Efficiency Analysis and Optimization of Two-Speed-Region Operation of Permanent Magnet Synchronous Motor Taking into Account Iron Loss Based on Linear Non-Equilibrium Thermodynamics

Abstract

In this article, the linear non-equilibrium thermodynamic approach is used to mathematically describe the energy regularities of an interior permanent magnet synchronous motor (IPMSM), taking into account iron loss. The IPMSM is considered a linear power converter (PC) that is multiple-linearized at operating points with a given angular velocity and load torque. A universal description of such a PC by a system of dimensionless parameters and characteristics made it possible to analyze the perfection of energy conversion in the object. For IPMSM, taking into account iron loss, a mathematical model of the corresponding PC has been built, and an algorithm and research program have been developed, which is valid in a wide range of machine speed regulations. This allows you to choose the optimal points of PC operation according to the maximum efficiency criteria and obtain the efficiency maps for IPMSM in different speed regions. The results of the studies demonstrate the effectiveness of the proposed method for determining the references of the d and q components of the armature current for both the loss-minimization strategy at the constant torque range of motor speed and the flux-weakening strategy in the constant power range.

1. Introduction

Permanent magnet synchronous machines (PMSMs) are widely used in various industries, transport, renewable energy, etc. This is due to their high specific power and torque indicators, high efficiency, reliability, good controllability, rapid torque response and low maintenance requirements [1,2]. In cases where it is necessary to adjust the angular velocity only down from the rated value, in the region of its regulation with a constant torque, a simpler surface-mounted PMSM (SPMSM) is mainly used, which is easier to implement since the permanent magnets (PMs) are placed on the rotor surface. However, this significantly limits the maximum angular velocity of the SPMSM in the range higher than the nominal one. For those applications where a wide range of angular velocity regulations is required, in particular, in the region of its regulation above the nominal value with constant power, the possibility of flux weakening (FW) in the machine is necessary. A special type of these machines is used here—interior permanent magnet synchronous machines (IPMSMs). They have the PMs placed inside the rotor, thanks to the magnetic conductivity along the d and q axes related to the rotor being different, which makes it possible to weaken the PM magnetic flux by adjusting the d-component of the armature current [3]. This is ensured through the vector control of components d and q of the armature current [4]. Many works have been devoted to the study of the peculiarities of such control and limitations on the angular velocity and electromagnetic torque due to the FW, which are reviewed in [5].

For all applications of PMSMs, and especially for EVs, an important indicator of an electric machine operation is its high efficiency and in the widest possible range of changes in angular velocity and electromagnetic torque. Research on ensuring the high efficiency of PMSMs is carried out in two interconnected directions—the justification of optimal machine coordinates and the implementation of control algorithms. These directions differ among themselves for the cases of an angular velocity adjustment from zero to the machine-rated value (a constant torque region) and its adjustment above the rated value (a constant power region).

For the constant torque region, the main optimal strategy from the point of view of loss minimization is the strategy of maximum torque per ampere (MTPA) [6,7,8,9]. In its classic form, this strategy ensures minimum loss in the copper of the armature winding. This is explained by two factors: the first one is the clear predominance of copper loss in the total power losses of the machine during its operation at low speed; the second one is the possibility of analytically obtaining expressions for the assignment of components d and q of the armature current, which are obtained from the basic mathematical model of the PMSM. However, due to the relative complexity of these mathematical expressions, their online calculations require complex controllers, so the main method of control is the use of an offline method—pre-calculated look-up tables (LUTs) [10]. This approach enables the practical implementation of a more accurate preliminary calculation of the trajectories of optimal control of PMSM operation, also taking into account loss in steel [11], magnetic saturation and cross-saturation effects [12]. As shown in [13], optimal control based on the minimum of total losses in the machine has significant advantages over the classic MTPA method. Various modern approaches are used for the offline determination of these losses: the use of a refined resistive model to justify optimal control coordinates from an energy point of view [14], calculation of loss in steel based on particle swarm optimization (PSO) and a recurrent neural network (NN) [15], a hybrid analytical model for predicting the electromagnetic loss, which is based on the combination of analytical calculations with the results of the FEM analysis of the machine magnetic field [16].

In contrast to offline methods, more complex online methods of IPMSM energy efficiency control in the constant torque region of their operation have recently been used to implement the MTPA strategy [6,9]. Among them are the online MTPA tracking of an IPMSM based on min–max optimization [17], Perturb and Observe (P&O) online algorithm [18], fractional-order extremum seeking method for MTPA [19], NN-based with cloud-based training for MTPA [20] and NN-based P&O algorithm [21].

In the constant power range of IPMSM angular velocity regulation, the main limitation is included in the justification of the optimal control coordinates—according to the armature voltage, which is determined by the DC supply voltage of the voltage inverter. In addition, there is also a limitation of the armature current, which is caused by the permissible heating of the machine in a long-term mode operation [22]. Therefore, the optimal from the point of view of energy efficiency control coordinates −d and q components of the armature current, in the steady state should be, as a rule, within the specified limits. As in the constant torque region, these coordinates are determined analytically for the classical approach, taking into account only copper loss in the machine [4,5,22,23,24]. However, in the constant power region, due to the increase in the IPMSM angular velocity, the frequency of the armature current also increases proportionally, which leads to an approximately quadratic increase in iron loss [5]. Already at a relatively low angular velocity, the iron loss begins to dominate the copper loss. Therefore, ignoring those leads to a significant error in the justification of the optimal components of the armature current for the given coordinates of the angular velocity at the load torque of the machine. In order to take into account the iron loss, as well as other factors, such as magnetic saturation and cross-saturation effects, in the constant power region of IPMSM speed regulation, the above-mentioned offline methods of substantiating optimal coordinates [11,14,15,16] and online methods of optimal control are also used [17,18,19,20,21,25]. However, offline methods are characterized by the complexity of preliminary research, and online methods are differed based on the complexity of real-time calculations and hence the need for powerful and expensive controllers. Methods that combine offline and online approaches have certain advantages [26].

Linear non-equilibrium thermodynamics (LNTDs) can be considered one of the productive directions for increasing the energy efficiency of various objects, which works by combining physical phenomena of different natures [27,28]. The peculiarity of LNTD methods are the ability to describe behavior of complex processes regardless of their nature by means of the linear relationship between their input and output power coordinates. For this, the system is considered a power converter (PC) with a certain number of its inputs and outputs, however the most often, with a single input and single output (SISO). In this case, the PC can be described by a system of dimensionless parameters and characteristics of its performance [29,30]. Such an important indicator as a degree of coupling between the PC input and output unambiguously determines the maximum achievable value of the PC efficiency, as well as the operating point, at which this maximum value is reached. Based on this indicator, it is possible to optimize the PC operation points depending on the adopted optimization criteria [30]. This powerful and universal method is successfully applied to describe free energy transformations in bio systems [28,29]. However, approaches of LNTD are also promising for other systems, especially those that combine processes of different physical natures. For example, in [31], it was successfully applied to analyze the efficiency of energy conversion in an electromechanical converter—an induction motor—and in [32], it was applied to an aeromechanical converter—a wind turbine.

In this work, the system of linear equations was obtained first for the description of steady-state modes of IPMSM taking into account the iron loss with the aim of model-based optimization of the machine operation in both ranges of its angular velocity regulation—with constant torque and constant power. In addition to the main task, the proposed approach makes it possible to analyze energy regularities and explore the possibilities of increasing the efficiency of the IPMSM in different operating modes. Taking into account the non-linear nature of steady-state processes in the machine, the system of linear equations is obtained by linearizing the dependence of the module of the input voltage vector on the angular velocity of the IPMSM at different operating points depending on the d-component of the armature current. The obtained universal dependencies in p. u. for the PC, which simulates the IPMSM operation taking into account iron loss, made it possible to analyze the perfection of energy conversion in such an object and understand its limitations.

The novelty and contribution of this work is the proposal of a new method for modeling the energy regularities of the operation of such a rather complex nonlinear object as an IPMSM taking into account iron loss. Within the framework of this method, the necessary mathematical models were obtained, and a research algorithm was developed and implemented in the MathCad environment. With the help of the created program, the analysis of the IPMSM operation in two ranges of its angular velocity regulation with different possibilities was carried out.

The paper is structured as follows. In Section 2, the research method is presented; in particular, the research object is shown in the form of a circular model of the IPMSM with the consideration of iron loss, and the main provisions of the LNTD and the universal method of describing and evaluating the performance indicators of linear PC are briefly highlighted. In Section 3, the main results of the research are presented and discussed: the mathematical models of SPMSM and IPMSM were obtained, taking into account iron loss as linear PCs; their research was carried out and the energy regularities of operation in both ranges of the machine angular velocity regulation were analyzed; the optimal coordinates of the armature current components were obtained to implement model-based vector control. An example of applying the obtained results to develop an IPMSM control system is shown in Section 4. Concluding remarks are given in Section 5.

