research article steady-state analysis and comparison of ...research article steady-state analysis...

Research ArticleSteady-State Analysis and Comparison of ControlStrategies for PMSM

Jyoti Agrawal1 and Sanjay Bodkhe2

1Department of Electrical Engineering, G. H. Raisoni College of Engineering, CRPF Gate No. 3, Hingna Road,Digdoh Hills, Nagpur, Maharashtra 440016, India2Department of Electrical Engineering, Shri Ramdeobaba College of Engineering & Management,601/K Choti Dhantoli, Nagpur, Maharashtra 440012, India

Correspondence should be addressed to Jyoti Agrawal; [email protected]

Received 20 September 2015; Accepted 17 November 2015

Academic Editor: ShengKai Yu

Copyright © 2015 J. Agrawal and S. Bodkhe. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

PermanentMagnet SynchronousMotor (PMSM) has been considered as the best choice for numerous applications. Tomake PMSMa high performance drive, effective control system is required. Vector control is accepted widely due to its decoupling effect but itis not the only performance requirement. Additional control methods such as constant torque angle control (CTAC), optimumtorque per ampere control (OTPAC), unity power factor control (UPFC), constant mutual flux linkages control (CMFLC), andangle control of air gap flux and current phasor (ACAGF) can also be implemented. This paper therefore presents some importantcontrol strategies for PMSM along withmerits and limitations which provide a wide variety of control choices inmany applications.The performance characteristics for each strategy under steady state are modelled and simulated in MATLAB environment. Basedon the simulation results, a conclusion is drawn that OTPAC is superior in normalized torque per unit normalized stator current(𝑇𝑒𝑛/𝑖𝑠𝑛) ratio whereas UPFC yields very low𝑇

𝑒𝑛/𝑖𝑠𝑛ratio. In addition, performances of these control strategies are compared, which

is a key to select optimum strategy depending on requirements. Based on the comparative study, it can be concluded that CMFLCis superior to CTAC, ACAGF, OTPAC, and UPFC. Hence, it can be a good control strategy to consider.

1. Introduction

Recently, PMSM drive has emerged as a top competitoramongst AC drives for industrial servo drives, hybrid electricvehicles, and other applications due to features like highspeed, low power waste, large starting torque, high powerfactor, and high efficiency [1–4]. Also control of PMSM iscomparatively simpler than that of induction motor andhigh performance of PMSM can be achieved by means ofvector control as it provides decoupled control of torqueand flux [5, 6]. But decoupled control of torque and flux isnot only the performance requirement for PMSM drive [7].Therefore, in this paper, different control strategies such asconstant torque angle control, optimum torque per amperecontrol, unity power factor control, constant mutual airgap flux linkages control, and angle control of air gap fluxand current phasor are considered in detail for the variable

speed motor drive. For the speeds lower than base speed,the control strategies for PMSM are constant torque anglecontrol, optimum torque per ampere control, unity powerfactor control, constant mutual air gap flux linkages control,andmaximum efficiency control, while, for the speeds higherthan base speed, control strategies are six-step voltage andconstant back emf [8].The comprehensive analysis of controlstrategies for the speeds lower than base speeds is made andcompared in this paper. With the help of phasor diagrams,this paper analyses the characteristics of both surface andinterior mounted permanent magnet motors. Each of thesecontrol strategies has its own merits and limitations. Forexample, the constant torque angle control forces the elec-tromagnetic torque to be proportional to the stator currentmagnitude but results in low power factor, optimum torqueper ampere current control strategy provides maximumelectromagnetic torque for a given stator current, a unity

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2015, Article ID 306787, 11 pageshttp://dx.doi.org/10.1155/2015/306787

2 Modelling and Simulation in Engineering

power factor control strategy optimizes the volt ampere (VA)requirement of the system, and a constant mutual air gapflux linkages control limits the flux linkage of the air gapequal to rotor permanent magnet flux linkage which helps toavoid the saturation of core. Similarly, a maximum efficiencycontrol reduces the net loss in the motor and is appropriatefor applications where saving the energy is important [8].A detail analysis and comparison of these control strate-gies have been made so as to choose the control strategythat optimizes the operation of a particular speed controlsystem.

This paper is organized in the following manner:Section 1 begins by providing a brief introduction aboutPMSMs and a study of different existing control strate-gies. In Section 2, the dynamic model and decoupled con-trol of PMSM are explained shortly. Section 3 presentsthe detailed derivation and implementation of five con-trol strategies for PMSM drive. In Section 4, simulationresults are presented to verify the unique feature and capa-bility of the control strategies introduced in the paperemphasizing their merits. The comparison of control strate-gies based on current, voltage, VA rating, and powerfactor requirement as a function of torque is describedin Section 5. Finally, the conclusions are summarized inSection 6.

