A novel Control Algorithm Expressions Set for not Negligible Resistive Parameters PM Brushless AC Motors
^{1 }Department of Electrical Engineering  University of Naples Federico II
Via Claudio 21, Naples, Italy
Emails: renato.rizzo@unina.it; andrea.delpizzo@unina.it; ivanspina@gmail.com
^{* }Corresponding author: Phone: +39 081 7683231
Abstract
This paper deals with Permanent Magnet Brushless Motors. In particular is proposed a new set of control algorithm expressions that is realized taking into account resistive parameters of the motor, differently from simplified models of this type of motors where these parameters are usually neglected. The control is set up and an analysis of the performance is reported in the paper, where the validation of the new expressions is done with reference to a motor prototype particularly compact because is foreseen for application on tram propulsion drives. The results are evidenced in the last part of the paper.
Keywords
Permanent magnets; Brushless motor.
Introduction
Permanent Magnet brushless motor drives (PMSM) are widely demanded in many and various application such as generation by renewable sources like wind [1,2], and especially in industrial and traction applications [36], thanks to their compactness and good dynamic features. As all motors utilized in electric drives or any other complex system, nowadays also smart energy distribution systems [79], it is necessary to feed them by power electronic converter and set up proper control of the whole drive according to the specific requested operation and performance [1013].
In order to improve their performance, reliability and availability, many control algorithms have been proposed in literature, starting from the value of the electrical parameters in the Park’s reference system.
These algorithms are mainly based on the hypothesis to neglect motor resistive parameters. This hypothesis is not acceptable in case of very compact motors, i.e. light traction motors, in these cases is preferred to have a lower efficiency and dissipate the loss energy adopting a more efficient liquid cooling system. Such simplified control algorithms can introduce unacceptable errors that can produce damages to the machine or reduce their lifetime [14].
The influence of resistive drops on machines voltage depends in particular on the load and speed. A quick method to evaluate the negligibility of resistances could be the evaluation of the influence of resistive drop on the rated voltage. Acceptable values are 12%, for higher values is better to redefine the operative limits and control trajectories.
In the paper is proposed a control algorithm which is realized taking into account the resistive parameters of the motor, the control is validated by a numerical analysis with reference to a motor prototype for tram propulsion application.
Preliminary Definition and Traditional PMSM Expressions
In order to see how the several PMSM mathematical expression get modified ones the resistive parameters are introduced, it is convenient to recall them. The rotor reference system mathematical model is reported in (1) together with the torque expression.
_{} 
(1) 
The traditional control of the motor is based on the model (1) and the reference current components i_{sd}^{*}, i_{sq}^{*} must ensure the satisfaction of the two main limits of operation (2) and (3) regarding respectively the maximum current (I_{s,L}) and the maximum voltage (V_{L})
_{} 
(2) 
_{} 
(3) 
in particular (3) is written from the first of (1) in steadystate operation, separating the d,q components and using the approximation of negligible resistive parameters. In Park’s reference system the (3) represents the equation of a circle centered in (Φ_{r}/L_{s},0) and with radius (V_{L}/pω_{r}L_{s}). In fig.1a the (2) is depicted together with (3) for three specific value of speed called ‘characteristic speeds’ of the motor.
The expression of the characteristic speeds ω_{r,B}, ω_{r,0} and ω_{r,L} is:
_{} 
(4) 
The maximum i_{q} current component is indicated with i_{q,max}_{ }and is:
_{} 
(5) 
with:
_{} 
(6) 
if is known the desired torque T_{e}^{*} at the fixed speed ω_{r}, the reference current component i_{sd}^{*}, i_{sq}^{*} are expressed by:
_{} 
(7) 
with:
_{} 
(8) 
_{} 
(9) 
A geometrical representation of i_{q0l}, i_{df} is reported in fig.1b.