2. Research Method

2.1. Mathematical Description of IPMSM Dynamics Taking into Account Iron Loss

The IPMSM mathematical model is traditionally presented in the rotating orthogonal d-q reference frame linked with the rotor, in which the d axis is directed along the flux linkage vector caused by the PM. We will assume the absence of saturation of the magnetic circuit and independence of the inductances of the motor armature winding from the rotor position. Under such conditions, the substitute scheme of the IPMSM, taking into account iron loss, will take the view shown in Figure 1 [33,34], where v_d and v_q are the projection on the d i q axis of the voltage vector applied to the armature windings, R is the resistance of the armature winding, R_c is the resistance simulating iron loss, L_d and L_q are the inductances of the armature windings relative to the d and q axes, p_p is the number of pairs of poles of the IPMSM rotor, ω is the angular velocity of the motor rotor and ψ_pm is the amplitude of the flux linkage created by one PM pole.

Based on the substitute scheme presented in Figure 1, the equations of the electrical balance of the IPMSM in the rotating reference frame, oriented along the rotor field, can be described by the next systems of equations

$$\left\{\begin{array}{l}{i}_d=i_{d0}+{\displaystyle \frac{1}{R_{\mathrm{c}}}}\left(L_d{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{d0}-p_{\mathrm{p}}\omega L_q{i}_{q0}\right)\\ i_q=i_{q0}+{\displaystyle \frac{1}{R_{\mathrm{c}}}}\left(L_q{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{q0}+p_{\mathrm{p}}\omega L_d{i}_{d0}+p_{\mathrm{p}}\omega {\psi }_{\mathrm{pm}}\right)\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{v}_d=i_{d}R+L_d{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{d0}-p_{\mathrm{p}}\omega L_q{i}_{q0}\\ v_q=i_{q}R+L_q{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{q0}+p_{\mathrm{p}}\omega L_d{i}_{d0}+p_{\mathrm{p}}\omega {\psi }_{\mathrm{pm}}\end{array}\right.$$

(2)

Substituting (1) into (2), we get

$$\left\{\begin{array}{l}{L}_d{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{d0}=-L_d{i}_{d0}{\displaystyle \frac{R}{L_{d}A}}+p_{\mathrm{p}}\omega L_q{i}_{q0}+{\displaystyle \frac{1}{A}}{v}_d\\ L_q{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{q0}=-L_q{i}_{q0}{\displaystyle \frac{R}{L_{\mathrm{q}}A}}-p_{\mathrm{p}}\omega \left(L_d{i}_{d0}+{\psi }_{\mathrm{pm}}\right)+{\displaystyle \frac{1}{A}}{v}_q\end{array}\right.$$

(3)

where

$$A=1+{\displaystyle \frac{R}{R_{\mathrm{c}}}}$$

(4)

In addition to the system of equations of electromagnetic balance (3), the IPMSM mathematical model also includes the differential equation of the balance of moments on the motor shaft

$$J_{\mathrm{m}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\omega =T-T_{\mathrm{L}}$$

(5)

where J_m is the moment of inertia of the drive brought to the motor shaft, T is the electromagnetic torque of the motor and T_L is the static load torque on the motor shaft.

The electromagnetic torque of the studied IPMSM is described by the equation [3,4]

$$T={\displaystyle \frac{3}{2}}{p}_{\mathrm{p}}\left[{\psi }_{\mathrm{pm}}{i}_{q0}+\left(L_d-L_q\right)i_{d0}{i}_{q0}\right]$$

(6)

Thus, the total dynamic mathematical model of IPMSM taking into account iron loss has the following view:

$$\left\{\begin{array}{l}{L}_d{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{d0}=-i_{d0}{\displaystyle \frac{R}{A}}+p_{\mathrm{p}}\omega L_d{i}_{q0}+{\displaystyle \frac{1}{A}}{v}_d\\ L_q{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{q0}=-i_{q0}{\displaystyle \frac{R}{A}}-p_{\mathrm{p}}\omega \left(L_d{i}_{d0}+{\psi }_{\mathrm{pm}}\right)+{\displaystyle \frac{1}{A}}{v}_q\\ J_{\mathrm{m}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\omega ={\displaystyle \frac{3}{2}}{p}_{\mathrm{p}}\left[{\psi }_{\mathrm{pm}}{i}_{q0}+\left(L_d-L_q\right)i_{d0}{i}_{q0}\right]-T_{\mathrm{L}}\end{array}\right.$$

(7)

2.2. Steady-State Equations

For the mathematical description of the steady-state operating mode of the IPMSM taking into account the iron loss, system (7) is transformed into the form

$$\left\{\begin{array}{l}{v}_d=R{i}_{d0}-A{p}_{\mathrm{p}}\omega L_q{i}_{q0}\\ v_q=R{i}_{q0}+A{p}_{\mathrm{p}}\omega \left(L_d{i}_{d0}+{\psi }_{\mathrm{pm}}\right)\\ T_{\mathrm{L}}={\displaystyle \frac{3}{2}}{p}_{\mathrm{p}}\left[{\psi }_{\mathrm{pm}}{i}_{q0}+\left(L_d-L_q\right)i_{d0}{i}_{q0}\right]\end{array}\right.$$

(8)

Useful dependences between the currents in the steady-state operation of the IPMSM taking into account iron loss can be obtained from the original systems of Equations (1) and (2). Having determined $L_d{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{d0}$ from the first equation of system (1) and substituting it into the first equation of system (2), as well as having determined $L_q{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{q0}$ from the second equation of system (1) and substituting it into the second equation of system (2), after transformations we obtain

$$\left\{\begin{array}{l}{i}_d={\displaystyle \frac{1}{A}}{i}_{d0}+{\displaystyle \frac{1}{R_{\mathrm{c}}A}}{v}_d\\ i_q={\displaystyle \frac{1}{A}}{i}_{q0}+{\displaystyle \frac{1}{R_{\mathrm{c}}A}}{v}_q\end{array}\right.$$

(9)

2.3. Initial Positions of LNTD

The second half of the 19th and the first half of the 20th century was marked by the rapid development of various branches of science in both theoretical and applied directions. At the same time, the classical thermodynamics also developed, which, thanks to the phenomenological approach to the macroscopic description of systems, is fundamental and allows for an analysis of energy transformations in phenomena of various natures (physical, chemical, biological) and even researching regularities in information, social systems, etc. [27]. However, the objects of mechanics, electrical engineering and lighting engineering, with few exceptions, did not become the subjects of thermodynamic research, since their own methods and laws were developed for them, which, as a rule, have generalized counterparts in thermodynamics. Thermodynamics was mainly used to describe reversible processes and isolated systems with a significant share of thermal energy, which is where the name of the science comes from.

The thermodynamics of non-equilibrium processes or LNTD, which developed from the classical thermodynamics in the second half of the 20th century, brought the general theoretical foundations under the energy regularities for not only isolated, but also closed and open systems, with real irreversible processes taking into account the speed of their flow [29]. As a result, a clear mathematical apparatus was developed, especially for linear systems, which allows for the description of energy transformations to be unified.

For any non-isolated system, which exchanges based on energy and matter with the external environment, the change in entropy is expressed by two terms: the external d_eS caused by exchanges with the external environment (entropy flow) and the internal d_iS caused by non-equilibrium processes within the system (production of entropy). Based on this, the generalized form of the second law of thermodynamics for non-isolated systems is expressed by the following dependence of the rate of entropy production in the system [28]:

$${\sigma }_{\mathrm{s}}={\displaystyle \frac{{\mathrm{d}}_{\mathrm{i}}S}{\mathrm{d}t}}\ge 0$$

(10)

Since every real irreversible process is accompanied by some internal flow, ${\overrightarrow{J}}_{\mathrm{k}}$ generated by the corresponding driving force ${\overrightarrow{X}}_{\mathrm{k}}$, which expresses non-equilibrium, it has been proven [28] that equality takes place under conditions of a local equilibrium

$${\sigma }_{\mathrm{s}}={\displaystyle \sum _{\mathrm{i}}{\overrightarrow{J}}_{\mathrm{k}}{\overrightarrow{X}}_{\mathrm{k}}}$$

(11)

The principle of local equilibrium or microscopic reversibility, which is the basis of LNTD, consists of the fact that in the non-equilibrium state of the body during not very rapid processes, the state of its very small elements can be considered to be in equilibrium, and the equations of classical thermodynamics can be applied to it [29]. The product of σ_s and the absolute temperature of the system is called the dissipative function, because it reflects the rate of dissipation (scattering, degradation) of the system free energy.