2. Dynamic Model and DecoupledControl of PMSM

In general, the dynamic equations of 𝑑- and 𝑞-axes statorvoltages of a PMSM in rotor reference frame are [9]

V𝑟𝑑𝑠= (𝑅𝑠+ 𝐿𝑑𝑝) 𝑖𝑟

𝑑𝑠− 𝜔𝑟𝐿𝑞𝑖𝑟

𝑞𝑠, (1)

V𝑟𝑞𝑠= (𝑅𝑠+ 𝐿𝑞𝑝) 𝑖𝑟

𝑞𝑠+ 𝜔𝑟(𝐿𝑑𝑖𝑟

𝑑𝑠+ 𝜆af) . (2)

The stator voltage phasor magnitude is given by

𝑉𝑠= √(V𝑟

𝑑𝑠)2

+ (V𝑟𝑞𝑠)2

. (3)

The phase voltages in 𝑎-𝑏-𝑐 frame are obtained from theabove 𝑑-𝑞 voltages by using the inverse Park transformationas defined in the following:

[[[[

[

V𝑎𝑠

V𝑏𝑠

V𝑐𝑠

]]]]

]

=

[[[[[[[

[

cos 𝜃𝑟

sin 𝜃𝑟

1

cos(𝜃𝑟−2𝜋

3) sin(𝜃

𝑟−2𝜋

3) 1

cos(𝜃𝑟+2𝜋

3) sin(𝜃

𝑟+2𝜋

3) 1

]]]]]]]

]

[[[[

[

V𝑟𝑞𝑠

V𝑟𝑑𝑠

V0

]]]]

]

. (4)

Similarly, the relationship between 𝑑-𝑞-𝑜 and 𝑎-𝑏-𝑐 currentsis obtained through the Park transformation as defined in thefollowing:

[[

[

𝑖𝑟

𝑞𝑠

𝑖𝑟

𝑑𝑠

𝑖0

]]

]

=

[[[[[[

[

cos 𝜃𝑟cos(𝜃

𝑟−2𝜋

3) cos(𝜃

𝑟+2𝜋

3)

sin 𝜃𝑟

sin(𝜃𝑟−2𝜋

3) sin(𝜃

𝑟+2𝜋

3)

1

2

1

2

1

2

]]]]]]

]

[[

[

𝑖𝑎𝑠

𝑖𝑏𝑠

𝑖𝑐𝑠

]]

]

.

(5)

In order to achieve linear transformation in modeling,analysis, and simulations, the power input to the three-phasemachine has to be equal to the power input to the two-phasemachine.

The 𝑑- and 𝑞-axes currents in the rotor frame of referenceare obtained as [10]

[

𝑖𝑟

𝑑𝑠= 𝑖𝑓

𝑖𝑟

𝑞𝑠= 𝑖𝑇

] = 𝑖𝑠[

cos 𝛿sin 𝛿

] = [

0

𝑖𝑠

] , (6)

where “𝑖𝑓” is the flux producing and “𝑖

𝑇” is the torque

producing component.Electromagnetic torque is the most important variable as

it determines the rotor position and speed.The expression forthe electromagnetic torque developed by the machine can beobtained from the input power and other quantities as givenin the following [11]:

𝑇𝑒=3

2

𝑃

2[𝜆af + (𝐿𝑑 − 𝐿𝑞) 𝑖

𝑟

𝑑𝑠] 𝑖𝑟

𝑞𝑠. (7)

By substituting the value of 𝑖𝑟𝑑𝑠

and 𝑖𝑟𝑞𝑠from (6), (7) can be

expressed as

𝑇𝑒=3

2

𝑃

2[𝜆af + (𝐿𝑑 − 𝐿𝑞) 𝑖𝑠 cos 𝛿] 𝑖𝑠 sin 𝛿. (8)

From (8), it can be seen that the air gap torque is the sum ofreluctance torque (𝑇

𝑒𝑟) and synchronous torque (𝑇

𝑒𝑠). From

the loci (refer to Figure 1), it is observed that the peak of airgap torque (𝑇

𝑒) occurs at an angle between 90∘ and 180∘ and

reduces between 0∘ and 90∘. Hence, the preferred angle is90∘ < 𝛿 < 180∘ [10].

3. Control Strategies for PMSM

The most commonly used five different control strategiesapplicable to PM synchronous machines are discussed in thissection:

(1) Constant torque angle control (CTAC).(2) Optimum torque per ampere control (OTPAC).(3) Unity power factor control (UPFC).(4) Constant mutual air gap flux linkages control

(CMFLC).

Modelling and Simulation in Engineering 3

TeTer

Tes

20 40 60 80 100 120 140 160 1800𝛿

−0.1

00.10.20.30.40.50.60.70.8

Tes,T

er, a

ndTe

(p.u

.)

Figure 1: Synchronous, reluctance, and air gap torques versus torqueangle (𝛿).

(5) Angle control of air gap flux and current phasors(ACAGF).

Such control strategies are important as they provide a widevariation of control choices in many applications. To obtainbetter performance, these control strategies are analyzed andderived step by step for the steady-state operations onlywherethe state rate of change of current is zero.

