<sup>e</sup> curve was shifted to the left by 90◦, as shown in Figure 5. It can be seen that there are three intersections between the two curves, and the torque at the rightmost intersection is in the unstable range, so it will not be discussed. At the two intersections on the left, the two torque curves correspond to the same angle, and the power factor is 1. Between the crossing points and on both sides, the two curves of the same torque correspond to different angles, and the power factor is less than 1. This shows that there will be a valley in the power factor curve in the middle of two maximum values.

**Figure 5.** The *θ*-*T*<sup>e</sup> and (*β* − 90◦)-*T*<sup>e</sup> curves with per unit value.

At the same time, the *θ* and *β* relationship curve and the power factor curve can be drawn as the torque increases. As shown in Figure 6, the valley in the power factor curve is apparent.

**Figure 6.** The power factor curve.

From the above analysis, it can be seen that the power factor curve valley is caused by two torque curves with two intersection points in the stable operating interval under matching parameters. It can also be understood that *θ* and *β* increase at different speeds.

### **3. The Influence of Parameter Matching on Power Factor Curve Valley of PMaSynRM and Its Corresponding Load Rate**

### *3.1. The Condition of Power Factor Curve Valley and the Principle of Parameter Matching*

The analysis in the previous section shows that the power factor curve valley is caused by the increasing speed of *θ* and *β* being different as load increases. Therefore, the condition is that there is a *β* − *θ* > 90◦ state during load increases. Whether there is a state of *β* − *θ* > 90◦ depends on the parameters of the motor. Starting with the matching of *λ* and *ρ*, the principle that produces power factor curve valley are analyzed in this section.

The change curves of *β* with *θ* under different salient ratio with *λ* = 0.2 are shown in Figure 6. With the increase in salient ratio, the middle part of curve stretches and protrudes to the upper left corner. The slope of the front part increases, but the slope of the back part decreases. This shows that as the salient ratio increases and as the load increases, the growth rate of *β* of the low load zone increases significantly, while the growth rate of *β* of the high load zone decreases. Throughout the whole load range, the value of *β* − *θ* increases first and then decreases. There must be a salient ratio state that makes a certain load point of *β* − *θ* = 90◦, which can be called the critical point of the power factor curve valley. Then, there will be a valley on the power factor curve with increasing salient ratio. As can be seen from the rectangular box in Figure 7, the value of *θ* at the starting point

of the curve is around 80◦. As the salient ratio increases, the range of change is relatively small, but the range of change of *β* is relatively large.

**Figure 7.** The curves of different *ρ* at *λ* = 0.2.

The change curves of *β* with *θ* are shown in Figure 8 for the condition of *λ* = 0.5. The state of the curves as the salient ratio increases is similar to Figure 7. The difference is that the initial value of *θ* in the rectangular box is about 60◦.

**Figure 8.** The curves of different *ρ* at *λ* = 0.5.

The change curves of *β* with *θ* are shown in Figure 9 for the condition of *λ* = 0.8. The overall state of the curve is similar to Figures 7 and 8, and the initial value of *θ* is further reduced to about 30◦.

**Figure 9.** The curves of different *ρ* at *λ* = 0.8.

As can be seen from Figures 7–9 as *λ* increases, the initial value of *θ* gradually decreases. The smaller the value of *θ*, the less difficult it is to reach *β* − *θ* > 90◦, which means that

it is easier for a valley to be formed on the power factor curve. At the same time, the distribution range of entire curve *β* and *θ* increases as *λ* increases.

The above analysis shows that there are different parameter matching principles that result in a valley in the power factor curve. The states of *β* − *θ* = 90◦ can be obtained by calculation, as shown in Figure 10. The required salient ratio increases as *λ* decreases. As the corresponding critical point of the power factor curve valley value of *θ* increases, the distance to the initial point is closer and the initial value of *β* and *θ* are larger.

**Figure 10.** The curves under different parameter matching.

By linking the critical point of the power factor curve valley in the *λ*-*ρ* coordinate plane, the power factor curve valley area can be obtained. The parameter matching principles can be obtained as shown in Figure 11.

**Figure 11.** Diagram of power factor curve valley area.

The dividing line is not a straight line, and the required *ρ* increases nonlinearly. The linear relationship is basically between *λ* = 0.6 and 0.9, and the required *ρ* value increases sharply in the interval less than 0.5. When the value is low, it is difficult to reach the state of power factor curve valley.

### *3.2. The Influence of Parameter Matching on the Load Rate of the Critical Point of Power Factor Curve Valley*

In the previous section, the conditions and parameter matching of power factor curve valley were analyzed. From Figure 10, it can be seen that the critical points of different states correspond to different *θ* values, indicating that the load rates are different. In order to obtain the principles, the torque curve in different states were calculated, and the *β-T*<sup>e</sup> curve was moved to the left by 90◦ to make the relationship clearer, as shown in Figures 12–16.

**Figure 12.** The curve of per unit value of torque at *λ* = 0.2.

**Figure 13.** The curve of per unit value of torque at *λ* = 0.3.

**Figure 14.** The curve of per unit value of torque at *λ* = 0.5.

**Figure 15.** The curve of per unit value of torque at *λ* = 0.6.

**Figure 16.** The curve of per unit value of torque at *λ* = 0.8.

As can be seen from the above figures, the greater the value of *λ*, the greater the load rate corresponding to the critical point of the power factor curve valley. When the value of *λ* is small, the *β-T*<sup>e</sup> curve is distributed in a smaller *β* angle range, while the torque range is relatively large. At the same time, the slope of the rising interval of the *θ*-*T*<sup>e</sup> curve is relatively small, so the interval between two curves is smaller. When the value of *λ* is larger, the interval between two curves is larger, indicating that the interval of high power factor is larger. The load rate curve corresponding to the critical point of the power factor curve valley under different *λ* is shown in Figure 17.