For the conditions of local equilibrium, the phenomenological law is also valid, which expresses the intensity of any flow in the form of products of the thermodynamic forces ${\overrightarrow{X}}_{\mathrm{i}}$ and the corresponding kinetic coefficients $L_{\mathrm{ki}}$ (Onsager’s principle of linearity) [29]:

$${\overrightarrow{J}}_{\mathrm{k}}={\displaystyle \sum _{\mathrm{i}}{L}_{\mathrm{ki}}{\overrightarrow{X}}_{\mathrm{i}}}.$$

(12)

Expression (12) shows that different thermodynamic driving forces can drive one flow. For PC of any nature, the driving forces at its input and output directly affect the flows at the input and output. According to the Curie principle [29], different driving forces can simultaneously affect the transfer if they are tensors of the same rank, for example, only scalars or only vectors. In (11) and (12), flows and forces have a vector character, which often happens in practice. Expression (12) is the basis of the construction of the theory of linear PCs that made it possible to theoretically describe a large number of complex processes, in which phenomena of different natures are combined. In the case of nonlinear systems, this theory is also true near the operating point of the PCP, where, as a rule, the principle of local equilibrium takes place.

2.4. Performance Indicators of the Universal PC

According to LNTD, for two conjugate processes between the PC output (o—output) and input (i—input), the following system of equations can be written [27]:

$$\left\{\begin{array}{l}{J}_{\mathrm{i}}=L_{\mathrm{ii}}{X}_{\mathrm{i}}+L_{\mathrm{io}}{X}_{\mathrm{o}}\\ J_{\mathrm{o}}=L_{\mathrm{oi}}{X}_{\mathrm{i}}+L_{\mathrm{oo}}{X}_{\mathrm{o}}\end{array}\right.$$

(13)

where the kinetic or “phenomenological” coefficients are described by the expressions

$$L_{\mathrm{jk}}={\left({\displaystyle \frac{\partial J_{\mathrm{j}}}{\partial X_{\mathrm{k}}}}\right)}_{X_{\mathrm{j}}=\mathrm{const}}$$

(14)

By definition, the forces X_i and X_o and the flow J_i are positive, and the flow J_o is negative, which means the absorption of energy at the PC input and its formation at its output; therefore, L_ii, L_oo and (-L_io) must be positive.

According to Onsager’s principle of reciprocity [29], L_jk = L_kj. That is, in the neighborhood of the equilibrium state, the linear dependence of the flow J_j on the force X_k coincides with the analogous dependence of the flow J_k on the force X_j. The ratios of Onsager reciprocity significantly reduce the number of independent kinetic coefficients in systems of type (13).

The above-described main provisions of LNTD are the basis of the PC thermodynamic analysis [29,30].

An analysis of the system (13) shows that the more negative value of L_io compared to other kinetic coefficients testifies about the higher the degree of influence of the input force on the output flow. In addition, the suppression of the input flow by the output force increases if the coefficient L_io is more negative. As follows, the kinetic coefficient L_io determines the degree of coupling q between the processes at the PC input and output. To avoid the effect on q, the values of other kinetic coefficients, the degree of coupling was defined as follows [30]:

$$q={\displaystyle \frac{L_{\mathrm{io}}}{\sqrt{L_{\mathrm{ii}}{L}_{\mathrm{oo}}}}}$$

(15)

According to the definition, q is negative and will acquire values from −1 to 0.

To obtain universal PC characteristics in p. u., two indicators are introduced [30]:

• relation of forces

$$\chi =X_{\mathrm{o}}/X_{\mathrm{i}}$$

(16)

• phenomenological relationship

$$Z=\sqrt{L_{\mathrm{oo}}/L_{\mathrm{ii}}}$$

(17)

Having the normalized input and output flows to the input force at zero output force (short circuit), the system of Equation (13) can be rewritten using the introduced indicators (16) and (17)

$$\left\{\begin{array}{l}{\displaystyle \frac{J_{\mathrm{i}}}{L_{\mathrm{ii}}{X}_{\mathrm{i}}}}=q \left(Z\chi \right)+1\\ {\displaystyle \frac{J_{\mathrm{o}}}{Z{L}_{\mathrm{ii}}{X}_{\mathrm{i}}}}=\left(Z\chi \right)+q\end{array}\right.$$

(18)

and the flow ratio is determined as follows:

$$j={\displaystyle \frac{J_{\mathrm{o}}}{J_{\mathrm{i}}}}=Z{\displaystyle \frac{\left(Z\chi \right)+q}{q·\left(Z\chi \right)+1}}$$

(19)

In (18) and (19), the product Zχ is a dimensionless quantity that expresses the reduced force ratio. The essence of q can be explained as follows. When q = −1, the output flow differs from the input flow by −Z times. Therefore, the input and output flows are rigidly coupled. When q = 0, the input and output flows are caused only by their own forces that means that the flows are not coupled at all. For values of q from −1 to 0, the output flow is supported by the input flows in the “downhill” direction. The ratio of these flows changes along with the change in Zχ, which means that the input and output processes in this case are partially coupled [32].

The thermodynamic efficiency η is the ratio of the rate, at which a PC produces free energy on its output, to the rate, at which it consumes free energy on its input. This is one of the most important characteristics of PC operation. The expression for η can be easily obtained from (19) and (16) in the form

$$\eta =-{\displaystyle \frac{J_{\mathrm{o}}{X}_{\mathrm{o}}}{J_{\mathrm{i}}{X}_{\mathrm{i}}}}=-j\chi =-\left(Z\chi \right){\displaystyle \frac{\left(Z\chi \right)+q}{q·\left(Z\chi \right)+1}}$$

(20)

Figure 2 shows graphical dependences (20) at different values of q. From (20), the optimal ratio of forces that provides maximum thermodynamic efficiency is

$${\left(Z\chi \right)}_{\mathrm{opt}-\mathsf{\eta }}=-{\displaystyle \frac{q}{1+\sqrt{1-q^2}}}$$

(21)

and this efficiency maximum value equal to

$${\eta }_{\max }={\left(Z\chi \right)}_{\mathrm{opt}-\mathsf{\eta }}^2$$

(22)

Other important indicators of PC operation are the normalized output flow j_o and normalized output power p [30]:

$$j_{\mathrm{o}}=-{\displaystyle \frac{J_2}{Z{L}_{11}{X}_1}}=-\left(Z\chi \right)-q$$

(23)

$$p=-{\displaystyle \frac{J_{\mathrm{o}}{X}_{\mathrm{o}}}{L_{\mathrm{ii}}{X}_{\mathrm{i}}^2}}=-\left(Z\chi \right)\left[q+\left(Z\chi \right)\right]$$

(24)

Based on of the obtained universal dependencies of the criteria for the operation of a universal PC (20), (23) and (24), it is possible to easily form other criteria for the optimal operation appropriate for specific PCs by combining two indicators, for example [29]: the maximum output flow at optimal efficiency, the maximum output power at optimum efficiency, etc.

Using two relative parameters of the universal PC—q and Zχ—it is also possible to determine in p. u. all components of power—the input power P_in^*, the output power P_out^* and the total power losses ΔP_Σ^*. Moreover, we can also define separate components of the total power losses: the part caused by the nonequilibrium of the process ΔP_R^*, that is, the load of the system, and the part caused by its incomplete coupling ΔP_q^*. To obtain these components in p. u., all absolute values of powers are normalized to L_iiX_i², as can be seen from (24). For example, we will be interested in the power loss components:

$$\Delta {P_{\mathrm{R}}}^{\ast }={\displaystyle \frac{\Delta P_{\mathrm{R}}}{L_{\mathrm{ii}}{X}_{\mathrm{i}}^2}}={\left[1+q\left(Z\chi \right)\right]}^2$$

(25)

$$\Delta {P_{\mathrm{q}}}^{\ast }={\displaystyle \frac{\Delta P_{\mathrm{q}}}{L_{\mathrm{ii}}{X}_{\mathrm{i}}^2}}={\left(Z\chi \right)}^2\left(1-q^2\right)$$

(26)

3. Results and Their Discussion

3.1. Parameters of the Studied Machine

For further research, a virtual IPMSM with the parameters listed in

Table 1. The parameters of the studied machine.

Parameter	Value
Rated power, Pn (kW)	10
Rated DC voltage, VDC.n (V)	150
Based angular velocity, ωn (s−1)	100
Maximum angular velocity, ωn (s−1)	200
Rated torque, TG.n (Nm)	100
Rated rms phase current, in (A)	63
Number of pole pairs, pp	2
PM flux linkage, ψpm (Vs)	0.35
Winding resistance, R (Ω)	0.1
d-axis winding inductance, Ld (mH)	1
q-axis winding inductance, Lq (mH)	3
Rated equivalent iron loss resistance, Rc.n (Ω)	14.1

was taken. According to the parameters of the machine specified in Table 1, copper loss in the nominal mode of its operation is 7% of the nominal machine power.