Limit Expressions Modified by Resistive Parameters Introduction
Since the resistive parameters are not always negligible, they should be taken into account when defining the control algorithm trajectories. In particular the approximation used in the maximum voltage (3) has to be removed and a complete expression has to be considered:
_{} 
(10) 
in comparison with (3), (10) still represents the equation of a circle, but with different center and radius, the radius R_{VL} and the coordinates of the center c_{d}, c_{q} are:
_{} 
(11) 
_{} 
(12) 
The sign of inequality in (10) represents the inner area of the circle. If R_{s}=0, as it can be noticed, relations (1012) yield the expressions traditionally used to define the voltage boundary of a PMSM. Since the resistance is a physical parameter, it cannot be equal to zero, this leads to the following considerations:
1. At null speed, the maximum current which satisfies the voltage limit is not infinitive, but equal to V_{L}/R_{s} (the order of magnitude of this value is quite high, typical of short circuit current values);
2. At fixed speed, the value of current for not overcoming the voltage limit becomes equal to (11), which can be seen as the ratio between the voltage V_{L} and an impedance with R_{s} as real part and pw_{r}L_{s} as imaginary part;
3. The coordinates c_{d}, c_{q} are speed dependent.
From the (12) it is possible to write:
_{} 
(13) 
where pw_{r} can be calculated from the second of (12):
_{} 
(14) 
To obtain the final relation between the coordinates c_{d} and c_{q}, (14) is substituted into (13), this yields:
_{} 
(15) 
then it is possible to understand how the center of the circle (10) moves when speed increases. In fact, (15) is another equation of a circle centered in the point (−F_{r}/2L_{s}) and with radius F_{r}/2L. When speed is zero, c_{d} = c_{q} =0, and the center of the circle (10) is situated in the origin of axis; with the approximation (3) this center was always placed in (−F_{r}/2L_{s}, 0), independently from the speed. By increasing the speed, c_{d} and c_{q} become different from zero and both negative, if w_{r} is assumed positive (for positive w_{r} eq.15 describes a semicircle). Thus, with the increase of the speed the center of the circle moves on the semicircle (15), as depicted in fig.2.
In fig.2 is represented (15) with the semicircle in dashed line, the arrow (clockwise) is coherent with the positive increasing of w_{r}, the point (−F_{r}/2L_{s},0) has been reported for w_{r}→∞; only for w_{r}→∞ the center of the approximated expression (3) coincides with the one of (10). The circle in continuous line indicates the maximum current I_{s},_{L}.
If the motor parameters are known, for any value of w_{r} the coordinates c_{d}, c_{q} are univocally defined, together with the radius R_{VL}, this allows to individuate the working area on the d,q domain; an example is given in Fig.3.
The working area is colored in grey in fig.3 for a fixed value of the speed w_{r}^{*}. On the same figure it is also possible to see how the working area gets enlarged or contracted for respectively higher and lower value of speed.
With respect to the approximated expression (3), (10) yields a reduced working area on the first quadrant of d,q domain. This happens because the circle (10) has a shorter radius and its center is moved in the negative direction of qaxis.
Same considerations can be done also with reference to the PMSM mathematical model (1): the addition of the resistive voltage drop (R_{s}^{ }I_{s}) makes the voltage V_{s} reaching its own limit for lower values of current I_{s}, at fixed speed w_{r}, and/or for lower value of w_{r}, fixing the current I_{s}.
The modification of the working area in d,q axis reference is reflected on the T_{e }−w_{r} domain as well, changing the definition of characteristic speeds and the values of maximum d,q current components which satisfy the limit of the voltage. This influences also the control algorithm, modifying the reference trajectories.
New Control Reference Trajectories
All the characteristic speeds above defined, considering the new maximum voltage expression (10), have to be recalculated. Also their representation on d,q domain appears different than the one of fig.1, and it is proposed in fig.4.
The characteristic speed w_{r},_{B} has been calculated starting from the maximum voltage expression (3) and imposing i_{d}=0, i_{q}=I_{s}_{,L}. The base speed w_{r},_{B} has now to be redefined as:
_{} 
((16) 
Applying the same procedure, the limit speed changes in:
_{} 
(17) 
Differently from w_{r},_{B} and w_{r},_{L} the characteristic speed w_{r},_{0} does not change with the introduction of resistive parameters, because it is calculated imposing zero i_{d}, i_{q} current components in the maximum voltage expression; w_{r},_{0} remains defined by (4).
By adding the resistive parameters in the maximum voltage expression, also the maximum current changes, in particular the component i_{q}_{0l}, i_{qfl}, have to be recalculated, while their composition remain the one of (8) and (5) in the different ranges of speed. For sake of simplicity the following positions are adopted:
_{} 
(18) 
In fig.5 (10) and (2) are depicted in d,q domain for a fixed value or w_{r} >w_{r},_{B}, and i_{q}_{0l} is the ordinate of the point A.
By applying Pythagoras theorem, i_{q}_{0l} is calculated:
_{} 
(19) 
The same can be done for the quantity i_{qfl} (ordinate of point A in fig.6).