**Figure 17.** The load rate curve with different *λ*.

The load rate and value of *λ* basically change linearly, and the minimum current state at any load point can be obtained by selecting the reasonable parameters according to the curve.

### *3.3. Adjustment of Load Rate Corresponding to the Minimum Current Point*

The load rate of the power factor curve valley can also be understood as the minimum current at a specific load rate. According to the analysis in the previous section, the minimum current under different load rates can be achieved under specific *λ* and *ρ* matching. This is a method of adjusting the load rate corresponding to the minimum current point. Its characteristic is to achieve the minimum current with the minimum *ρ* under each *λ*. The high load point requires a larger *λ* and a smaller *ρ*, and the low load point requires a smaller *λ* and a larger *ρ*. This can save the amount of permanent magnets. This is the most reasonable method in theory. However, its disadvantage is that low load requires a large *ρ*, which is difficult to achieve with the existing rotor manufacturing technology in engineering.

From the analysis, it can be seen that two minimum current load points are generated under the power factor curve valley. One point tends toward low load, and the other point

tends toward high load. In this way, there can be a second method of adjusting the load rate corresponding to the minimum current point. The curves are shown in Figures 18 and 19 when *λ* is 0.8.

**Figure 18.** The curve of per unit value of torque at *λ* = 0.8 *ρ* = 7.

**Figure 19.** The curve of per unit value of torque at *λ* = 0.8 *ρ* = 20.

The minimum current point gradually moves to low load as the salient ratio increases, so the minimum current at any load point can also be achieved. Compared with the first method, the second method can achieve the minimum current at the low load point with a smaller salient ratio. The disadvantage of this is that the load point current between two minimum points of current is large.

### **4. Influence of Power Factor Curve Valley of PMaSynRM on Whole Load Efficiency**

The analysis of efficiency involves losses and needs to be targeted at specific research objects. A 5.5 kW motor was taken as an example for the present analysis.

### *4.1. The Influence of the Power Factor Curve Valley on the Efficiency of Load Rate Point*

Different parameter matching can realize the power factor curve valley of any load point. In this state, further analysis is needed to determine whether the efficiency of the load point is optimal. *λ* = 0.5 and *λ* = 0.8 were selected to calculate the efficiency under different salient ratios. The load rate efficiency under the power factor curve valley is shown in Figures 20 and 21.

As shown in Figure 20, the efficiency of the critical point of the power factor curve valley is not the highest. As the salient ratio increases, the efficiency changes from high to low.

As can be seen in Figure 21, the power factor curve valley occurs at a critical point with the highest efficiency, and the efficiency on both sides gradually decreases. The highest efficiency in different states is different. This is because the corresponding load point is the high load point when *λ* is 0.8. Here, copper loss accounts for a larger proportion of the total loss, and the magnitude of the current determines efficiency. Therefore, the power factor curve valley occurs at the critical point with the highest efficiency. The corresponding load point is the low load point when *λ* is 0.5. The copper loss accounts for a smaller proportion of the total loss, and the current cannot completely determine efficiency. The demagnetization field increases as the salient ratio decreases, the iron loss gradually decreases, and the efficiency gradually increases. By combining the loss rate of each load point, a reasonable design of motor parameters can obtain optimal efficiency of any load.

**Figure 20.** Efficiency curve with different *ρ* at *λ* = 0.5.

**Figure 21.** Efficiency curve with different *ρ* at *λ* = 0.8.

*4.2. The Influence of the Corresponding Load Rate Point of Power Factor Curve Valley on Whole Load Efficiency*

Although a reasonable design of motor parameters can obtain optimal efficiency of any load, the effect of the size of the high-efficiency zone in the whole load range needs further research. The conditions of *λ* = 0.5 and *λ* = 0.8 were again selected for analysis. The power factor and efficiency of the whole load were calculated and are shown in Figures 22–24.

Compared with the state of *λ* = 0.5, the power factor of low load is higher and the power factor of high load is lower, as shown in Figure 22. Compared with the state of *λ* = 0.5, the low load efficiency is high, and the high load efficiency is low, as shown in Figures 23 and 24. The copper loss accounts for a large proportion of total loss in the high load area. The current state of *λ* = 0.8 is small and has high efficiency. The current of *λ* = 0.5 in the low load area is small and has high efficiency, but the copper loss is small. Therefore, the low load efficiency difference between the two states is less than that for high load efficiency. As can be seen, the larger the *λ*, the larger the range for high efficiency, and the smaller the *λ*, the higher the efficiency range for low load.

**Figure 22.** Power factor curves.

**Figure 23.** Efficiency curves.

**Figure 24.** Magnification diagram of efficiency.

### **5. Prototype Test**

A 5.5 kW PMaSynRM was designed and manufactured. The parameters of the motor are shown in Table 1, and the structure of the rotor is shown in Figure 25. The inductance of the prototype was tested as shown in Figure 26. The d-axis inductance was 41.1 mH, the q-axis inductance was 127.8 mH, and the salient pole rate of the motor was 3.11.

**Table 1.** Main parameters of the prototype.


**Figure 25.** The prototype of the rotor.

**Figure 26.** The inductance test of the prototype.

The rated phase voltage of the prototype was 220 V, and the back-EMF was 160.5 V by experimental test. *λ* and *ρ* were not on the curve of the critical point of the power factor curve valley discussed above, and the back-EMF and *ρ* could not be changed after the prototype was produced. In order to verify the correctness of the above conclusions, the input voltage was only changed for testing. According to the previous curve, when input voltage is adjusted to 174.3 V, the motor is in a critical state of the power factor curve valley. The input voltage was adjusted to 174.3 V, and the power factor was tested as shown in Figure 27. The PMaSynRM was used as the tested motor linked to a torque sensor, and the

load motor was a DC generator. The test power factor curve is shown in Figure 28, and the efficiency curve is shown in Figure 29.