A separate important task for this study is the modeling of iron loss in the IPMSM, which are especially manifested at high angular velocities of the machine. When modelling iron loss, it is assumed that the magnetic flux in the core of the armature has a sinusoidal character.

Let us find with a reasonable approximation the value of resistance R_c, which simulates iron loss, given the power of iron loss at a level of 5% of the nominal machine power, that is, the value of relative power of iron loss is equal to δ_Fe = 0.05. From the substitute scheme of the IPMSM shown in Figure 1, this relative power can be approximately written as

$${\delta }_{\mathrm{Fe}}={\displaystyle \frac{{\left(1.2p_{\mathrm{p}}{\omega }_{\mathrm{n}}{\psi }_{\mathrm{pm}}\right)}^2}{R_{\mathrm{c}.\mathrm{n}}{P}_{\mathrm{n}}}}$$

(27)

where R_c.n is the value of resistance R_c in the nominal mode of the IPMSM operation; the coefficient of 1.2 makes it possible to approximately estimate the influence of the flux linkage of the armature reaction at a level of 20% of the flux linkage from the PMs.

For the studied IPMSM with the parameters given in Table 1, the value of R_c calculated by (24) is 14.1 Ohms. The main components of iron power loss—due to hysteresis and eddy currents—depend on the machine angular velocity differently. Therefore, the value of the resistance that models iron loss decreases as the machine speed decreases. As shown in [34], such a trend can be modeled by the following equation:

$$R_{\mathrm{c}}=R_{\mathrm{c}.\mathrm{n}}{\displaystyle \frac{\frac{K_{\mathrm{f}}}{K_{\mathrm{h}}}+1}{\frac{K_{\mathrm{f}}}{K_{\mathrm{h}}}+\frac{1}{{\omega }^{\ast }}}}$$

(28)

where K_h and K_f are the coefficients characterizing, respectively, hysteresis losses and eddy currents at nominal values of magnetic flux and frequency; ω^* is the relative value of the angular velocity of the machine, which is equal to one of its nominal values.

According to experimental data [35], K_f/K_h = 0.5694 is accepted.

3.2. PMSM as Linear PC

As a PC, the PMSM converts electrical power at the input of the armature winding into mechanical power on the motor shaft. A force acts in the form of an armature voltage vector $X_{\mathrm{i}}={\overrightarrow{v}}_{\mathrm{a}}$ at the input of this PC, which determines the input flow in the form of an armature current vector $J_{\mathrm{i}}={\overrightarrow{i}}_{\mathrm{a}}$. At the output of the PC, the force in the form of the angular velocity of the PMSM X_o = ω determines the electromagnetic torque of the machine, which in the steady-state is equal to the machine load torque J_o = T_L. The output power of the PC is equal to −P_o = ω T_L, and the active power at its input is the dot product of the armature voltage and current vectors. The latter looks like this:

$$P_{\mathrm{i}}=\left({\overrightarrow{v}}_{\mathrm{a}}·{\overrightarrow{i}}_{\mathrm{a}}\right)={\displaystyle \frac{3}{2}}\left(v_d i_d+v_q i_q\right)$$

(29)

where the factor 3/2 is due to the transformation of the two-phase reference frame to the three-phase one.

The dot product of the armature voltage and current vectors, in order to separate the voltage and current, can also be represented as

$$P_{\mathrm{i}}=\left({\overrightarrow{v}}_{\mathrm{a}}·{\overrightarrow{i}}_{\mathrm{a}}\right)=\left|{\overrightarrow{v}}_{\mathrm{a}}\right|\left|{\overrightarrow{i}}_{\mathrm{a}}\right|\cos \phi ={\displaystyle \frac{3}{2}}\sqrt{v_d^2+v_q^2}\sqrt{i_d^2+i_q^2}\cos \left(\mathrm{atan}{\displaystyle \frac{v_d}{v_q}}-\mathrm{atan}{\displaystyle \frac{i_d}{i_q}}\right)$$

(30)

where φ is the angle between the voltage and armature current vectors.

From (30), the input power and flow of the PC can be chosen as follows:

$$X_{\mathrm{i}}=\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{v_d^2+v_q^2},J_{\mathrm{i}}=\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{i_d^2+i_q^2}\cos \left(\mathrm{atan}{\displaystyle \frac{v_d}{v_q}}-\mathrm{atan}{\displaystyle \frac{i_d}{i_q}}\right)$$

(31)

Similarly to (13), the system of linear equations that describes the operation of the PMSM as a linear PC will have the form

$$\left\{\begin{array}{r}\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{i_d^2+i_q^2}\cos \left(\mathrm{atan}{\displaystyle \frac{v_d}{v_q}}-\mathrm{atan}{\displaystyle \frac{i_d}{i_q}}\right)=L_{\mathrm{ii}}\left(\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{v_d^2+v_q^2}\right)+L_{\mathrm{io}}\omega \\ -T=L_{\mathrm{oi}}\left(\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{v_d^2+v_q^2}\right)+L_{\mathrm{oo}}\omega \end{array}\right.$$

(32)

Control of the PMSM electromagnetic torque is carried out, most often, in a vector method using a vector invertor, which is implemented by forming the armature current vector, respectively, the EMF vector according to the corresponding law. This law depends on the type of PMSM and its angular velocity control region and directly affects the machine efficiency.

3.3. Efficiency Analysis of SPMSM Operation in the Constant-Torque Region

In the SPMSM, the magnets are fixed on the surface of the rotor; therefore, taking into account the fact that the magnetic permeability of PMs is close to the magnetic permeability of air, the inductances along the d and q axes have the same value: L_d = L_q = L = 2 mH. This leads to the presence of only an active component in the electromagnetic torque (6):

$$T={\displaystyle \frac{3}{2}}{p}_{\mathrm{p}}{\psi }_{\mathrm{pm}}{i}_{q0}$$

(33)

In this regard, the energy-efficient SPMSM vector control strategy is the condition i_d = 0 [6]. Under this condition, the input active power (29) will take the form

$$P_{\mathrm{i}}=\left(\sqrt{{\displaystyle \frac{3}{2}}}{v}_q\right)\left(\sqrt{{\displaystyle \frac{3}{2}}}{i}_q\right)$$

(34)

Based on (34), the input power and flow of the PC will be equal to $X_{\mathrm{i}}=\sqrt{{\displaystyle \frac{3}{2}}}{v}_q$, $J_{\mathrm{i}}=\sqrt{{\displaystyle \frac{3}{2}}}{i}_q$, and the system of linear equations describing the operation of SPMSM as PC will have the form

$$\left\{\begin{array}{r}\left(\sqrt{{\displaystyle \frac{3}{2}}}{i}_q\right)=L_{\mathrm{ii}}\left(\sqrt{{\displaystyle \frac{3}{2}}}{v}_q\right)+L_{\mathrm{io}}\mathsf{\omega }\\ -T_{\mathrm{L}}=L_{\mathrm{oi}}\left(\sqrt{{\displaystyle \frac{3}{2}}}{v}_q\right)+L_{\mathrm{oo}}\mathsf{\omega }\end{array}\right.$$

(35)

Let us determine the kinetic coefficients in the system of Equation (35) using the mathematical model of the PMSM steady-state operation (8) and (9).

Under the condition of i_d = 0, from the first equation of system (9), we obtain

$$i_{\mathrm{d}0}={\displaystyle \frac{1}{R_{\mathrm{c}}}}{v}_d$$

(36)

Substituting (36) into the first equation of system (9), we get

$$v_d=-p_{\mathrm{p}}\omega L_q{i}_{q0}$$

(37)

Substituting (37) into (36) and then the resulting expression into the second equation of system (8), after transformations we get

$$i_{q0}\left(1+A{\left({\displaystyle \frac{p_{\mathrm{p}}\omega }{R}}\right)}^2{L}_d{L}_q\right)=-{\displaystyle \frac{1}{R}}{v}_q-{\displaystyle \frac{A}{R}}{p}_{\mathrm{p}}{\psi }_{\mathrm{pm}}\omega$$

(38)

The second term in parentheses can be neglected due to its smallness.