3.1. Constant Torque Angle Control. In this control strategy,the torque angle (𝛿) which is the angle between the rotorfield and current phasor is maintained at 90∘. Hence, bymaintaining 𝛿 = 90∘ from (6), we can conclude that theflux producing component is equal to 0 and the torqueproducing component is equal to the supply current makingPMSM operate like a separately excited DC machine [7].Therefore, this strategy is also referred to as zero directaxis current (ZDAC) control. The main advantage of ZDACcontrol strategy is that it gives the simplest and easiest controlfor PMSM.

Hence, the following relevant equations hold for thisstrategy is derived in the following.

Since 𝑖𝑟𝑑𝑠= 0, from (6) and (7), the electromagnetic torque

becomes

𝑇𝑒=3

2

𝑃

2𝜆af 𝑖𝑟

𝑞𝑠=3

2

𝑃

2𝜆af 𝑖𝑠. (9)

From (1) and (2), the steady-state𝑑- and 𝑞-axes stator voltagesare

V𝑟𝑑𝑠= −𝜔𝑟𝐿𝑞𝑖𝑠,

V𝑟𝑞𝑠= (𝑅𝑠+ 𝐿𝑞𝑝) 𝑖𝑠+ 𝜔𝑟𝜆af .

(10)

3.2. Optimum Torque per Ampere Current Control. Thisstrategy is one of the most widely used control strategiesfor PMSM [12–16]. Application of the optimum torque perampere (OTPA) control strategy ensures maximum torquefor a minimum possible value of current which in turnminimizes the ohmic losses [17–24].Themathematicalmodel

of this strategy is developed as follows. Consider the electro-magnetic torque equation of PMSM given in (8):

𝑇𝑒=3

2

𝑃

2[𝜆af + (𝐿𝑑 − 𝐿𝑞) 𝑖𝑠 cos 𝛿] 𝑖𝑠 sin 𝛿,

𝑇𝑒=3

2

𝑃

2[𝜆af 𝑖𝑠 sin 𝛿 + (𝐿𝑑 − 𝐿𝑞) 𝑖

2

𝑠cos 𝛿 sin 𝛿] ,

𝑇𝑒=3

2

𝑃

2[𝜆af 𝑖𝑠 sin 𝛿 +

1

2(𝐿𝑑− 𝐿𝑞) 𝑖2

𝑠sin 2𝛿] .

(11)

The normalized torque expression can be obtained as

𝑇𝑒𝑛=𝑇𝑒

𝑇𝑏

=

(3/2) (𝑃/2) [𝜆af 𝑖𝑠 sin 𝛿 + (1/2) (𝐿𝑑 − 𝐿𝑞) 𝑖2

𝑠sin 2𝛿]

(3/2) (𝑃/2) 𝜆af𝐼𝑏,

𝑇𝑒𝑛=

𝑖𝑠[𝜆af sin 𝛿 + (1/2) (𝐿𝑑 − 𝐿𝑞) 𝑖𝑠 sin 2𝛿]

𝜆af𝐼𝑏.

(12)

Let 𝜆af = 𝐼𝑏𝐿𝑏, 𝑖𝑠𝑛 = 𝑖𝑠/𝐼𝑏, 𝐿𝑑𝑛 = 𝐿𝑑/𝐿𝑏, and 𝐿𝑞𝑛 = 𝐿𝑞/𝐿𝑏.Rewrite (12) as follows:

𝑇𝑒𝑛= 𝑖𝑠𝑛[sin 𝛿 + 1

2(𝐿𝑑𝑛− 𝐿𝑞𝑛) 𝑖𝑠𝑛sin 2𝛿] . (13)

From (13), the torque per unit stator current is defined as

𝑇𝑒𝑛

𝑖𝑠𝑛

= [sin 𝛿 + 1

2(𝐿𝑑𝑛− 𝐿𝑞𝑛) 𝑖𝑠𝑛sin 2𝛿] . (14)

The torque angle where the PMSM produces maximumtorque per unit stator current is obtained by differentiating(14) with respect to 𝛿 and equating it to zero; that is, thefollowing equation should be satisfied [8]:

𝑑

𝑑𝛿[sin 𝛿 + 1

2(𝐿𝑑𝑛− 𝐿𝑞𝑛) 𝑖𝑠𝑛sin 2𝛿] = 0. (15)

The solution of the above equation gives

{cos 𝛿 + 1

2(𝐿𝑑𝑛− 𝐿𝑞𝑛) 𝑖𝑠𝑛2 cos 2𝛿} = 0. (16)

Using the double-angle identities, cos(2𝛿) = 2cos2(𝛿) − 1 in(16) can be rewritten as

{cos 𝛿

+1

2(𝐿𝑑𝑛− 𝐿𝑞𝑛) 𝑖𝑠𝑛[cos (2𝛿) = 2cos2 (𝛿) − 1]}

= 0.

