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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; jyotigovindagrawal@gmail.com
Received 20 September 2015; Accepted 17 November 2015
Academic Editor: ShengKai Yu
Copyright Β© 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(πππ/ππ π) ratio whereas UPFC yields very lowπ
ππ/ππ πratio. 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 [1β4]. 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 π- and π-axes statorvoltages of a PMSM in rotor reference frame are [9]
Vπππ = (π π + πΏππ) ππ
ππ β πππΏπππ
ππ , (1)
Vπππ = (π π + πΏππ) ππ
ππ + ππ(πΏπππ
ππ + πaf) . (2)
The stator voltage phasor magnitude is given by
ππ = β(Vπ
ππ )2
+ (Vπππ )2
. (3)
The phase voltages in π-π-π frame are obtained from theabove π-π voltages by using the inverse Park transformationas defined in the following:
[[[[
[
Vππ
Vππ
Vππ
]]]]
]
=
[[[[[[[
[
cos ππ
sin ππ
1
cos(ππβ2π
3) sin(π
πβ2π
3) 1
cos(ππ+2π
3) sin(π
π+2π
3) 1
]]]]]]]
]
[[[[
[
Vπππ
Vπππ
V0
]]]]
]
. (4)
Similarly, the relationship between π-π-π and π-π-π currentsis obtained through the Park transformation as defined in thefollowing:
[[
[
ππ
ππ
ππ
ππ
π0
]]
]
=
[[[[[[
[
cos ππcos(π
πβ2π
3) cos(π
π+2π
3)
sin ππ
sin(ππβ2π
3) sin(π
π+2π
3)
1
2
1
2
1
2
]]]]]]
]
[[
[
πππ
πππ
πππ
]]
]
.
(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 π- and π-axes currents in the rotor frame of referenceare obtained as [10]
[
ππ
ππ = ππ
ππ
ππ = ππ
] = ππ [
cos πΏsin πΏ
] = [
0
ππ
] , (6)
where βππβ is the flux producing and βπ
πβ 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]:
ππ=3
2
π
2[πaf + (πΏπ β πΏπ) π
π
ππ ] ππ
ππ . (7)
By substituting the value of ππππ
and ππππ from (6), (7) can be
expressed as
ππ=3
2
π
2[πaf + (πΏπ β πΏπ) ππ cos πΏ] ππ sin πΏ. (8)
From (8), it can be seen that the air gap torque is the sum ofreluctance torque (π
ππ) and synchronous torque (π
ππ ). From
the loci (refer to Figure 1), it is observed that the peak of airgap torque (π
π) occurs at an angle between 90β and 180β and
reduces between 0β and 90β. Hence, the preferred angle is90β < πΏ < 180β [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 1800πΏ
β0.1
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 (πΏ).
(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 (πΏ) which is the angle between the rotorfield and current phasor is maintained at 90β. Hence, bymaintaining πΏ = 90β 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 ππππ = 0, from (6) and (7), the electromagnetic torque
becomes
ππ=3
2
π
2πaf ππ
ππ =3
2
π
2πaf ππ . (9)
From (1) and (2), the steady-stateπ- and π-axes stator voltagesare
Vπππ = βπππΏπππ ,
Vπππ = (π π + πΏππ) ππ + πππaf .
(10)
3.2. Optimum Torque per Ampere Current Control. Thisstrategy is one of the most widely used control strategiesfor PMSM [12β16]. Application of the optimum torque perampere (OTPA) control strategy ensures maximum torquefor a minimum possible value of current which in turnminimizes the ohmic losses [17β24].Themathematicalmodel
of this strategy is developed as follows. Consider the electro-magnetic torque equation of PMSM given in (8):
ππ=3
2
π
2[πaf + (πΏπ β πΏπ) ππ cos πΏ] ππ sin πΏ,
ππ=3
2
π
2[πaf ππ sin πΏ + (πΏπ β πΏπ) π
2
π cos πΏ sin πΏ] ,
ππ=3
2
π
2[πaf ππ sin πΏ +
1
2(πΏπβ πΏπ) π2
π sin 2πΏ] .
(11)
The normalized torque expression can be obtained as
πππ=ππ
ππ
=
(3/2) (π/2) [πaf ππ sin πΏ + (1/2) (πΏπ β πΏπ) π2
π sin 2πΏ]
(3/2) (π/2) πafπΌπ,
πππ=
ππ [πaf sin πΏ + (1/2) (πΏπ β πΏπ) ππ sin 2πΏ]
πafπΌπ.