By substituting the obtained value i_q0 into the second equation of system (9), as well as into the third equation of system (9), we obtain

$$i_q={\displaystyle \frac{1}{R}}{v}_q-{\displaystyle \frac{p_{\mathrm{p}}{\psi }_{\mathrm{pm}}}{R}}\omega$$

(39)

$$T_{\mathrm{L}}={\displaystyle \frac{3}{2}}{\displaystyle \frac{p_{\mathrm{p}}{\psi }_{\mathrm{pm}}}{R}}{v}_q-{\displaystyle \frac{3}{2}}{\displaystyle \frac{A{\left(p_{\mathrm{p}}{\psi }_{\mathrm{pm}}\right)}^2}{R}}\omega$$

(40)

Equations (39) and (40) are reduced to system of Equation (35):

$$\left\{\begin{array}{l}\left(\sqrt{{\displaystyle \frac{3}{2}}}{i}_q\right)={\displaystyle \frac{1}{R}}\left(\sqrt{{\displaystyle \frac{3}{2}}}{v}_q\right)-\sqrt{{\displaystyle \frac{3}{2}}}{\displaystyle \frac{p_{\mathrm{p}}{\psi }_{\mathrm{pm}}}{R}}\omega \\ -T_{\mathrm{L}}=-\sqrt{{\displaystyle \frac{3}{2}}}{\displaystyle \frac{p_{\mathrm{p}}{\psi }_{\mathrm{pm}}}{R}}\left(\sqrt{{\displaystyle \frac{3}{2}}}{v}_q\right)+{\displaystyle \frac{3}{2}}{\displaystyle \frac{A{\left(p_{\mathrm{p}}{\psi }_{\mathrm{pm}}\right)}^2}{R}}\omega \end{array}\right.$$

(41)

From the equations of system (41), we obtain the expressions for the kinetic coefficients of the PC, which describe the operation of the vector-controlled SPMSM:

$$L_{\mathrm{ii}}={\displaystyle \frac{1}{R}},L_{\mathrm{io}}=-\sqrt{{\displaystyle \frac{3}{2}}}·{\displaystyle \frac{p_{\mathrm{p}} {\psi }_{\mathrm{pm}}}{R}},L_{\mathrm{oo}}={\displaystyle \frac{3}{2}}·{\displaystyle \frac{A}{R}} {\left(p_{\mathrm{p}}{\psi }_{\mathrm{pm}}\right)}^2$$

(42)

Based on kinetic coefficients (42), expressions for the main dimensionless parameters of this PC can be obtained from (15)–(17):

$$q=-{\left(A\right)}^{-0.5},Z=\sqrt{{\displaystyle \frac{3A}{2}}} p_{\mathrm{p}} {\psi }_{\mathrm{pm}},\chi =\sqrt{{\displaystyle \frac{2}{3}}}·{\displaystyle \frac{\omega }{v_q}},Z\chi =\sqrt{A} p_{\mathrm{p}} {\psi }_{\mathrm{pm}}{\displaystyle \frac{\omega }{v_q}}$$

(43)

As can be seen from the obtained results (43), the degree of coupling q of the PC modeling the SPMSM vector controlled by the i_d = 0 method, for the case without taking into account iron loss, is equal to −1 and does not depend on the mode parameters of the machine. In this case, the SPMSM is modeled by only one substitute scheme determined by the coordinate q, but without resistance R_c (Figure 1b). Therefore, the current i_q completely forms an electromagnetic torque; that is, it is directly linked with the output current. For the case of taking into account iron loss, the resistance R_c appears in the substitute circuit, through which a part of the input current is closed. In this way, there is already an incomplete coupling between the input and output of the PC, and the degree of coupling is determined using the value A according to (43). With a decrease in the angular velocity of the machine, A decreases because, according to (28), the value of the resistance R_c decreases. For the machine under study, when the relative angular velocity ω^* decreases from 1 to 0.25, the value of q changes from −0.9965 to −0.9898. The maximum energy efficiency of such a machine, according to (20), also decreases from 0.845 to 0.751 (Figure 3). The value of (Zχ)_opt also decreases from 0.919 to 0.867.

The value of the SPMSM operating point Zχ, in turn, depends on the values of two main variables—the angular velocity of the machine ω and the load torque T_L, which directly affect the value of v_q. The value of the latter was calculated from Equations (6), (8) and (9) under the following conditions:

$$i_{q0}={\displaystyle \frac{2T_{\mathrm{L}}}{3p_{\mathrm{p}} {\psi }_{\mathrm{pm}}{i}_{q0}}},v_d=- p_{\mathrm{p}} \omega L_q{i}_{q0},i_{d0}=-{\displaystyle \frac{v_d}{R_{\mathrm{c}}}},v_q=R{i}_{q0}+A{p}_{\mathrm{p}}\omega \left(L_d{i}_{d0}+{\psi }_{\mathrm{pm}}\right)$$

(44)

Figure 4 shows how the dimensionless indicator Zχ of the operating mode of the studied machine depends on the fixed values of the main variables ω and T_L. As can be seen from this figure, the optimal value of (Zχ)_opt, and therefore the maximum efficiency of the machine, is achieved only with a certain combination of values ω and T_L. The following relationship between the variables ω and T_L corresponds to the (Zχ)_opt curve:

$$T_{\mathrm{L}}=-14.03+1.442\omega +0.006377{\omega }^2$$

(45)

In Figure 4 both above and below the curves shown (Zχ)_opt, the energy efficiency of the SPMSM decreases, and the lower part corresponds to the left side of the η(Zχ) characteristics shown in Figure 3, and the upper part is on the right side of the η(Zχ) characteristics. As can be seen from Figure 4, the greatest deviation of the obtained dependences Zχ(ω) and Zχ(T_L) from the corresponding optimal curves (Zχ)_opt is observed at low values of both ω and T_L, and the best approximation of the specified dependences to (Zχ)_opt occurs for the average values of ω and T_L. The obtained regularity can be clearly seen from the efficiency map, which is constructed using the dependencies obtained above for the studied SPMSM in Figure 5.

Therefore, the energy efficiency of SPMSM taking into account iron loss and the operating according to the criterion i_d = 0 directly depends on the parameters of the drive—angular velocity and load torque. High efficiency can be ensured only with the indicated combinations of these parameters. Unfortunately, there are no other options to change the SPMSM performance map.

3.4. Efficiency Optimization of the IPMSM Operation in the Constant-Torque Region

As can be seen from the nonlinear mathematical model of IPMSM for its steady-state operation—from systems of Equations (8) and (9), as well as from complex expressions for the input force and flow (31)—analytical expressions are obtained for the kinetic coefficients of the PC that describe the operation of this machine, which is impossible. Therefore, the research method for this case remains only numerical. Since the system is clearly nonlinear, the PC parameters will change depending on the working point of the machine—the set values of the angular velocity ω₀ and the load torque T_L0. For a given pair of values of these variables, it is necessary to linearize the system and obtain the PC parameters.

As is known [9], already in the constant torque control region of the IPMSM angular velocity regulation, the energy efficiency of its operation also depends on the value of the armature current component i_d. Even for the classic case—without taking into account iron loss—it is not possible to obtain analytical expressions for the dependencies of the optimal values of the components of the armature current on the value of only the load torque—the MTPA curve [35]. If iron loss is also taken into account, even with the help of such a simple model, which is used in this work, it will not be possible to obtain such analytical dependencies. Therefore, for the purpose of numerical energy optimization of the IPMSM taking into account iron loss, it is necessary to introduce one more variative variable. As the analysis showed, it is easiest to choose the component of the armature current i_d0, which determines the reactive component of the electromagnetic torque. Therefore, further calculations should be carried out as a function of this variable.

Therefore, it is advisable to build the IPMSM mathematical model taking into account iron loss, for successive numerical calculations, based on the systems of Equations (8) and (9) and expressions (31) in the following view:

$$i_{q0}\left(i_{d0}\right)={\displaystyle \frac{2T_{\mathrm{L}}}{3p_{\mathrm{p}}\left[{\psi }_{\mathrm{pm}}+\left(L_d-L_q\right)i_{d0}\right]}}$$

(46)

$$v_d\left(i_{d0}\right)=R{i}_{d0}-A p_{\mathrm{p}}\omega L_q i_{q0}\left(i_{d0}\right)$$

(47)

$$v_q\left(i_{d0}\right)=R{i}_{q0}+A p_{\mathrm{p}}\omega \left(L_d{i}_{d0}+{\psi }_{\mathrm{pm}}\right)$$

(48)

$$i_d\left(i_{d0}\right)={\displaystyle \frac{1}{A}}{i}_{d0}+{\displaystyle \frac{1}{R_{\mathrm{c}}A}}{v}_d\left(i_{d0}\right)$$

(49)

$$i_q\left(i_{d0}\right)={\displaystyle \frac{1}{A}}{i}_{q0}\left(i_{d0}\right)+{\displaystyle \frac{1}{R_{\mathrm{c}}A}}{v}_q\left(i_{d0}\right)$$

(50)

$$v_{\mathrm{a}}\left(i_{d0}\right)=\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{v_d^2\left(i_{d0}\right)+v_q^2\left(i_{d0}\right)}$$

(51)

$$i_{\mathrm{a}}\left(i_{d0}\right)=\sqrt{{\displaystyle \frac{3}{2}}}\sqrt{i_d^2+i_q^2}\cos \left(\mathrm{atan}{\displaystyle \frac{v_d\left(i_{d0}\right)}{v_q\left(i_{d0}\right)}}-\mathrm{atan}{\displaystyle \frac{i_d\left(i_{d0}\right)}{i_q\left(i_{d0}\right)}}\right)$$