(17)

Solving (17) for 𝛿 gives

𝛿 = cos−1{

{

{

−1

4 (𝐿𝑑𝑛− 𝐿𝑞𝑛) 𝑖𝑠𝑛

± √1

2+ [

1


]

2

}

}

}

.

(18)


In (18), 90∘ < 𝛿 < 180∘ so as to minimize field in the air gap;hence, only positive sign is considered [7].

Finally, the expression for torque angle is given as

𝛿 = cos−1{

{

{

−1


+ √1

2+ [

1


]

2

}

}

}

.

(19)

3.3. Unity Power Factor Control. Power factor can be definedas the cosine of the phase angle between voltage and currentas given in the following:

p.f . = cos𝜙, (20)

where p.f . is the power factor and “𝜙” denotes the anglebetween voltage and current. In some applications, the maingoal is to have a unity power factor during the operation ofmotor [25–27]. Unity power factor control implies the voltampere (VA) requirement of the inverter can be reduced bymaintaining the power factor at unity [28]. The performanceequations in this strategy are derived and given below.

In UPF control strategy, the phase angle has to be zerowhich implies the following relationship:

tan 𝛿 =V𝑟𝑞𝑠

V𝑟𝑑𝑠

=

𝑖𝑟

𝑞𝑠

𝑖𝑟

𝑑𝑠

. (21)

Substituting (1), (2), and (6) into (21) results in

tan 𝛿 =(𝑅𝑠+ 𝐿𝑞𝑝) 𝑖𝑟

𝑞𝑠+ 𝜔𝑟(𝐿𝑑𝑖𝑟

𝑑𝑠+ 𝜆af)

(𝑅𝑠+ 𝐿𝑑𝑝) 𝑖𝑟

𝑑𝑠− 𝜔𝑟𝐿𝑞𝑖𝑟𝑞𝑠

,

tan 𝛿 =(𝑅𝑠+ 𝐿𝑞𝑝) 𝑖𝑠sin 𝛿 + 𝜔

𝑟(𝐿𝑑𝑖𝑠cos 𝛿 + 𝜆af)

(𝑅𝑠+ 𝐿𝑑𝑝) 𝑖𝑠cos 𝛿 − 𝜔

𝑟𝐿𝑞𝑖𝑠sin 𝛿

,

tan 𝛿 =𝑅𝑠𝑖𝑠sin 𝛿 + 𝜔

𝑟𝐿𝑑𝑖𝑠cos 𝛿 + 𝜔

𝑟𝜆af

𝑅𝑠𝑖𝑠cos 𝛿 − 𝜔

𝑟𝐿𝑞𝑖𝑠sin 𝛿

,

sin 𝛿cos 𝛿

=1 + 𝐿𝑑𝑛𝑖𝑠𝑛cos 𝛿 + (𝑅

𝑠𝑛𝑖𝑠𝑛/𝜔𝑟𝑛) sin 𝛿

𝑅𝑠𝑛𝑖𝑠𝑛cos 𝛿/𝜔

𝑟𝑛− 𝐿𝑞𝑛𝑖𝑠𝑛sin 𝛿

.

(22)

Solving for 𝛿,

𝛿 = cos−1{{

{{

{

−1 + √1 − 4𝐿𝑞𝑛𝑖2𝑠𝑛(𝐿𝑑𝑛− 𝐿𝑞𝑛)

2𝑖𝑠𝑛(𝐿𝑑𝑛− 𝐿𝑞𝑛)

}}

}}

}

. (23)

From (23), it is evident that 𝛿 is independent of rotor speed.Positive sign in (23) and (𝐿

𝑑𝑛< 𝐿𝑞𝑛) should be considered

so as to utilize maximum possible torque under UPF controlstrategy [29].

3.4. Constant Mutual Flux Linkages Control. In constantmutual flux linkage control (CMFLC), the mutual flux

linkages are maintained constant and usually set equal torotor flux linkages. The reason behind this is that machineis protected against magnetic saturation [30]. Limiting themutual flux linkages, the stator voltage requirement can bekept consonantly low. This is the main advantage of CMFLstrategy. In addition, for the speeds higher than base speed,this strategy provides flux weakening as compared to theother schemes that are limited for operation at speeds lowerthan the base speed [31]. In this case, themagnitude ofmutualflux linkage is expressed as follows:

𝜆𝑚= √(𝜆

𝑟

𝑑𝑠+ 𝜆𝑟𝑞𝑠)2

= √(𝜆af + 𝐿𝑑𝑖𝑟

𝑑𝑠)2

+ (𝐿𝑞𝑖𝑟𝑞𝑠)2

. (24)

In (24), the magnitude of mutual flux linkage is kept constantand equal to 𝜆af . Also, substituting (6) into (24) gives

𝜆af = √(𝜆af + 𝐿𝑑𝑖𝑠 cos 𝛿)2

+ (𝐿𝑞𝑖𝑠sin 𝛿)

2

, (25)