**Figure 27.** The load test of the prototype.

**Figure 28.** The power factor curve (*U* = 174.3 V).

**Figure 29.** The efficiency curve.

The test results showed that, when the power factor reached 1, the load rate was between 0.3 and 0.4, which is the critical state of the power factor curve valley.

### **6. Conclusions**

From the LSPMaSynRM mathematical model, the principle of the power factor curve valley was derived, and its parameter matching principle was analyzed. Then, the characteristics of the corresponding load rate and the influence on whole load efficiency were further analyzed. The conclusion reached in this study is not confined to any specific rotor structure. As long as the parameter matching conforms to the analysis carried out here, the corresponding power factor characteristics can be obtained, thus providing a certain

theoretical basis for the detection and judgment of the motor running state. The conclusions are as follows:


**Author Contributions:** Conceptualization, J.W. and Z.Y.; methodology, J.W. and Y.L.; software, S.W. and L.C.; validation, J.W., Y.L. and Z.Y.; formal analysis, J.W. and Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Smart Grid Joint Fund of National Natural Science Foundation of China, No.U2166213.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


## *Article* **Parameter Identification of Asynchronous Load Nodes**

**Andrey Kryukov 1,2, Konstantin Suslov 2,3,\*, Pavel Ilyushin 3,4 and Azat Akhmetshin <sup>5</sup>**


**Abstract:** Asynchronous loads (AL), because of their low negative-sequence resistance, produce the effect of reduced unbalance at their connection points. Therefore, proper modeling of unbalanced load flows in power supply systems requires properly accounting for AL. Adequate models of the induction motor can be realized in the phase frame of reference. The effective use of such models is possible only if accurate data on the parameters of induction motor equivalent circuits for positive and negative sequences are available. Our analysis shows that the techniques used to determine these parameters on the basis of reference data can yield markedly disparate results. It is possible to overcome this difficulty by applying parameter identification methods that use the phase frame of reference. The paper proposes a technique for parameter identification of models of individual induction motors and asynchronous load nodes. The results of computer-aided simulation allow us to conclude that by using parameter identification, we can obtain an equivalent model of an asynchronous load node, and such a model provides high accuracy for both balanced and unbalanced load flow analysis. By varying load flow parameters, we demonstrate that the model proves valid over a wide range of their values. We have proposed a technique for the identification of asynchronous load nodes with such asynchronous loads, including electrical drives equipped with static frequency converters. With the aid of the AL identification models proposed in this paper, it is possible to solve the following practical tasks of management of electric power systems: increasing the accuracy of modeling their operating conditions; making informed decisions when taking measures to reduce unbalance in power grids while accounting for the balancing adjustment effect of AL.

**Keywords:** power supply systems; unbalanced load flows; unbalanced load; parameter identification

### **1. Introduction**

The electric power system (EPS) is a set of complex devices that generate, transmit, distribute, and consume electric power. Improving the reliability of operation and efficiency of power system use is impossible without solving a set of problems relating to load dispatching in ordinary and emergency states. Due to the introduction of transient monitoring tools, it became possible to determine the parameters of power system elements in real-time.

Solving the problems of load dispatching in EPSs is based on the use of mathematical models. They are used for steady-state load flow analysis, optimization, state estimation, transient analysis, etc. The basis of the mathematical model of an EPS is an equivalent circuit, which is formed from the circuits of individual components, namely: power transmission lines, power transformers, generating equipment, load nodes, etc. The parameters of the equivalent circuit of each component are usually determined by reference data or

**Citation:** Kryukov, A.; Suslov, K.; Ilyushin, P.; Akhmetshin, A. Parameter Identification of Asynchronous Load Nodes. *Energies* **2023**, *16*, 1893. https://doi.org/ 10.3390/en16041893

Academic Editors: Yuling He, David Gerada, Conggan Ma and Haisen Zhao

Received: 9 January 2023 Revised: 9 February 2023 Accepted: 11 February 2023 Published: 14 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

nameplate data and are considered immutable, although they depend on wear and tear of components, weather conditions, and other factors. The errors in determining these parameters on the basis of reference data are quite significant. For example, the error of the active resistance of a line can be in the range of +16–−20%, and the representation of corona discharge losses by a constant value of active conductivity can lead to an error of 1.5–3 times the actual value when determining the losses. The most significant errors can occur in the process of building the models of load nodes. This is due to the considerable uncertainty in the mix of consumers served and their operating conditions. The problem of adequate modeling of load nodes can be solved on the basis of identification methods.

Issues of parameter identification of induction machines and nodes with predominantly unbalanced loads have been addressed in a sizeable number of studies. For example, article [1] proposed a technique for identifying the parameters of a three-phase induction motor in the case when the initial values of the estimates change in a wide range. Study [2] presented the results of an experimental study of the efficacy of the technique of adaptive identification of electrical parameters of the induction machine under steady-state conditions on the basis of the power balance. Article [3] dealt with the issues of the identification of load nodes and their stability control. In [4–6], the results of the modeling and identification of an induction machine were presented. Article [7] discussed the identification of the parameters of an induction motor in its operational mode. Study [8] described an algorithm proposed by its authors for the identification of induction machine parameters by a recursive least-squares method. Article [9] studied mathematical methods of identification of lumped parameters of electrical machines. It discussed the basic principles and mathematical foundations of lumped parameter identification methods for various types of electrical machines, including induction machines. Article [10] proposed an approach to parameter identification of induction machines. The excitation input signal was determined by optimization methods. Instrumental variable estimation was introduced to improve the quality of identification by least-squares estimation. A method for magnetizing curve identification of induction machines was described in [11]. The study proposed an experimental method for determining the magnetization curve specifically designed for vector-controlled drives. The method employed an indirect vector controller and a PWM inverter, which were used during normal operation of the drive. The method was verified by extensive experimentation. A novel parameter identification method for the induction motor (IM) was proposed in [12]. The study pointed out that the effect of vector control depends largely on the accuracy of setting its parameters, which change with temperature variation. Based on the relationship between motor winding resistance and temperature, the authors presented a method for calculating dynamic resistance by on-line detection of the winding temperature. The experiment results attested to the high identification accuracy of the approach. Article [13] considers real-time parameter identification algorithms for effective control of electrical machines. Study [14] proposed a technique of flux estimation of induction machines with the linear parameter-varying system identification method. The identification algorithm was tested on data obtained from a nonlinear simulation model with continuous time. Paper [15] described a method for parameter identification of nine-phase induction machines with concentrated windings. Studies [16,17] considered a method of induction machine parameter identification suitable for self-commissioning. An algorithm for parameter identification of electrical machines using numerical simulations was proposed in articles [18,19]. The problem of reducing the electrical energy consumption of fans through parameter identification of the drive was solved in [20]. A method for identification of induction machine parameters, including core loss resistance, using a recursive least mean square algorithm was proposed in [21].