From fig.6 it is simple to verify that:
_{} 
(20) 
Which gives the i_{qfl} expression:
_{} 
(21) 
In flux weakening region, the daxis current component i_{df} expressed by (9) has now to be redefined. In fig.7 the reference current component i_{sq}^{*} is supposed to be known and the speed w_{r} assigned higher than w_{r},_{B}. The component i_{df} is necessary to assure that the voltage keep within the maximum value V_{L}.
The current i_{df} can be easily calculated by means of geometrical consideration and is:
_{} 
(22) 
Ones assigned the reference torque T_{e}^{*} at fixed speed w_{r}, the reference current components is formally defined, but the quantities w_{r},_{B}, w_{r},_{L}, i_{q}_{0l}, i_{qfl}, i_{df} have to be calculated respectively as in (16), (17), (19), (21) and (22); while w_{r},_{0} remains defined by (4).
Numerical Analysis
A numerical analysis is carried out with reference to a traction advanced prototype motor whose parameters are reported in table1.
Rated values 

Power 
P_{n} 
67.5 kW 
Electromagnetic torque 
T_{e,n} 
2014 Nm 
Speed 
ω_{r,n} 
33.51 rad/sec 
Current (rms) 
I_{s,n} 
120 A 
Phase voltage (rms) 
V_{f,n} 
286 V 
Limit values 

Electromagnetic torque 
T_{e,L} 
2910 Nm 
Current (maximum) 
I_{s,L} 
170 A 
Phase voltage (maximum) 
V_{L} 
286 V 


Stator phase resistance 
R_{s} 
332 mΩ 
Stator self inductance 
L_{s} 
5.40 mH 
Permanent magnets fluxlinkage 
Ф_{r} 
0.998 Wb 
This watercooled motor is intended to replace the actual induction motors on tram application and, in order to eliminate the speed reducer and lowering the floor, it has been designed with the specification of high compactness [15]. This makes the resistance a not negligible parameter and the resistive voltage at rated current turns to be about 14% of V_{f,n}.
With reference to the motor of tab.1, in fig.8a the working area on the T_{e } w_{r} domain is delimited by two different curves. The outer curve refers to i_{q,max} with i_{qfl} calculated as in (6) based on the approximation (3). The inner curve, instead, refers to the complete expression (21) of i_{qfl}.
Figure 8. T_{e}ω_{r} domain for approximated and complete expression:
a)when using ‘flux weakening strategy’ b) when using ‘maximum torque/current strategy’
The portion of the domain where the ‘maximum torque/current strategy’ is possible is represented in fig.8b; as before the outer curve refers to the approximated i_{q}_{0l} (9), while the inner curve refers to the complete expression (19) of i_{q}_{0l}. As it can be noticed, the resistive parameters contract the working area in order to ensure that the voltage keeps within its limit. If vice versa the control algorithm was not based on the complete expression of the limits, it would force the motor to work on the darker portion of fig.8 overcoming the limit voltage. On the other hand, the simplified expressions are easier to be implemented and require a lower computational burden from the control system. A different solution to ensure the voltage within proper limits is using the approximated expression but choosing a value for V_{L} lower than the rated voltage V_{R}. In fig.9 the working area – divided in the part of ‘maximum torque/current strategy’ and ‘flux weakening strategy’’ – is reported for two different set of equations: the red line refers to the complete expression and with V_{L} equal to V_{R}; the black line refers to the approximated expression with V_{L} nearly equal to 94% of V_{R}.
Figure 9. T_{e}ω_{r} domain for complete expression with the right value of V_{L} and for approximated expression with a lower value of V_{L}
The use of approximated expression with lower V_{L} drags the outer line of the calculated domain within the real limit, but produces a restriction of the ‘maximum torque/current strategy’ domain, as it can be seen in fig.9. This reduces the motor performances and increase the loss of power.
Conclusions
A novel control algorithm expressions set for not negligible resistive parameters PM Brushless AC motors has been proposed in the paper. This algorithm is based on a new set of control algorithm expressions that is realized taking into account resistive parameters of the motor, differently from simplified models of this type of motors where these parameters are usually neglected. The control has been presented and explained including a critical analysis of the performance, the validation of the new expressions is done with reference to a motor prototype particularly compact because is foreseen for application on tram propulsion drives. The results complete the paper evidencing the effectiveness of the novel algorithm proposed for light trains application.
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