(12)
Let πaf = πΌππΏπ, ππ π = ππ /πΌπ, πΏππ = πΏπ/πΏπ, and πΏππ = πΏπ/πΏπ.Rewrite (12) as follows:
πππ= ππ π[sin πΏ + 1
2(πΏππβ πΏππ) ππ πsin 2πΏ] . (13)
From (13), the torque per unit stator current is defined as
πππ
ππ π
= [sin πΏ + 1
2(πΏππβ πΏππ) ππ πsin 2πΏ] . (14)
The torque angle where the PMSM produces maximumtorque per unit stator current is obtained by differentiating(14) with respect to πΏ and equating it to zero; that is, thefollowing equation should be satisfied [8]:
π
ππΏ[sin πΏ + 1
2(πΏππβ πΏππ) ππ πsin 2πΏ] = 0. (15)
The solution of the above equation gives
{cos πΏ + 1
2(πΏππβ πΏππ) ππ π2 cos 2πΏ} = 0. (16)
Using the double-angle identities, cos(2πΏ) = 2cos2(πΏ) β 1 in(16) can be rewritten as
{cos πΏ
+1
2(πΏππβ πΏππ) ππ π[cos (2πΏ) = 2cos2 (πΏ) β 1]}
= 0.
(17)
Solving (17) for πΏ gives
πΏ = cosβ1{
{
{
β1
4 (πΏππβ πΏππ) ππ π
Β± β1
2+ [
1
4 (πΏππβ πΏππ) ππ π
]
2
}
}
}
.
(18)
4 Modelling and Simulation in Engineering
In (18), 90β < πΏ < 180β 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
πΏ = cosβ1{
{
{
β1
4 (πΏππβ πΏππ) ππ π
+ β1
2+ [
1
4 (πΏππβ πΏππ) ππ π
]
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 . = cosπ, (20)
where p.f . is the power factor and βπβ denotes the anglebetween voltage and current. In some applications, the maingoal is to have a unity power factor during the operation ofmotor [25β27]. 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 πΏ =Vπππ
Vπππ
=
ππ
ππ
ππ
ππ
. (21)
Substituting (1), (2), and (6) into (21) results in
tan πΏ =(π π + πΏππ) ππ
ππ + ππ(πΏπππ
ππ + πaf)
(π π + πΏππ) ππ
ππ β πππΏπππππ
,
tan πΏ =(π π + πΏππ) ππ sin πΏ + π
π(πΏπππ cos πΏ + πaf)
(π π + πΏππ) ππ cos πΏ β π
ππΏπππ sin πΏ
,
tan πΏ =π π ππ sin πΏ + π
ππΏπππ cos πΏ + π
ππaf
π π ππ cos πΏ β π
ππΏπππ sin πΏ
,
sin πΏcos πΏ
=1 + πΏππππ πcos πΏ + (π
π πππ π/πππ) sin πΏ
π π πππ πcos πΏ/π
ππβ πΏππππ πsin πΏ
.
(22)
Solving for πΏ,
πΏ = cosβ1{{
{{
{
β1 + β1 β 4πΏπππ2π π(πΏππβ πΏππ)
2ππ π(πΏππβ πΏππ)
}}
}}
}
. (23)
From (23), it is evident that πΏ is independent of rotor speed.Positive sign in (23) and (πΏ
ππ< πΏππ) 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:
ππ= β(π
π
ππ + ππππ )2
= β(πaf + πΏπππ
ππ )2
+ (πΏπππππ )2
. (24)
In (24), the magnitude of mutual flux linkage is kept constantand equal to πaf . Also, substituting (6) into (24) gives
πaf = β(πaf + πΏπππ cos πΏ)2
+ (πΏπππ sin πΏ)
2
, (25)
π2
af = (πaf + πΏπππ cos πΏ)2
+ (πΏπππ sin πΏ)
2
. (26)
Using the formulae, π2 + π2 = (π + π)2
β 2ππ in (26) can berewritten as
= [(πaf + πΏπππ cos πΏ) + (πΏπππ sin πΏ)]2
β 2 (πaf
+ πΏπππ cos πΏ) (πΏ
πππ sin πΏ) = [(πaf + πΏπππ cos πΏ)
2
+ 2 (πaf + πΏπππ cos πΏ) (πΏπππ sin πΏ) + (πΏπππ sin πΏ)2
]
β 2 (πaf + πΏπππ cos πΏ) (πΏπππ sin πΏ) .
(27)
Using trigonometric-Pythagorean identities, that is, cos2πΏ +sin2πΏ = 1, the above equation can be rewritten as
2πafπΏπππ cos πΏ + (πΏπππ cos πΏ)2
+ (πΏπππ sin πΏ)
2
= 0. (28)
In order to determine the magnitude of πΏ, two different casesarise depending upon the saliency ratio, that is, πΏ
π/πΏπ.
Case 1 (for surface mounted PMSM πΏπ/πΏπ= 1). Solving (28)
for πΏ yields
πΏ = cosβ1 {βπΏπππ
2πaf} . (29)
In normalized form, the torque angle πΏ is derived as
πΏ = cosβ1 {βππ πΏπ
2πΌππΏπ
} = cosβ1 {βππ ππΏππ
2} , (30)
where πaf = πΌππΏπ.
Modelling and Simulation in Engineering 5
Case 2 (for interior mounted PMSM πΏπ/πΏπ
= 1). Solving(28) for πΏ yields
πΏ = cosβ1{{{{
{{{{
{
1
πΏππππ π[1 β (πΏ
π/πΏπ)2
]
Β± β
{{
{{
{
1
πΏππ[1 β (πΏ
π/πΏπ)2
] ππ π
}}
}}
}
2
β1
[1 β (πΏπ/πΏπ)2
]
}}}}
}}}}
}
.