𝜆2

af = (𝜆af + 𝐿𝑑𝑖𝑠 cos 𝛿)2


2

. (26)

Using the formulae, 𝑎2 + 𝑏2 = (𝑎 + 𝑏)2

− 2𝑎𝑏 in (26) can berewritten as

= [(𝜆af + 𝐿𝑑𝑖𝑠 cos 𝛿) + (𝐿𝑞𝑖𝑠 sin 𝛿)]2

− 2 (𝜆af

+ 𝐿𝑑𝑖𝑠cos 𝛿) (𝐿

𝑞𝑖𝑠sin 𝛿) = [(𝜆af + 𝐿𝑑𝑖𝑠 cos 𝛿)

2

+ 2 (𝜆af + 𝐿𝑑𝑖𝑠 cos 𝛿) (𝐿𝑞𝑖𝑠 sin 𝛿) + (𝐿𝑞𝑖𝑠 sin 𝛿)2

]

− 2 (𝜆af + 𝐿𝑑𝑖𝑠 cos 𝛿) (𝐿𝑞𝑖𝑠 sin 𝛿) .

(27)

Using trigonometric-Pythagorean identities, that is, cos2𝛿 +sin2𝛿 = 1, the above equation can be rewritten as

2𝜆af𝐿𝑑𝑖𝑠 cos 𝛿 + (𝐿𝑑𝑖𝑠 cos 𝛿)2


2

= 0. (28)

In order to determine the magnitude of 𝛿, two different casesarise depending upon the saliency ratio, that is, 𝐿

𝑞/𝐿𝑑.

Case 1 (for surface mounted PMSM 𝐿𝑞/𝐿𝑑= 1). Solving (28)

for 𝛿 yields

𝛿 = cos−1 {−𝐿𝑑𝑖𝑠

2𝜆af} . (29)

In normalized form, the torque angle 𝛿 is derived as

𝛿 = cos−1 {−𝑖𝑠𝐿𝑑

2𝐼𝑏𝐿𝑏

} = cos−1 {−𝑖𝑠𝑛𝐿𝑑𝑛

2} , (30)

where 𝜆af = 𝐼𝑏𝐿𝑏.


Case 2 (for interior mounted PMSM 𝐿𝑞/𝐿𝑑

= 1). Solving(28) for 𝛿 yields

𝛿 = cos−1{{{{

{{{{

{

1

𝐿𝑑𝑛𝑖𝑠𝑛[1 − (𝐿

𝑞/𝐿𝑑)2

]

± √

{{

{{

{

1

𝐿𝑑𝑛[1 − (𝐿

𝑞/𝐿𝑑)2

] 𝑖𝑠𝑛

}}

}}

}

2

−1

[1 − (𝐿𝑞/𝐿𝑑)2

]

}}}}

}}}}

}

.

(31)

For CMFLC strategy, 𝛿 has to be greater than 90∘. TheCMFLC is preferred over UPF control strategy as it providessignificant torque [32].

3.5. Angle Control of Air Gap Flux and Current Phasors. Inthis strategy, the air gap torque expression may be derived asfollows.

Consider (7)

𝑇𝑒=3

2

𝑃

2[𝜆af + (𝐿𝑑 − 𝐿𝑞) 𝑖

𝑟

𝑑𝑠] 𝑖𝑟

𝑞𝑠

=3

2

𝑃

2[𝜆af 𝑖𝑟

𝑞𝑠+ (𝐿𝑑− 𝐿𝑞) 𝑖𝑟

𝑑𝑠𝑖𝑟

𝑞𝑠] .

(32)

The above equation can be written in the following form:

𝑇𝑒=3

2

𝑃

2[𝜆af 𝑖𝑟

𝑞𝑠+ 𝐿𝑑𝑖𝑟

𝑑𝑠𝑖𝑟

𝑞𝑠− 𝐿𝑞𝑖𝑟

𝑑𝑠𝑖𝑟

𝑞𝑠] . (33)

Rearrange (33) as follows:

𝑇𝑒=3

2

𝑃

2[(𝜆af + 𝐿𝑑𝑖

𝑟

𝑑𝑠) 𝑖𝑟

𝑞𝑠− 𝐿𝑞𝑖𝑟

𝑑𝑠𝑖𝑟

𝑞𝑠] . (34)

From (24) and (6), the above expression can be written in thefollowing form:

𝑇𝑒=3

2

𝑃

2[𝜆𝑟

𝑑𝑠𝑖𝑟

𝑞𝑠− 𝜆𝑟

𝑞𝑠𝑖𝑟

𝑑𝑠] =

3

2

𝑃

2𝜆𝑚𝑖𝑠sin 𝜃𝑚𝑠, (35)

where 𝜆𝑟𝑑𝑠= 𝜆𝑚cos 𝜃𝑚𝑠

and 𝜆𝑟𝑞𝑠= 𝜆𝑚sin 𝜃𝑚𝑠.

Also, angle between the air gap flux phasor and current is𝜃𝑚𝑠= 𝛿 − 𝜃

𝜆.