An analysis of the research contributions reviewed above allows us to conclude that the relevance of problems with parameter identification of IMs and asynchronous load nodes has been well established. However, most of the reviewed works dealt with the determination of parameters of individual IMs, whereas load nodes were considered only in [3]. The study relied on a single-line representation of the power system, which significantly hindered the modeling of unbalanced load flows. Adequate models of IMs and asynchronous load nodes in the phase frame of reference, which is the most natural form of representation of multiphase circuits, were proposed in [22,23]. However, effective use of such models is possible only if accurate data on the parameters of IM equivalent circuits for positive and negative sequences are available. The parameter identification techniques discussed below can be used to solve this problem.

### **2. Modeling of Unbalanced Loads in the Phase Frame of Reference**

On the basis of phase coordinates, adequate models [22] of asymmetric modes of complex electric power systems at the fundamental frequency and frequencies of higher harmonics can be implemented. So, for example, an experimental verification of the EPS modeling technique in phase coordinates, performed on the basis of comparison with synchronized measurement data for a model containing 619 nodes and 2996 branches, showed that the differences between the calculated and measured values of the asymmetry coefficients do not exceed 0.6%, and for the phase values stresses 2.3%.

Compared to the static elements of an EPS, such as power lines and transformers, an asynchronous load, including a large number of motors, is a more complex object.

The asymmetry of the resistance matrix corresponding to the motor poses challenges to simulation based on the lattice circuit with RLC elements. The difficulties are due to the presence of two magnetic fields rotating clockwise and counterclockwise. When the supply voltages are unbalanced, the induction motor has sine wave processes running at three frequencies: 50 Hz, the frequency of the slip *s*, and about 100 Hz.

The behavior of induction motors under balanced three-phase voltage, when the motor can be represented by a single-line equivalent circuit, has been thoroughly researched. Induction motors can have different parameters of equivalent circuits during starting up and operation with low slip values. Furthermore, there are several variants of equivalent circuits. From the standpoint of load flow analysis in the phase frame of reference, when it is necessary to consider motor parameters at low slip values as well as at slip values close to 2 (electromagnetic brake mode), it is advisable to make the following assumptions.

First, it is convenient to use the equivalent circuit of an induction motor with the external magnetizing circuit placed on the primary terminals, according to Figure 1a. It is assumed that at start-up and the slip value of 2 − *s* (for negative sequence voltage), the equivalent circuit will have different parameters of the rotor circuit, Figure 1b. Figure 1 shows the magnetization branch components *R*μ, *X*μ, stator resistances *R*1, *X*1, and equivalent rotor resistances referred to the stator *<sup>R</sup>*<sup>2</sup> *<sup>s</sup>* , *X*2, as well as the corresponding starting parameters *<sup>R</sup>*2*<sup>P</sup>* <sup>2</sup>−*s*, *<sup>X</sup>*2*P*.

**Figure 1.** Positive (**a**) and negative (**b**) sequence equivalent circuits.

Second, it was assumed that in the start-up and electromagnetic brake modes (for the negative voltage sequence), the square of the reactive resistance is much greater than the square of the active resistance.

Third, with respect to the magnetization branch, a dual approach was adopted. When the no-load parameters are known (cosϕ*<sup>x</sup>* and active power *Px*), its parameters are determined on the basis of the relations presented below, and when the parameters are unknown, the magnetization branch is ignored.

Fourth, the parameters of the circuit components in Figure 1 are determined from the rated values of efficiency η, power factor cos ϕ*H*, current *IGH*, and the current flowing through the part of the circuit that determines the load flow.

Fifth, the values of the positive and negative sequence voltages and the given mechanical power of the motor are used to determine the positive and negative sequence currents. In this case, the motor is modeled by current sources connected in a star (Figure 2). The values of the source currents are adjusted at each step of the iterative process. The motor neutral is considered to be insulated, and no zero-sequence currents occur in its circuits.

**Figure 2.** Equivalent circuit in the phase frame of reference. (A, B, C—Phase A, Phase B, Phase C, respectively, N—neutral point).

According to Figure 1a, the parameters of the circuit at rated power settings for the positive sequence are determined from the values of efficiency η, rated current *IGH*, and power factor cos ϕ*H*.