(52)

Having the set parameters of the working point of the machine ω = ω₀ and T_L = T_L0, as a result of cyclic calculations according to expressions (46)–(52) for points in the range i_d0 = 0–(−100) A, the final dependences v_a(i_d0) and i_a(i_d0), which characterize the input power of the PC depending on the value of the component of the armature current i_d0, are obtained at a given output power P_o(ω₀, T_L0). Thus, for each value of i_d0, there is a pair of the PC input and output coordinates—flow X and force J. However, these values are not enough to determine three kinetic coefficients from two equations in system (32). To obtain one more point of the linear PC, it is necessary to linearize the obtained dependences of the input force v_a(i_d0) and the input flow i_a(i_d0) of the PC based on the output coordinates of the force ω and the flow T_L at the given point (ω₀, T_L0). To obtain such linear dependences, similar calculations were carried out according to expressions (46)–(52), but for points equidistant to the left and right of the given point at some insignificant distances: once by the value of ω₀ for T_L = T_L0 and the second time by the value of T_L0 for ω = ω₀. Four linear dependences of the input flow and force on the output flow and force at the given point (ω₀, T_L0) were found based on the pairs of values v_a(i_d0) and i_a(i_d0) obtained at these points and depending on i_d0. Attempts to apply points from the obtained linear dependencies to determine the kinetic coefficients showed that if even two points from different dependencies are taken, then the values of the kinetic coefficients L_io and L_oi determined from the two equations of system (32) will be different. Therefore, only one linearized dependence was taken—v_a(i_d0) as the function of ω, which the least of all obtained differs from the linear character that leads to the smallest calculation error. The angular coefficient of the tangent to the curve at the point ω₀ is determined based on the received values of the armature input voltage module (48) from the left $v_{\mathrm{a}\_\mathrm{minus}}\left(i_{d0}\right)$ and right $v_{\mathrm{a}\_\mathrm{plus}}\left(i_{d0}\right)$ of the given value ω₀:

$$k_{\mathrm{v}\left(\mathsf{\omega }\right)}\left(i_{d0}\right)={\displaystyle \frac{v_{\mathrm{a}\_\mathrm{plus}}\left(i_{d0}\right)-v_{\mathrm{a}\_\mathrm{minus}}\left(i_{d0}\right)}{2\Delta \omega }}.$$

(53)

Using the obtained value $k_{\mathrm{v}\left(\mathsf{\omega }\right)}\left(i_{d0}\right)$, it is possible to determine the necessary one more operating point of the PC—the value of the input voltage v_a(i_d0) at ω = 0:

$$v_{\mathrm{a}.\mathsf{\omega }=0}\left(i_{d0}\right)=v_{\mathrm{a}}\left(i_{d0}\right)-k_{\mathrm{v}\left(\mathsf{\omega }\right)}\left(i_{d0}\right) {\omega }_0.$$

(54)

The value of the kinetic coefficients was carried out in the following order.

From the second equation of system (32) for ω = 0, the following is obtained:

$$L_{\mathrm{oi}}\left(i_{d0}\right)={\displaystyle \frac{-T_{\mathrm{L}0}}{v_{\mathrm{a}.\mathsf{\omega }=0}\left(i_{d0}\right)}}$$

(55)

From the first equation of system (32) for the working point of the PC, the following is found:

$$L_{\mathrm{ii}}\left(i_{d0}\right)={\displaystyle \frac{i_{\mathrm{a}}\left(i_{d0}\right)-L_{\mathrm{oi}}\left(i_{d0}\right){\omega }_0}{v_{\mathrm{a}}\left(i_{d0}\right)}}$$

(56)

From the second equation of system (32) for the working point of the PC, the following is determined

$$L_{\mathrm{oo}}\left(i_{d0}\right)={\displaystyle \frac{-T_{\mathrm{L}0}-L_{\mathrm{oi}}\left(i_{d0}\right)v_{\mathrm{a}}\left(i_{d0}\right)}{{\omega }_0}}$$

(57)

With the use of kinetic coefficients (55)–(57) according to expressions (15)–(17), dependences on the main parameters of the PC and the indicator of the energy efficiency of the studied IPMSM taking into account iron loss were obtained: q(i_d0), Zχ(i_d0) and η(i_d0).

According to the described algorithm, a program was created in the Mathcad environment that allows you to calculate the specified parameters and performance indicators of the PC, which models the operation of the studied IPMSM taking into account iron loss for different operating points of the machine depending on the component of the armature current i_d0. The flow chart of this algorithm is presented in Figure 6. The algorithm works as follows.

In block 1, the operating parameters of the machine are specified—the set values of the angular velocity ω₀ and the load torque T_L0. In blocks 2–4, in cycles for a given range of values of the d-component of the armature current, identical calculations of the dependencies on id0 of the main electrical variables of the machine are carried out according to expressions (46)–(52). These calculations differ only in the values of the angular velocity—for a given value of ω₀ and for the values shifted from it to the left and right by the amount Δω (in the work, Δω = 5 s⁻¹ is used). According to the results of these calculations, in block 5, a multiple, i_d0-dependent, linearization of the dependence of the armature voltage module based on the machine angular velocity is carried out according to expression (53), and the coordinates of another working point of the PC, which models the IPMSM, are found based on the obtained linear dependence according to (54). Next, in block 6, according to expressions (55)–(57), the i_d0-dependent values of the kinetic coefficients of the PC are found. Based on these values, block 7 also determines the main parameters of the PC—q and Zχ, which also depend on i_d0, as well as the energy efficiency of the machine operation η(i_d0). From the latter, in block 8, the optimal value of i_d0.opt, for which the efficiency is maximum, can be obtained.

To check the correctness of the algorithm, the first computational experiment was conducted for the variant for which the results can be determined analytically—for the IPMSM without taking into account iron loss. Having set three orders of magnitude higher than the real value of R_c in the model, the results were obtained for the nominal angular velocity of the machine and for the values of the load torque of 0.25 T_n, 0.5 T_n, 0.75 T_n and 1.0 T_n, which are shown in Figure 7.

From the dependences η(Zχ) shown in Figure 7, the value of the current component i_d0, at which the maximum energy efficiency is achieved, were obtained, which are listed in

Table 2. Comparison of the optimal values of the i_d current component obtained by the developed algorithm with the corresponding values obtained analytically.

TL, Nm	25	50	75	100
id0, A	−3.0	−11.5	−21.5	−31.5
T*	0.2041	0.4082	0.6122	0.8163
id*	−0.0123	−0.0719	−0.1306	−0.1885
id, A	−2.15	−12.58	−22.86	−32.99

. Since in the case of not taking into account iron loss, i_d0 = i_d, these values should correspond to the values of the MTPA curve. In [24], the dependence on p. u. between the electromagnetic torque and the component of the armature current i_d for the IPMSM at the points of the MTPA curve is given as

$$T^{\ast }=1.5\sqrt{-i_d^{\ast }{\left(1-i_d^{\ast }\right)}^3}$$

(58)

where the relative values are normalized to the corresponding base values

$$T_{\mathrm{b}}=p_{\mathrm{p}} {\psi }_{\mathrm{pm}} i_{\mathrm{b}},i_{\mathrm{b}}={\psi }_{\mathrm{pm}}/\left(L_q-L_d\right)$$

(59)

For the calculation, a numerical dependence was obtained from expression (55) in [24]:

$$i_d^{\ast }=0.01T^{\ast 2}-0.298T^{\ast }+0.0481$$

(60)

Calculated using (58)–(60) for the studied IPMSM, i_d values from the MTPA curve are given in Table 2. The deviation obtained according to the developed algorithm from the corresponding values obtained analytically does not exceed 1.5 A for medium and high loads, which is less than 10% for medium loads and less than 5% for the nominal motor load torque. This is an acceptable result for these studies and indicates the adequacy of the proposed research method and algorithm for calculating the parameters and performance indicators of the IPMSM taking into account iron loss.

From Figure 7, it can be seen that even without taking into account in the IPMSM iron loss, the degree of coupling decreases. q = −1, i.e., full coupling for each loads occurs only for the one value of i_d0, i.e., for a certain optimal ratio between the components of the armature current i_d0 and i_q0. With other ratios between these components, which also provide the required electromagnetic torque of the machine, the coupling between the input and output will already be incomplete. This can be explained by the different contribution of the components of the armature current to the production of the electromagnetic torque according to (6) and the two substitute circuits that work in this case (Figure 1). With differences from the optimal ratios between the components of the armature current, there is a “slippage” of power in one of the circuits or in two circuits together.

In Figure 8, similar to those shown in Figure 7, dependencies were obtained from the developed calculation program for the studied IPMSM taking into account iron loss.