The air gap flux of PMSM cannot be kept constant forall values of current. So the main concept of this strategyis to maintain 𝜃

𝑚𝑠at 90∘ which is analogous to the control

of separately excited DC machine [33]. This is the mainadvantage of this strategy as it permits a simple controlwithout a position sensor. The drawback with this strategyis that it cannot be used in the applications where low/zerospeed is required as themagnitude of induced emf is very low[10].

4. Simulation Studies and Discussion

The performance characteristics of PMSM under differentcontrol strategies for rated speed (1 p.u.) are realized in

p.f.VA

MFLn

Ten/isnPn

Vsn

Ten

0

0.5

1

1.5

2

2.5

3

3.5

4

0.5 1 1.5 20isn , p.u.

Figure 2: Performance characteristics of PMSM for constant torqueangle control.

MATLAB environment. Simulation results for five controlstrategies under which PMSM is operating are presentedahead. The plotted variables are in normalized units (p.u.).The parameters and rating of PMSM used to plot the curvesin the simulation are given in the Appendix. Also, all thechosen quantities such as power factor, stator voltage requiredelectromagnetic torque, apparent power, mutual flux linkage,and input power are plotted on the same scale.

4.1. Constant Torque Angle Control. The performance char-acteristics for this control strategy are shown in Figure 2.From Figure 2, it is observed that the power factor (cos𝜙)deteriorates as the stator current rises. The normalized statorvoltage (𝑉

𝑠𝑛) required to drive the motor in this control

strategy is presented in the following figure. Under thiscontrol strategy, the PMSM is able to produce a torqueup to 2 p.u. The torque versus stator current curve showsthat the electromagnetic torque (𝑇

𝑒𝑛) is directly proportional

to the magnitude of stator current which is analogous toDC motor. Also, from the normalized mutual flux linkage(MFLn) characteristics, it is seen that it cannot reduce below1 p.u. but can vary from 1 p.u. to a point greater than 1 p.u.This is only possible till torque angle is kept constant at 90∘[7]. Due to this, it is limited to the applications which do notrequire flux weakening operation. In addition, the apparentpower (VA) is also plotted so as to evaluate the VA ratingrequirement of the inverter.

4.2. OptimumTorque per Unit Current Control. Figure 3 plotsthe optimum torque per ampere (OTPA) locus which appears


OTPA curve

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0−1 −0.5−1.5−2irdsn , p.u.

ir qsn

, p.u

.

Figure 3: OTPA locus in 𝑖𝑟𝑑𝑠and 𝑖𝑟𝑞𝑠frame.

like a hyperbola in the rotor 𝑖𝑟𝑑𝑠and 𝑖𝑟𝑞𝑠frame. For plotting the

OTPA locus for different values of commanded torque, the 𝑞-axis current is calculated first. Then, from (7), it is observedthat 𝑑-axis current is the function of 𝑞-axis current fromwhich the 𝑑-axis current is determined. These minimumcurrent points for a given torque when connected togethermake a hyperbola which is referred to as OTPA trajectory. Indetermining the curves of Figure 4, it has been assumed thedifference (𝐿

𝑑𝑛− 𝐿𝑞𝑛) should be positive. The magnitude of

𝑇𝑒𝑛is proportional to 𝑖

𝑠𝑛.The𝑇

𝑒𝑛/𝑖𝑠𝑛envelope for this strategy

is slightly higher than unity. The OTPAC strategy results inreasonable p.f . varying from unity to roughly 0.65.

4.3. Unity Power Factor Control. Figure 5 shows the perfor-mance characteristics with the UPF control strategy.

Power versus current envelope shows the real power atany value of stator current. At the beginning, 𝑇

𝑒𝑛increases

with the increase in 𝑖𝑠𝑛and attains to its peak value 𝑇

𝑒𝑛(max)

at 𝑖𝑠𝑛(max). Afterwards, if 𝑖

𝑠𝑛is increased further beyond

𝑖𝑠𝑛(max), 𝑇

𝑒𝑛(max) decreases. Also, the magnitude of 𝑉

𝑠𝑛is

decreasing with increase in the value of 𝑖𝑠𝑛. But from the plot

of 𝑇𝑒𝑛/𝑖𝑠𝑛, it is seen that its value is less than 1, indicating that

UPF control is not optimum in terms of torque generation asthe maximum torque offered in this control is smaller whencompared to other control methods.This feature is needed inapplications demanding extended speed range.

4.4. Constant Mutual Flux Linkages Control. The perfor-mance characteristics of constantmutual flux linkages controlfor surface mounted (SM) and interior mounted (IM) PMSMare shown in Figures 6 and 7, respectively. On limiting themagnitude of mutual flux linkage to the rotor permanentmagnet flux, the torque producing capability of PMSM is alsolimited. For the SMPMSM, 𝑉

𝑠𝑛is maintained approximately

0 0.5 1 1.5 2

MFLn

delp.f.VA

Ten/isn

Pn

Vsn

Ten

0

0.5

1

1.5

2

2.5

3

3.5

4

isn , p.u.