If the active power *Px* and cos ϕ*<sup>x</sup>* of the no-load operation of the motor are known, the parameters of the magnetizing branch and the current flowing through it can be determined from them:

$$Z\_{\mu} = \frac{3\,\,\,\,\,\,\,\,\,\,L\_1^2 \cos\varphi\_{\underline{x}}\,\,\,\,\,R\_{\mu} = Z\_{\mu}\cos\varphi\_{\underline{x}\prime}\,\,\,X\_{\mu} = \sqrt{Z\_{\mu}^2 - R\_{\mu}^2} \tag{1}$$

The mechanical shaft power of the motor at rated settings is determined by the active power dissipated in the resistance *<sup>R</sup>*2(1−*sH*) *sH* , where *sH* is the rated slip. The efficiency factor is made up of the following loss components:


The efficiency at rated power settings is defined as the ratio of shaft power to gross power:

$$\eta = \frac{P\_H}{P\_H + \Delta P\_M + \Delta P\_d + 3I\_{CH}^2 (R\_1 + R\_2) + \frac{3I\_1^2}{Z\_\mu^2} R\_\mu} \,'$$

$$R\_0 = R\_1 + R\_2 \, \tag{2}$$

$$R\_0 = \frac{P\_H}{3I\_{CH}^2} \left(\frac{1}{\eta} - 1 - \frac{\Delta P\_M + \Delta P\_d}{P\_H} - \frac{3I I\_1^2 R\_\mu}{P\_H Z\_\mu^2} \right) \,.$$

The magnetizing branch current when the vectors are counted from the voltage . *U*<sup>1</sup> is

$$\dot{I}\_{\mu} = \frac{\mathcal{U}\_{1}}{R\_{\mu} + j \ \mathcal{X}\_{\mu}} = I\_{\mu}^{\prime} + j \ I\_{\mu}^{\prime} = \frac{\mathcal{U}\_{1} R\_{\mu}}{R\_{\mu}^{\prime} + X\_{\mu}^{\prime}^{2}} - j \frac{\mathcal{U}\_{1} X\_{\mu}}{R\_{\mu}^{\prime}^{2} + X\_{\mu}^{\prime}^{2}}. \tag{3}$$

At rated load, the motor current is

$$\dot{I}\_{H} = I\_{H}^{\prime} + j^{\prime} I\_{H}^{\prime\prime} = \frac{P\_{H}}{\Im \, U\_{1} \, \eta} - j \frac{P\_{H}}{\Im \, U\_{1} \, \eta} \sqrt{\frac{1}{\cos^{2} \varphi\_{H}} - 1},\tag{4}$$

where *PH* <sup>η</sup> is the active power consumed at rated power settings. The rated current and power factor are determined by the expressions:

$$I\_{GH} = \sqrt{(I\_H '-I\_{\mu} ')^2 + (I\_{H''} ' - I\_{\mu''})^2};\tag{5}$$

$$\cos\varphi\_{GH} = \frac{I\_H{\!\!\! }^{\!\!\! } -I\_{\!\!\! }^{\!\!\! }}{I\_{GH}}.\tag{6}$$

However, if the no-load operation parameters of the motor are unknown, it is possible to assume, by way of approximation, that *IGH* = *PH* <sup>3</sup> *<sup>U</sup>*<sup>1</sup> <sup>η</sup> cos <sup>ϕ</sup>*<sup>H</sup>* neglecting the magnetizing current and considering that cos ϕ*GH* = cos ϕ*H*.

According to the equivalent circuit of Figure 1a, we can write:

$$\dot{I}\_G = \frac{\dot{U}\_1}{\left(R\_1 + \frac{R\_2}{s}\right) + j \ X\_k}, \ X\_k = X\_1 + X\_{2\prime}$$

$$I\_{GH} = \frac{U\_1}{\sqrt{\left(R\_1 + \frac{R\_2}{s\_{li}}\right)^2 + X\_k^2}},\tag{7}$$

$$\cos\varphi\_{GH} = \frac{R\_1 + R\_2/s\_H}{\sqrt{\left(R\_1 + \frac{R\_2}{s\_H}\right)^2 + X\_k^2}} = \frac{R}{\sqrt{R^2 + X\_k^2}},\tag{8}$$

$$R = R\_1 + R\_2 / s\_H.\tag{9}$$

The system of Equations (2), (7) and (8) is solved by simple substitution. From Equation (8) we determine

$$X\_k^2 = R^2 \left(\frac{1}{\cos^2 \varphi\_{GH}} - 1\right).$$

and it follows from Equation (7) that *Z*<sup>2</sup> = *<sup>U</sup>*<sup>1</sup> *IGH* <sup>2</sup> = *R*<sup>2</sup> + *Xk* 2, so that *R* = *Z* cos ϕ*GH*. From relations (2) and (9) we can determine components *R*<sup>1</sup> and *R*2:

$$R\_2 = \frac{s\_H(R - R\_0)}{1 - s\_H};\ R\_1 = R\_0 - R\_2.$$

When determining *R*<sup>0</sup> from relation (3), it can be assumed that the added losses are 0.5% of the input power and that the mechanical losses are 1.0% of the rated power.

Denoting *<sup>X</sup>*<sup>1</sup> + *<sup>X</sup>*2*<sup>p</sup>* = *Xkp* and assuming that *R*<sup>1</sup> + *R*2*<sup>p</sup>* <sup>2</sup> << *Xkp* 2, we obtain

$$\mathbf{X}\_{kp} = \frac{U\_1}{\mathcal{K}\_P I\_{GH}}$$

,

where *KP* is the locked-rotor current ratio. From the equation of the electromagnetic locked-rotor torque when the magnetization branch is ignored, we get the relation

$$M\_P = \frac{3\ \mathcal{U}\_1^2 R\_{2p} p}{X\_{kp}^2 \omega}, \text{ or } R\_{2p} = \frac{\omega}{p} \frac{MX\_p^2}{3\ \mathcal{U}\_1^2}.$$

where *p* is the number of motor pole pairs.