A comparison of the results shown in Figure 8 with the similar ones shown in Figure 6, which makes it possible to draw the following conclusions. Taking into account the iron loss leads to a decrease in the modulus of the PC degree of coupling, which simulates the operation of the machine (Figure 8a) and a shift of its maximum towards higher values of the id0 current component. The values of the maximum energy efficiency under medium and high loads of the drive (T_L^* = 0.5–1.0) are reduced by taking into account the iron loss by 5–9% compared to the option without taking into account these losses (Figure 8c). In the case of low drive load (T_L^* = 0.25), η is reduced by as much as 17%, which is explained by the relative increase in the effect on η of losses in steel compared to reduced losses in copper. The points of maximum energy efficiency are also shifted towards larger values of the i_d0 current component, which are very close to similar values of this current, at which the maximum value of the degree of coupling is ensured. Similar regularities also occur for other values of the IPMSM angular velocity.

Thus, the presence of one more variable coordinates—the d-component of the armature current—makes it possible to ensure the operation of the IPMSM taking into account the iron loss at the points of maximum energy efficiency. The developed technique makes it possible to obtain optimal values of i_d0.opt for each pair of values of the operating point of the drive—angular velocity and load torque.

The developed algorithm (Figure 6) and the corresponding program in Mathcad make it possible to obtain the optimal values of i_d0.opt for each pair of the machine variables that characterizes the drive operating point—angular velocity and load torque. As a result of the calculations carried out for 20 operating points of the studied IPMSM, the dependencies i_d0.opt(ω,T_L) are constructed in Figure 9a.

Given the smooth character of the curves obtained in Figure 9, the amount of research conducted was enough to create a look-up table (LUT) from which, through interpolation, it will be possible to determine the optimal value of i_d0.opt for any pair of ω and T_L values in the range from zero to nominal. Using such an LUT-ME, as well as the created program, the values of the studied IPMSM’s operation efficiency were calculated in the entire range of changes in angular velocity and load torque, in which the amplitude of the linear armature voltage V_l-l.m does not exceed the specified nominal value V_DC = 150 V. The value of this voltage depending on the angular velocity and load torque of the studied IPMSM are shown in Figure 9b. As can be seen from this figure, the motor can operate at an angular velocity greater than the nominal value of 100 s⁻¹, but with a small load, and the lower the load, the higher the angular velocity it can achieve. So, with a load torque of T_L = 25 Nm, the motor can develop an angular velocity up to 125 s⁻¹ with maximum efficiency.

So, as can be seen from the obtained results, the developed method of energy efficiency analysis allows one to evaluate the capabilities of the IPMSM with respect to the operating region with a constant torque, but with an angular velocity greater than the nominal one, as shown in [36].

Figure 10 presents the efficiency map obtained for the studied IPMSM taking into account iron loss, which operates at the points of ME at optimal values of the i_d0.opt current component. A comparison of the obtained efficiency map for the IPMSM with the efficiency map for SPMSM (Figure 5) shows the advantages of the first over the second in higher efficiency especially with loads close to the nominal one.

Based on the given results of the IPMSM operation with different loads at the nominal angular velocity, it is possible to easily calculate, based on (25) and (26), the power loss components caused by the imbalance in the power conversion process ΔP_R and incomplete coupling of the PC ΔP_q. The calculation procedure is as follows. From Figure 8b, the values of the current component i_d0.opt for each load are obtained. For each i_d0.opt from Figure 8a,b, the values of q and Zχ are defined, which are used to find the power loss components in p. u. using (25) and (26). For each of the loads, the value of the input force X_i = v_s(i_d0.opt) and the kinetic coefficient L_ii(i_d0.opt) are determined from the program. Based on the obtained data, the absolute values of the components of power losses were found using the same expressions (25) and (26). Their dependence on the IPMSM load is shown in Figure 11.

As can be seen from the obtained results, the losses from incomplete coupling ΔPq, which are caused primarily by losses in steel, at a constant motor angular velocity, depend very little on the drive load, while the losses from the imbalance of the process ΔP_R, which are the heating losses in the armature windings, rapidly increase with an increase in the load torque of the drive. At the rated values of the IPMSM angular velocity and the load torque, the total losses ΔP_Σ amount of 1973 W, which is the nominal value of the losses in the studied machine.

3.5. Efficiency Analysis of the IPMSM Operation in the Constant-Power Region

The numerical simulation algorithm of the IPMSM taking into account iron loss operation developed with the use of the LNTD is quite suitable for modeling their operation in the region of angular velocity regulation with a constant power. The difference between the IPMSM operation in this region compared to the one of constant torque is the presence of a limitation on the armature voltage that is caused by the value of the DC voltage V_DC of the voltage invertor power supply. In addition, there is also in place here a limitation of the armature current in the long-term operating mode that is caused by the heating of the machine. These restrictions are described by the following inequalities [5]:

$$\sqrt{v_d^2+v_q^2}\le V_{\mathrm{s}.\max }$$

(61)

$$\sqrt{i_d^2+i_q^2}\le I_{\mathrm{s}.\max }$$

(62)

where v_s.max is the maximum value of the armature voltage amplitude in the two-phase reference frame and i_s.max is the maximum value of the armature current amplitude in the two-phase reference frame in the long-term motor operating mode.

The v_s.max value is related to V_DC based on the following relationship:

$$V_{\mathrm{DC}}=\sqrt{3} V_{\mathrm{s}.\max }$$

(63)

Simulation of the studied IPMSM operation in the constant-power region was conducted with the use of the developed calculation program in the MathCad 15 environment similarly to the constant torque region.

At the beginning, for the nominal operating mode of the machine—the nominal values of the angular velocity and the load torque—the V_s.max and I_s.max values were determined by (61) and (62): V_s.max = 87 V and I_s.max = 93.5 A. According to (63), the required value for this is V_DC = 150 V.

Further studies were carried out for the relative angular velocity of the IPMSM equal to ω^* = 1.5. The research methodology consisted of reducing the motor load torque to such a value that the voltage v_s decreased to v_s.max. As a result, the value of the load torque T_L = 38 Nm, which was permissible from the point of view of the current limitation, was obtained (Figure 12). At the same time, the optimal i_d0 value for ensuring the specified maximum torque was −90 A. Dependencies of the main parameters and characteristics for the PC, which models the studied IPMSM operation based on the value of such an armature current component i_d0 during machine operation in the constant power region with an angular velocity of 150 s⁻¹ for the maximum value of a load torque of 38 Nm are shown in Figure 13. As can be seen from the figure, at the operating point, the degree of coupling is q = −0.9923, and the energy efficiency of the IPMSM is η = 0.750. At the same time, for the optimal value i_d0 = −23 A, for this load, η = 0.828. However, for such a regime, the required value of v_s is 107 V, which is significantly more than the nominal value (Figure 12a).

Analyzing the dependencies in Figure 13, it can be concluded that the operation of the IPMSM in the constant-power control region is characterized by a significant shift in the operating point from the point of maximum energy efficiency. This is explained by the need to set such a value of the armature current component i_d0, which would provide the task of increased angular velocity while limiting the value of the armature voltage at the nominal level. When the angular velocity increases, the armature voltage remains unchanged, but the armature current increases (Figure 12b), which leads to a decrease in energy efficiency. The higher the given angular velocity, the larger the i_d0 value will be, and the lower the energy efficiency will be. That is, when working in the constant-power region, the i_d0 value is assigned a different task than energy optimization, as it was in the constant-torque control region. Hence, accordingly, there is a decrease in the energy efficiency of the machine. When the load torque decreases below the permissible value of 38 Nm for a given angular velocity, the energy efficiency decreases even more. Thus, at the same angular velocity of the machine of 150 s⁻¹, the decrease in T_L to 18 Nm leads to a decrease of i_d0 to −70 A, but η decreases already to 0.657.

According to the method described above, using the developed program, the values of the d-component of the armature current were obtained, which ensure the armature voltage v_s.max = 87 V for a number of operating points of the studied IPMSM during its operation at high angular velocities in the FW region. These points are shown in Figure 14, according to which, similar to the IPMSM operation in the region of angular velocity regulation with a constant torque (Figure 9a), the corresponding LUT FW is formed.

Using this LUT FW, as well as the created program, the efficiency values of the studied IPMSM were calculated, and a corresponding efficiency map of its operation in the FW region was obtained that is shown in Figure 15. From the side of high speeds and load torques of the machine, this efficiency map is limited by the curve that corresponds to the rated total power losses of the studied IPMSM for the steady state of its operation, which are 1973 W. Total power losses in copper and iron of the machine were calculated in the program using the expressions for the components of relative power losses (25) and (26):

$$\Delta P_{\mathsf{\Sigma }}=\Delta P_{\mathrm{R}}+\Delta P_{\mathrm{q}}=L_{\mathrm{ii}}{X}_{\mathrm{i}}^2\left[{\left(1+q·\left(Z\chi \right)\right)}^2+{\left(Z\chi \right)}^2·\left(1-q^2\right)\right].$$

(64)

In (61), all parameters L_ii, q and (Zχ) were calculated using the software as functions of the coordinates of the machine working point (ω,T_L), and the input force of the PC was $X_{\mathrm{i}}=\sqrt{3/2} V_{\mathrm{s}.\max }=\sqrt{3/2}·87=106.6$ V.