(31)
For CMFLC strategy, πΏ has to be greater than 90β. 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)
ππ=3
2
π
2[πaf + (πΏπ β πΏπ) π
π
ππ ] ππ
ππ
=3
2
π
2[πaf ππ
ππ + (πΏπβ πΏπ) ππ
ππ ππ
ππ ] .
(32)
The above equation can be written in the following form:
ππ=3
2
π
2[πaf ππ
ππ + πΏπππ
ππ ππ
ππ β πΏπππ
ππ ππ
ππ ] . (33)
Rearrange (33) as follows:
ππ=3
2
π
2[(πaf + πΏππ
π
ππ ) ππ
ππ β πΏπππ
ππ ππ
ππ ] . (34)
From (24) and (6), the above expression can be written in thefollowing form:
ππ=3
2
π
2[ππ
ππ ππ
ππ β ππ
ππ ππ
ππ ] =
3
2
π
2ππππ sin πππ , (35)
where ππππ = ππcos πππ
and ππππ = ππsin πππ .
Also, angle between the air gap flux phasor and current isπππ = πΏ β π
π.
The air gap flux of PMSM cannot be kept constant forall values of current. So the main concept of this strategyis to maintain π
ππ at 90β 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 (cosπ)deteriorates as the stator current rises. The normalized statorvoltage (π
π π) 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 (π
ππ) 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 90β[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
6 Modelling and Simulation in Engineering
OTPA curve
β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
0β1 β0.5β1.5β2irdsn , p.u.
ir qsn
, p.u
.
Figure 3: OTPA locus in ππππ and ππππ frame.
like a hyperbola in the rotor ππππ and ππππ frame. For plotting the
OTPA locus for different values of commanded torque, the π-axis current is calculated first. Then, from (7), it is observedthat π-axis current is the function of π-axis current fromwhich the π-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 (πΏ
ππβ πΏππ) should be positive. The magnitude of
πππis proportional to π
π π.Theπ
ππ/ππ πenvelope 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, π
ππincreases
with the increase in ππ πand attains to its peak value π
ππ(max)
at ππ π(max). Afterwards, if π
π πis increased further beyond
ππ π(max), π
ππ(max) decreases. Also, the magnitude of π
π πis
decreasing with increase in the value of ππ π. But from the plot
of πππ/ππ π, 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, π
π πis 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 cosπ = 1
control.
constant while, for IMPMSM, ππ π
increases with ππ π. 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 π
π π.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 π
π π.
The ratio of normalized torque per unit to normalized statorcurrent (π
ππ/ππ π) is decreasing but offers significant π
ππover a
greater current range when compared to the UPFC strategy.
Modelling and Simulation in Engineering 7
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(ππ π) [10]. The ratio π
ππ/ππ π
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 πππ. Furthermore, it can be observed from the plot of
8 Modelling and Simulation in Engineering
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 π
ππ; 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 (π
ππ/ππ π)
ratio among the five different control strategies whereas theUPF control yields a very low (π
ππ/ππ π) 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
Modelling and Simulation in Engineering 9
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
π π π= 0.1729 p.u.
πΏππ
= 0.4347 p.u.
πΏππ= 0.6986 p.u.
πΏπ= 0.0129H
ππ= 97.138V
πΌπ= 12A
ππ= 628.6 rad/s
π½ = 0.0012 kgβ m2
π΅ = 0.01Nβ mβ s/rad
π = 6
ππ= 5.5631Nβ m
πdc = 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
π΅: Damping constant, (N/rad/s)ππ
ππ , ππππ : π- and π-axes stator currents in rotor
reference frame, (A)πππ , πππ , πππ : Instantaneous stator phase currents, (A)
ππ : Stator current magnitude, (A)πΌπ: Base current, (A)
π½: Total moment of inertia, (kgm2)πΏπ, πΏπ: Stator π- and π-axes self-inductances, (H)
πΏππ, πΏππ: Normalized stator π- and π-axes
self-inductances, (H)πΏπ: Base inductance, (H)
π: Number of polesπ π : Stator resistance per phase, (Ξ©)
ππ: Electromagnetic torque, (Nβ m)
πππ: Normalized electromagnetic torque, (p.u.)
ππ: Load torque, (Nβ m)
ππ: Base torque, (Nβ m)
πΏ: Torque angleπaf : Armature flux linkages, (Vβ s)ππ: Mutual flux linkages, (Vβ s)
π: Differential operator, π/ππ‘ππ : Stator voltage phasor magnitude, (V)
Vππ , Vππ , Vππ : Input phase voltages, (V)
Vπππ , Vπππ : π- and π-axes stator voltages in rotor
reference frame, (V)ππ: Actual rotor position, (radians)
ππ: Angle between the mutual flux linkages
and the permanent magnet rotor fluxlinkage
ππ: Electrical rotor speed, (rad/s)
ππ: Base speed, (rad/s).
10 Modelling and Simulation in Engineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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