The locked-rotor torque was determined from the locked-rotor torque ratio *kMP* = *MP MH* and the rated torque *MH* = <sup>2</sup>*PH <sup>p</sup>* (<sup>1</sup> <sup>+</sup> <sup>η</sup>)2*<sup>π</sup> <sup>f</sup>* , from which it follows that

$$R\_{2p} = \frac{2 \, k\_{MP} P\_H X\_{kp}^2}{3 \, U\_1^2 (1 + \eta)}.$$

The multiplier <sup>2</sup> <sup>1</sup> <sup>+</sup> <sup>η</sup> allows one to convert the shaft power to the electromagnetic power of the motor with a small error.

Effective use of the described model is possible only if accurate data on the parameters *Xk*, *R*2, *XkP*, *R*2*P*, *X*<sup>μ</sup> of IM equivalent circuit for positive and negative sequences are available. This problem can be solved by applying the parameter identification technique [22], described below. Identification results can be used in the unbalanced load flow analysis of complex power supply systems. In this case, the asynchronous load node can be represented by an equivalent IM according to the technique detailed in [24].

### **3. Induction Motor Identification Technique**

The problem of adequate modeling of load nodes can be solved on the basis of identification methods. In the context of power system control, it is advisable for identification purposes to use information about parameters of operating conditions obtained from information and measurement systems built using PMUs (Figure 3).

**Figure 3.** Synchronized measurements.

The problem of topology and parameter identification of load nodes can be formalized as follows [22,25]. To this end, we can introduce a class of models <sup>=</sup> <sup>1</sup> <sup>2</sup> ... *<sup>m</sup>* describing the processes occurring at load nodes. Each of the models is represented as

$$\mathbb{G}\_i = \mathbb{G}\_i(\mathbf{X}, \mathbf{Z}, \mathbf{P}, \mathbf{Z}, \mathbf{L})$$

where *xk* ∈ **X**, *k* = 1 ... *nX*—state variables; *zk* ∈ **Z**, *k* = 1 ... *nZ*—input variables; *pk* ∈ **P**, *k* = 1... *nP*—model parameters subject to identification; *σ<sup>k</sup>* ∈ **Σ**, *k* = 1... *n*Σ—internal relations defining the model structure; *lk* ∈ **L**, *k* = 1 ... *nX*—functional relationships acting as mathematical relations operators allowing to find the parameters describing the object state *xk* ∈ **X**, *k* = 1... *nX* by inputs *zk* ∈ **Z**, *k* = 1... *nZ*, with the required degree of certainty. Then we can write

$$\mathbf{X} = \mathbf{L}(\mathbf{Y}, \mathbf{P}, \mathbf{Z}).\tag{10}$$

This relationship is called the rule governing the functioning of the model. To form the relationship (10), it is necessary to choose from a class of models <sup>=</sup> <sup>1</sup> <sup>2</sup> ... *<sup>m</sup>* a model *<sup>k</sup>* ∈ with the rule

$$
\Gamma\_\* : \left( \mathcal{O}\_{(\mathcal{O})} \right) \to \mathcal{O}\_k.
$$

The parentheses in the last relation denote that **L**∗ is a partially defined relation; that is, not all characteristics of the original (*O*) are captured by the model, but only those that are deemed significant in solving the stated problem of modeling power system conditions. The functional transformation **L**∗ can be chosen subject to the following condition

$$\begin{array}{c} \|\mathbf{X} - \mathbf{L}^\*(\mathbf{Y}, \mathbf{P}, \mathbf{E})\| \to \begin{array}{c} \min \\ p\_k \in \mathbf{P} \\ \sigma\_k \in \mathbf{E} \end{array} \end{array}$$

in some parts of the chosen class of functions.

In addition, the choice of **L**∗ can be made subject to the condition that there be a minimum of some criterion of discrepancy between the model and the original:

$$
\aleph\_{\mathcal{L}^\*} \to \min\_{\mathcal{L}^\* \in \mathcal{L}}.
$$

As a rule, the choice of the functional transformation **L**∗, carried out at the stage of structural identification, is subjective and does not lend itself easily to rigorous formalization. Figure 4 is a diagram showing possible types of load node models.

**Figure 4.** On the problem of structural identification of load nodes.

Parameters *Xk*, *R*2, *XkP*, *R*2*<sup>P</sup>* can be determined on the basis of measurements of complexes of currents consumed by the motor and voltages at its terminals, as well as its rotation speed. To solve this problem, it is necessary to know the resistance of the magnetizing branch *X*μ. This parameter can be found on the basis of additional measurements, e.g., under no-load conditions, or determined by the indirect technique described below.

If the value of *X*μ is known, the parameters of the equivalent circuit of the positive sequence can be found based on measurements of the moduli and phases of the IM currents and voltages, as well as the speed of rotation (slip *s*) based on the following relation:

$$\underline{Z}\_{D1} = \frac{jX\_{\mu}\underline{Z}\_{k}}{jX\_{\mu} + \underline{Z}\_{k}} \, \tag{11}$$

where *Zk* = *<sup>R</sup>*<sup>2</sup> *<sup>s</sup>* + *jXk*; *ZD*<sup>1</sup> = . *U*1 . *I*1 ; . *U*1, . *I*<sup>1</sup> are complexes of positive sequence voltage and current, determined on the basis of measurements of phase currents . *IA*, . *IB*, . *IC* and voltages . *UA*, . *UB*, . *UC* according to the known relations of the method of symmetrical components. Measurements can be made under both balanced and unbalanced load flows.

The main practical focus of the research presented in the article is to create methods for adequately taking into account load nodes when modeling stationary modes of electric power systems. Therefore, for further consideration, the model of the load node in phase coordinates in the form of an equivalent asynchronous electric motor was adopted. To solve the identification problem, it is necessary to measure the parameters of load nodes, which can be determined on the basis of PMU-WAMS devices (Figures 3 and 5).

**Figure 5.** A set of problems for identifying load nodes in an electric power system.

Based on (11), we can write the following expression:

$$\underline{Z}\_k = \frac{jX\_\mu \underline{Z}\_{D1}}{jX\_\mu - \underline{Z}\_{D1}}$$

.