From the obtained efficiency map, it can be seen that the operation of the studied IPMSM, taking into account the iron loss, is characterized by a fairly high efficiency in the FW region only at relatively low angular velocities and at the maximum load torques for these velocities.

To analyze the energy regularities of IPMSM operation in the FW region of angular velocity regulation, for the last experiment, the components of power losses due to the imbalance process ΔP_R and the PC incomplete coupling ΔP_q on the drive load torque T_L were calculated according to the algorithm described in Section 3.4 (Figure 16). For a given angular velocity of 150 s⁻¹, the T_L varied from 5 Nm to the maximum permissible value of 38 Nm.

The nature of the components of power losses in the IPMSM during its operation in the FW region with constant power regulation obtained in Figure 16 is radically different from the nature of these losses in the region with constant torque regulation shown in Figure 11. Although the total value of power losses at the maximum load of the drive differs slightly, in the constant power region, the iron loss clearly prevails over the copper loss. As in the previous case, the iron loss slightly depends on the drive load, since the motor angular velocity is unchanged, but in the constant power region, this loss is approximately twice as large as that in the constant-torque region. Loss in copper is also higher in the constant-power region than in the constant-torque region for the same value of the load torque. This is explained by the significant value of the current component i_d0, at the level of (−65)–(−90) A, compared to this component in the range (−13)–(−43) A in the constant-torque region. As a result of the significant losses and the low value of the electromagnetic torque of the motor, despite its higher angular velocity, the efficiency of the IPMSM in the constant-power region decreases (from 0.365 to 0.745 in the computational experiment) compared to the constant-torque region (from 0.795 to 0.845).

4. An Example of Applying the Obtained Results to Develop an IPMSM Control System

The results of the justification of IPMSM control coordinates obtained in the previous section can be applied to the implementation of the electric drive system based on this machine. As an example, Figure 17 shows the functional scheme of the vector-controlled according to the FOC strategy electric drive system based on the IPMSM. The well-known FOC strategy is implemented in the current control block, where according to the angle of the rotor position, which is fixed using the encoder E, a direct transformation of the current coordinates takes place, from the three-phase abc reference frame to the two-phase orthogonal reference frame dq. In this block, the controllers of d and q components of the armature current are implemented, at the output of which, after decoupling the cross connections, the references of d and q components of the armature voltage are obtained. After the reverse transformation of the reference frame, the task on real abc voltages is sent to the PWM controller, which is most often implemented using the space vector modulation (SVM) method. The PWM controller directly controls the switches of the voltage invertor, which converts the DC voltage of the battery B into the controlled three-phase voltages of the IPMSM armature.

A feature of the proposed electric drive system is the online formation of the references of the armature current components $i_{d0}^{\ast }$ and $i_{q0}^{\ast }$ entering the input of the block current control in different modes of the IPMSM operation depending on its working coordinates—the angular velocity ω and the load moment T_L. The reference values of the current component $i_{d0}^{\ast }$ are obtained from the corresponding LUTs, which are obtained as a result of offline study of a specific IPMSM taking into account iron loss according to the method of modeling the machine using a linearized PC based on the LNTD proposed in this work. At the same time, the LUT MEC is used for the operation of the machine with MEC in the region of its angular velocity regulation with a constant torque, and the LUT FW is used for the operation of the machine with FW in the region of the angular velocity regulation with a constant power. An electronic switch S switches over these two LUTs if the amplitude of the armature line voltage $\sqrt{2} v_{\mathrm{a}}$ reaches to the value of the invertor supply voltage v_DC (battery voltage). The inputs of both LUTs receive the values of the IPMSM working coordinates—the real angular velocity measured by the encoder E, and the reference value of the motor electromagnetic torque T^*, which is formed at the output of the PI speed controller SC.

Since the level of the invertor supply DC voltage can change significantly, this will significantly affect the reference value of the d-component of the armature current in the FW region, which is formed at the output of the LUT FW. The studies carried out in the work showed that the change in i_d0, caused by the change in the DC supply voltage of the invertor, can be sufficiently accurately compensated by the current correction coefficient k_i.corr according to the expression

$$\Delta i_{d0}=k_{\mathrm{i}.\mathrm{corr}}\left(\omega \right)\left(V_{\mathrm{DC}}^{\ast }-v_{\mathrm{DC}}\right)$$

(65)

To confirm this, Figure 18 shows the families of dependences of the necessary corrective value of the d-component of the armature current Δi_d0 based on the value of the deviation of the invertor voltage from the set value V^*_DC = 150 V for three values of the angular velocities of the studied IPMSM, which are shown in Figure 14. As can be seen from Figure 18, dependencies of the type (65) have a close-to-linear character in the studied range of deviations of the invertor voltage based on ±7.5 V (±5%). The magnitude of the angular velocity has a slight influence on the value of the angular coefficient of the given dependences k_i.corr, while the value of the load torque practically does not affect it. The influence of the angular velocity on the value of k_i.corr is sufficient to take into account the linear dependence of k_i.corr on ω, as shown in (65) and Figure 18. For this specific study, the following dependence was obtained: k_i.corr(ω) = 3.77–0.01142 ω.

At the end of this work, it should be noted that the conducted research is one of the first in this new direction—the application of LNTD capabilities to the analysis of patterns of energy transformation in electrical engineering objects. As for this interesting object—the IPMSM—there are significant reserves regarding the improvement of the circular model of a vector-controlled machine in the directions of taking into account the features of the machine design, the configuration of its magnetic field, the phenomena of saturation of the magnet wire and other losses, in particular, in permanent magnets [37,38]. The obtained refined simulation results will be of interest both for the development of new simulation computer models [39] and for the development of new energy-efficient IPMSM control systems [40].

5. Conclusions

For many applications, the requirements for the energy efficiency of PMSMs in a wide range of their angular velocities and torques are increasing. Therefore, the substantiation of optimal, from the maximum efficiency point of view, set coordinates—d and q components of the armature current, which is used for the vector control of the PMSMs electromagnetic torque—is an urgent task.

The LNTD approach, in particular, the method of universal description of objects as linear PCs, is promising for the analysis of energy processes and improving the efficiency of various systems, in particular such electromechanical converters as a PMSM. This approach makes it possible to evaluate the energy quality of systems of different natures from one universal point of view without delving into the physical, chemical or other features of the processes.

The method of mathematical modeling of the IPMSM operation, taking into account iron loss as a universal PC operating at a given point, developed in this work based on the LNTD, makes it possible to evaluate the quality of energy conversion in the system and, if possible, to choose the optimal, for loss minimization, values of regulating coordinates for creating the machine electromagnetic torque. Conducting research for SPMSM during its operation in the region with constant torque control according to the vector criterion i_d = 0 made it possible to obtain analytical expressions for PC indicators, which showed that the maximum possible values of efficiency are achieved only for certain ratios between the values of angular velocity and electromagnetic torque. The algorithm and research program have been developed for the IPMSM, taking into account iron loss, which makes it possible to obtain the indicators of the universal PC for each working point of the machine depending on the values of the armature current component i_d. For the operation of the machine at the constant-torque region, the optimal from the maximum efficiency point of view values of the d and q components of the armature current were obtained. For the constant-power region, the value of the armature current component i_d ensures the machine’s operation on the voltage limitation curve, where optimal operation in terms of energy efficiency is ensured. The proposed method also makes it possible to easily determine the main components of power losses—from process imbalance, which is the copper loss, and from incomplete coupling between the PC input and output, which in this case the iron loss is. For the IPMSM operation in the ranges below and above the nominal angular velocity, the structure of these losses is different: in the constant-torque range, as a rule, loss of copper prevails and increases with an increasing load, and in the constant-power region, loss of steel dominates and increases with an increasing angular velocity of the machine. The developed methodology based on the LNTD makes is possible to obtain the energy efficiency maps for the IPMSM operation, taking into account iron loss, in the entire range of changes in the coordinates of the electric drive—angular velocity and load torque.

Based on that obtained using the proposed method of optimal control coordinates, which are presented in the created LUTs, a vector control system is developed using the FOC strategy for an IPMSM-based electric drive with speed control in a wide range, covering the regions of constant torque and constant power.

Further research on this topic will be aimed at increasing the accuracy of modeling all types of losses in the IPMSM and improving, based on this, the model energetic indicators of machine’s steady-state operation. This will make it possible to increase the effectiveness of the proposed method of substantiation of the optimal control coordinates of the IPMSM based on LNTD.