Figure 4: Performance characteristics of PMSM for optimumtorque per unit current control.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

delp.f.VA Ten/isn

Pn

Vsn

Ten

isn(max)

Ten(max)

isn , p.u.

Figure 5: Performance characteristics of PMSM for cos𝜙 = 1

control.

constant while, for IMPMSM, 𝑉𝑠𝑛

increases with 𝑖𝑠𝑛. Also,

fromFigures 6 and 7, it is observed from the characteristics ofp.f . that it is near to unity up to 1 p.u. of 𝑖

𝑠𝑛.This indicates that

the CMFLC is closer to unity power factor when comparedwith CTACwhere the p.f . is near to unity up to 0.25 p.u. of 𝑖

𝑠𝑛.

The ratio of normalized torque per unit to normalized statorcurrent (𝑇

𝑒𝑛/𝑖𝑠𝑛) is decreasing but offers significant 𝑇

𝑒𝑛over a

greater current range when compared to the UPFC strategy.


0

1

2

3

4

5

6

delp.f.VA Ten/isn

Pn

Vsn

Ten

0.5 1 1.5 2 2.5 3 3.5 4 4.50isn , p.u.

Figure 6: Performance characteristics of constant mutual air gapflux linkages control for SMPMSM.

0

1

2

3

4

5

6

7

8

9

delp.f.VA Ten/isn

Pn

Vsn

Ten

0.5 1 1.5 2 2.5 3 3.5 4 4.50isn , p.u.

Figure 7: Performance characteristics of constant mutual air gapflux linkages control for IMPMSM.

4.5. Angle Control of Air Gap Flux and Current Phasors. Theperformance characteristics of angle control of air gap fluxand current phasors for PMSM are shown in Figure 8. Thesalient feature of this strategy is that VA requirement is low asMFLn is decreasing with the increase in magnitude of statorcurrent. Decrease in MFLn with the increase in magnitude

del MFLn

0

0.5

1

1.5

2

2.5

3

3.5

VA Ten/isn

VsnTen

0.5 1 1.5 2 2.50isn , p.u.

Figure 8: Performance characteristics of PMSM for angle control ofair gap flux and current phasor.

of stator current also limits the requirement of stator voltage(𝑉𝑠𝑛) [10]. The ratio 𝑇

𝑒𝑛/𝑖𝑠𝑛

is less than 1, indicating thatACAGF control is not optimal in terms of torque generation.All these features and characteristics closely resemble thecharacteristics of unity power factor control strategy.

5. Comparison of Control Strategies

For constant torque angle control, optimum torque perampere current control, unity power factor control, constantmutual air gap flux linkages control, and angle control of airgap flux and current phasors, the different quantities versustorque are plotted and realized in MATLAB environment tocompare the performances of these control strategies. Thefollowing simulation results presented ahead give compar-isons between these control strategies for the most importantcharacteristics, that is, current, voltage, VA rating, and powerfactor requirement versus normalized torque for rated speed(1 p.u.). This study will help to select the optimal controlstrategy depending upon the requirements.

5.1. Current Requirement as a Function of Torque. The per-formance characteristics of current requirement for differentcontrol strategies as a function of torque are shown inFigure 9. It should be noted that the OTPAC needs theminimumpossible value of current for a given value of torquewhen compared with CTAC, UPFC, CMFLC, and ACAGFas expected. However, for all these five strategies, there is nomajor difference for the requirement of current up to 1 p.u.of 𝑇𝑒𝑛. Furthermore, it can be observed from the plot of


0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.50

CTACOTPACUPFC

CMFLCACAGF

Ten , p.u.

i sn, p

.u.

Figure 9: Current requirement for five different control strategies asa function of normalized torque.

UPFC that, for each value of torque, there exist two operatingpoints. But the points with lower current requirement areconsidered rather than points with higher current due to thecurrent limitations [29], though theUPFneeds themaximumpossible value of current for a given value of torque higherthan 0.5 p.u. when compared with CTAC, OTPAC, CMFLC,and ACAGF.

5.2. Voltage Requirement as a Function of Torque. The per-formance characteristics of voltage requirement for differentcontrol strategies as a function of torque are shown inFigure 10.

It should be noted that the voltage requirement for CTACstrategy is the highest, whereas for both UPFC and ACAGFit is the lowest.

5.3. VA Rating Requirement as a Function of Torque. Theperformance characteristics of volt ampere requirement fordifferent control strategies as a function of torque are shownin Figure 11.

The comparison clearly reveals that the volt ampererequirement for CTAC strategy is the highest whereas forboth UPFC and ACAGF it is the lowest. This is because thecurrent and voltage requirement for CTAC are the highestand VA is the product of both. Also, the volt ampere require-ment for CMFLC strategy is the lowest when compared withCTAC and OTPAC. However, all strategies approximatelyrequire the same volt ampere up to 1 p.u. of 𝑇

𝑒𝑛; after that,

the requirements diverge significantly. Again, from the plot ofUPFC, it can be observed that for each value of torque there

CTACOTPACUPFC

CMFLCACAGF

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.50Ten , p.u.