If the slip *s* is known, the parameter *R*<sup>2</sup> can be determined from the above equation. The parameters of the equivalent circuit of the negative sequence can be found as per equations similar to those given above:

$$\underline{Z}\_{kp} = \frac{R\_{2p}}{2 - s} + jX\_{kp} = \frac{jX\_{\mu}\underline{Z}\_{D2}}{jX\_{\mu} - \underline{Z}\_{D2}}; \underline{Z}\_{D2} = \frac{\dot{U}\_2}{\dot{I}\_2}.$$

where . *U*2, . *I*2— complexes of negative sequence currents and voltages are determined by measurements of phase currents and voltages. To obtain acceptable accuracy, the parameters of start-up conditions should be found in the power flow with a voltage unbalance (*k*2*<sup>U</sup>* of about 10%).

The resistance *X*μ can be determined by the data provided in reference books. All that is required is information about the rated voltage and the rated motor power. An acceptable accuracy of calculation of *X*μ can be obtained on the basis of a nonlinear approximation of the following kind:

$$X\_{\mu\*} = X\_{\mu~0} [1 + \Delta X\_{\mu} (1 - e^{-\alpha' P})].\tag{12}$$

Parameters *X*μ0, Δ*X*μ, and α, for IM powers exceeding 5 kW are given in Table 1.


**Table 1.** Parameters of the approximation of the relationship *X*μ<sup>∗</sup> = *X*μ∗(*P*).

The obtained value *X*μ<sup>∗</sup> should be multiplied by the basic resistance, determined by the rated parameters of the IM.

### **4. Identification Results**

Input information in the form of moduli and angles of current and voltage, as well as those of slip, was formed on the basis of computer-aided simulation using the software package Fazonord [22]. For this purpose, an equivalent circuit of an IM with a rated power of 90 kW was created. In the obtained currents and voltages of the calculated load flow, the errors corresponding to the accuracy classes of measuring instruments (0.1, 0.2, 0.5, and 1) were introduced. The resistance *X*μ was calculated on the basis of expression (12). The results of the identification are shown in Figure 6. The parameter *R*<sup>2</sup> was determined with an error close to zero.

The results obtained show that in order to achieve acceptable identification accuracy, it is necessary to use measuring instruments with an accuracy class that provides a maximum error of no more than 0.2%.

The proposed technique can be used to solve the problem of parameter identification for a group of IMs connected to a node in an electrical network. To confirm this possibility, we performed identification of the AL node, the circuit of which is shown in Figure 7. The IM parameters are summarized in Table 2. The equivalent circuit is shown in Figure 8.

**Figure 6.** Identification errors.

**Figure 7.** Original circuit.

**Table 2.** Parameters of the nodal IM.


**Figure 8.** Equivalent circuit.

In the process of identification, the slip value was set on the basis of the data for the equivalent IM given in [25]. The load flow analysis errors that arise when using the equivalent model of an asynchronous load node, obtained on the basis of parameter identification, are shown in Figures 9–12.

**Figure 9.** Errors in determining the moduli of voltages (**a**) and currents (**b**).

**Figure 10.** Errors in determining phases of voltages (**a**) and currents (**b**).

**Figure 11.** Errors in determining the active and reactive power consumed by the AL node (**a**), and power losses in the transmission line (**b**).

**Figure 12.** Errors in determining the unbalance ratio in the negative sequence.

Table 3 and Figure 13 show the static characteristics of *P* = *P*(*UPH*) and *Q* = *Q*(*UPH*), where *P*, *Q* stands for active and reactive powers consumed by the AL node; *UPH* is the phase voltage. These dependencies were plotted for the original and equivalent AL node models. Our analysis of the results thus obtained allows us to conclude that the AL node model, formed on the basis of parameter identification, provides for valid simulation of the asynchronous load node within a wide range of changes in network operating conditions.

**Table 3.** Static load characteristics.


The results obtained also allow us to conclude that, with the aid of parameter identification, we can arrive at an equivalent model of an asynchronous load node that provides high accuracy for both balanced and unbalanced load flow analysis. It should be emphasized that the model was validated in a wide range of changing power flow parameters.

The proposed technique is also valid for AL circuits of a more general type, the models of which are shown in Figure 14. In these circuits, induction motors are connected to the buses of the node through cable lines. In addition, the node was powered by a busbar trunking system.

**Figure 14.** Models of AL node circuits (**a**)—first type of connection, (**b**)—second type of connection).

The simulation results are presented in Table 4, which shows that equivalent models with a structure similar to the one presented in Figure 8 provided acceptable accuracy for unbalanced load flow analysis.



Figures 15 and 16 show the results of parameter identification for the case when, in addition to the IM, a static load was connected to the nodal point of the network, with such a load specified by the amount of power . *SC* = *P* + *jQ* drawn by it. Percentage of stationary load

$$
\lambda = \frac{S\_{\mathbb{C}}}{S\_{AH}} \cdot 100
$$

ranged from 0–80%. Here *SAH* is the total power of the IM group.

**Figure 15.** Relationship *δ U* = *δ U*(*λ*).

**Figure 16.** Relationship *δ I* = *δ I*(*λ*).

The obtained relationships *δ U* = *δ U*(*λ*) and *δ I* = *δ I*(*λ*) attested to the fact that with an increase of the parameter *λ*, the errors of load flow analysis using the equivalent IM model increased but remained quite acceptable for practical applications over a sufficiently wide range of variation of *λ*.

The following conclusions can be drawn on the basis of the results obtained:

1. The technique of parameter identification of an asynchronous load node allows one to obtain adequate models of IMs that provide high accuracy for balanced load flow analysis. In the numeric example presented in the paper, the calculation error of phase voltage moduli for different motor connection schemes did not exceed 1.5%;

2. If a static load is present at the node, the error of the equivalent model increases; in the numeric example above, when the value of the parameter *λ* = was equal to 75%, the error of the voltage moduli increased to 5%, and the error of the currents increased to 3.3%.

### **5. Parameter Identification of Asynchronous Load Nodes with Variable Frequency Drives**

At modern production facilities, variable-frequency induction-motor drives equipped with static frequency converters (SFC) are widely used. Therefore, the problem of identifying AL nodes that contain, along with conventional IMs, frequency-controlled asynchronous electric drives that can create harmonic distortions in networks becomes relevant [26]. The technique described above can be used to solve this problem.

The verification of its efficacy and accuracy in the presence of electric drives at the load node that are equipped with an SFC was carried out for the circuit shown in Figure 17 as follows. Based on the SimPowerSystems MATLAB package, a load node model was generated.

**Figure 17.** Network circuit.

The power of the IM controlled by the SFC was assumed to be 22 kW. The power of the fixed-speed IM ranged from 22 kW to 45 kW. The power ratio was set by the coefficient

$$\alpha = \frac{P\_{spch}}{P\_{aed}}\prime$$

where *Pspch*—power of the motor equipped with an SFC; *Paed*—power of the fixed-speed IM. The power supply system was fed from a source with unbalanced voltage (*k*2*<sup>U</sup>* = 3%), which corresponded to real-life conditions that hold true for many facilities connected to district windings of traction substations on mainline AC railroads. The results of the simulation are presented in Table 5.


**Table 5.** Voltages and currents at load node buses.

In accordance with the identification technique described above, we determined the parameters of AL equivalent circuits (Table 6). Next, we created a model of the power supply system in the software package Fazonord, in which the load node was represented by an equivalent induction motor (see Figure 18). With the aid of this model, a load flow analysis was performed using the parameters of the AL node obtained as a result of identification. Comparative results of a simulation run in MATLAB and Fazonord are shown in Table 7. Errors in determining the active and reactive powers drawn from the network, as well as the unbalance ratio *k*2*U*, are shown in Figures 19 and 20.

**Table 6.** Equivalent circuit parameters.


**Figure 18.** Load node model formed based on identification results (1–6—numbers of nodes).

**Table 7.** Parameters characterizing the load flow of the load node.


**Figure 19.** Errors of active (**a**) and reactive (**b**) powers as functions of the parameter.

**Figure 20.** Unbalance ratio error as a function of the parameter *k*2*U*, % α.

In Figures 19 and 20, the index "*M*" refers to the results obtained using the SimPowerSystems package, and the index "*F*" refers to the data calculated using the Fazonord software package.

The results obtained allow us to draw the following conclusions:

1. Our technique of parameter identification in the phase frame of reference is based on the substitution of a load node with an equivalent induction motor. The technique allows one to obtain high accuracy in unbalanced load flow analysis in the presence of conventional induction machines and electric motors with variable-frequency drives at the node. The error in determining the unbalance ratio of the negative sequence did not exceed two percent.

2. As the share of static frequency converters at the load node increased, the errors in determining the unbalance ratios *k*2*<sup>U</sup>* also increased but remained within the limits deemed acceptable for solving practical problems;

3. The error in determining the reactive power, which reaches 6.5%, can be explained by differences in approaches to its determination adopted in the MATLAB and Fazonord software systems. In calculations aided by the Fazonord package, only one of its segments was used, which provided the fundamental harmonic for the unbalanced load flow analysis. When using the MATLAB system, the simulation was carried out so as to take into account the non-linear current-voltage characteristics of SFC components, and the reactive power was determined by factoring in the higher harmonics.

The reactive power error can be reduced by additional non-sine load flow modeling using the technique reported in [22]. After determining the higher harmonic voltages, it is possible to recalculate the reactive power, e.g., using the technique of equivalent sine waves.

### **6. Conclusions**

Based on the evidence provided in this study, we can claim we have solved a currently relevant scientific and engineering problem of enhancing the accuracy of modeling unbalanced load flows in electric power systems. Our solution is based on an adequate model of induction motors and a technique of parameter identification for asynchronous loads. The following results were obtained:

1. We have developed a technique for modeling complex load nodes. The technique stands out from other known solutions for its use of the phase frame of reference. Its application scope covers the problems of load dispatching in smart grids;

2. The study has contributed a technique for parameter identification of load nodes. The technique is applicable to the problems of power system load dispatching. A key defining feature of the technique is the structure of the model, which is made up of three sources of current with parameters that are refined in the process of iterative load flow analysis of the power system;

3. We have proposed a technique for the identification of asynchronous load nodes with such asynchronous loads, including electrical drives equipped with static frequency converters;

4. With the aid of the asynchronous load identification models proposed in this paper, it is possible to solve the following practical tasks of electric power system management: increasing the accuracy of load flow modeling; making informed decisions when taking measures to reduce unbalance in power grids; and accounting for the balancing adjustment effect of asynchronous loads.

**Author Contributions:** Conceptualization, A.K., P.I. and A.A.; methodology, A.K. and K.S.; software A.K.; validation, A.K., A.A., P.I. and K.S.; formal analysis, A.K. and K.S.; investigation, A.K., A.A., P.I. and K.S.; resources, K.S.; data curation, A.K.; writing—original draft preparation, A.K. and K.S.; writing—review and editing, A.K., A.A. and K.S.; visualization, A.A. and P.I.; supervision, A.K. and K.S.; project administration, A.A. and K.S.; funding acquisition, K.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was carried out within the framework of the state task "Conducting applied scientific research" on the topic "Development of methods, algorithms and software for modeling the modes of traction power supply systems for DC railways and electromagnetic fields at traction substations for AC railways".

**Data Availability Statement:** Data sharing not applicable. No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

### **References**


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