Vsn

, p.u

.

Figure 10: Voltage requirement for five different control strategiesas a function of normalized torque.

exist two operating points for volt ampere requirement. Butthe points with lower VA requirement are considered due tothe current limitations.

5.4. Variation of Power Factor Requirement as a Functionof Torque. The performance characteristics of power factorrequirement for different control strategies as a function oftorque are shown in Figure 12. The UPFC strategy yieldsunity power factor whereas for CTAC strategy it falls rapidlyroughly around 0.68 to 0.64 when compared with OTPACand CMFLC as the torque increases. Power factor require-ment for CMFLC and OTPAC is next to UPFC [29].

6. Conclusion

In this paper, different control strategies for PMSM arederived and presented in detail. The study based on thesimulation results reveals that OTPAC is superior in (𝑇

𝑒𝑛/𝑖𝑠𝑛)

ratio among the five different control strategies whereas theUPF control yields a very low (𝑇

𝑒𝑛/𝑖𝑠𝑛) ratio. Also, all the

performance characteristics for each strategy shown aboveare compared. And the comparative analysis reveals that themain advantage with UPFC is the voltage requirement whichis comparatively low but the drawback lies in torque pro-duction in the PMSM which is about 1.2 p.u. On comparingUPFC with CMFLC, it should be noted that the voltagerequirement for CMFLC is next to UPFC but can producemuch higher electromagnetic torque. Finally, from the abovecomparative study, it can be concluded that the CMFLC hasbetter steady-state performance characteristics and it can be


CTACOTPACUPFC

CMFLCACAGF

0.5 1 1.5 2 2.500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5VA

, p.u

.

Ten , p.u.

Figure 11: VA requirement for five different control strategies as afunction of normalized torque.

a good control strategy to consider when compared to theOTPAC, CTAC, UPFC, and ACAGF.

Appendix

𝑅𝑠𝑛= 0.1729 p.u.

𝐿𝑑𝑛

= 0.4347 p.u.

𝐿𝑞𝑛= 0.6986 p.u.

𝐿𝑏= 0.0129H

𝑉𝑏= 97.138V

𝐼𝑏= 12A

𝜔𝑏= 628.6 rad/s

𝐽 = 0.0012 kg⋅m2

𝐵 = 0.01N⋅m⋅s/rad

𝑃 = 6

𝑇𝑏= 5.5631N⋅m

𝑉dc = 285V (bus voltage)

Power = 3.5 kW.

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

p.f.

CTACOTPAC

UPFCCMFLC

0.5 1 1.5 2 2.5 30Ten , p.u.

Figure 12: Power factor requirement for four different controlstrategies as a function of normalized torque.

Nomenclature

𝐵: Damping constant, (N/rad/s)𝑖𝑟

𝑑𝑠, 𝑖𝑟𝑞𝑠: 𝑑- and 𝑞-axes stator currents in rotor

reference frame, (A)𝑖𝑎𝑠, 𝑖𝑏𝑠, 𝑖𝑐𝑠: Instantaneous stator phase currents, (A)

𝑖𝑠: Stator current magnitude, (A)𝐼𝑏: Base current, (A)

𝐽: Total moment of inertia, (kgm2)𝐿𝑑, 𝐿𝑞: Stator 𝑑- and 𝑞-axes self-inductances, (H)

𝐿𝑑𝑛, 𝐿𝑞𝑛: Normalized stator 𝑑- and 𝑞-axes

self-inductances, (H)𝐿𝑏: Base inductance, (H)

𝑃: Number of poles𝑅𝑠: Stator resistance per phase, (Ω)

𝑇𝑒: Electromagnetic torque, (N⋅m)

𝑇𝑒𝑛: Normalized electromagnetic torque, (p.u.)

𝑇𝑙: Load torque, (N⋅m)

𝑇𝑏: Base torque, (N⋅m)

𝛿: Torque angle𝜆af : Armature flux linkages, (V⋅s)𝜆𝑚: Mutual flux linkages, (V⋅s)

𝑝: Differential operator, 𝑑/𝑑𝑡𝑉𝑠: Stator voltage phasor magnitude, (V)

V𝑎𝑠, V𝑏𝑠, V𝑐𝑠: Input phase voltages, (V)

V𝑟𝑑𝑠, V𝑟𝑞𝑠: 𝑑- and 𝑞-axes stator voltages in rotor

reference frame, (V)𝜃𝑟: Actual rotor position, (radians)

𝜃𝜆: Angle between the mutual flux linkages

and the permanent magnet rotor fluxlinkage

𝜔𝑟: Electrical rotor speed, (rad/s)

𝜔𝑏: Base speed, (rad/s).


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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