Lecture 6: Control of Three-Phase Inverters

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1 Yoash Levron The Anrew an Erna Viterbi Faculty of Electrical Engineering, Technion Israel Institute of Technology, Haifa 323, Israel Juri Belikov Department of Computer Systems, Tallinn University of Technology, Akaeemia tee 15a, Tallinn, Estonia Lecture 6: Control of Three-Phase Inverters This lecture outlines the esign principles of three-phase inverters, focusing on their control. We introuce the concepts of gri forming, gri feeing, an gri supporting inverters, an explain how they are use in specific applications, such as renewable energy systems an microgris. We also present a basic control scheme for Permanent Magnet Synchronous Motors (PMSM). Note that the objective of this lecture is only to review the main approaches use in typical esigns. Many practical etails are omitte in orer to clearly present the main ieas. Basic Definitions Inverters convert DC power to AC power, as shown in Fig. 1. I DC V DC 3-phase inverter a b c To gri Figure 1: A three-phase inverter. The voltage an on the DC sie are v c an i c, the AC voltages are v a, v b, v c, an the AC s are i a, i b, i c. Throughout this lecture we will assume that the AC voltages an s are balance. In steay-state, inverters are characterize by five quantities: The frequency ω. The (single-phase) active power P, such that P ac = 3P. The (single-phase) reactive power Q. The voltage amplitue E. The voltage angle δ. An ieal inverter operates with zero loss, an stores negligible energy. Thus, in steay-state, p c = v c i c = v a i a v b i b v c i c = p ac = 3P. (1) As a result, assuming that the power an DC voltage are constant, then the DC is also constant. Moes of Operation Three typical moes of operation of gri connecte inverters are gri forming, gri feeing, an gri supporting. The selecte moe of operation etermines the inverter ynamics, an its steay-state characteristics. Series of lectures on power system ynamics. The lectures are freely available on

2 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 2 Gri forming inverters Gri forming inverters are typically use in small isolate networks, an their main objective is to regulate the network voltage an frequency. To this en, gri forming inverters are controlle as voltage sources, with fixe voltage amplitue E an frequency ω. The active power P an reactive power Q are not irectly controlle, an are etermine by the interaction of the inverter with the network. Typical applications are stanby UPS systems, an islane microgris. Since the frequency ω is set, the inverter usually cannot operate in parallel to other gri forming inverters, unless an aitional mechanism is use to match the frequencies. In power flow stuies, a gri forming inverter may be viewe as an infinite bus, an is represente as a reference bus, with a constant voltage amplitue E an angle δ =. Gri forming inverters are also calle Voltage Source Inverters (VSI). Gri feeing inverters Gri feeing inverters are operate as power sources, an are mainly esigne to eliver power to an energize gri. The active power P an reactive power Q are irectly controlle, while the frequency ω an voltage amplitue E are not irectly controlle, an are etermine by the interaction of the inverter with the gri. Typical applications are renewable energy systems an small gri-connecte generators, which operate with specific active an reactive powers. As an example, in photovoltaic systems the active power is typically set by the source, an the reactive power is often set to zero. Gri feeing inverters are esigne to operate in parallel to other inverters an generators. They cannot operate in isolation, an require aitional units to set the voltage magnitue an frequency. In power flow stuies, a gri feeing inverter is represente as a P-Q bus, with constant active power P an constant reactive power Q. Gri feeing inverter are also calle Gri Following Inverters, or inverters with P-Q control. Gri supporting inverters Gri supporting inverters operate somewhat similarly to synchronous generators: they eliver power to the gri, while contributing to the stability an reliability of the system. The frequency an voltage magnitue are controlle by a roop mechanism, such that there is a linear relationship between ω an P, an between E an Q. This type of control promotes fair sharing of the active an reactive powers among generators, while regulating the frequency an voltage. Gri supporting inverters may be connecte in parallel to other generators, an can operate in isolation. In power flow stuies, such an inverter cannot be represente as a stanar bus. However, if the frequency ω is known, an the voltage roop mechanism can be ignore, then the inverter is represente as a P-V bus with constant active power P an voltage amplitue E, similarly to a synchronous machine. Typical applications are energy storage systems, online UPS systems, an istribute generators that operate in isolate or weak gris. Several properties of the ifferent operation moes are summarize in Table 1.

3 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 3 Table 1: Typical operation moes of gri-connecte inverters Gri Forming Gri Feeing Gri Supporting Also calle Voltage Source Inverters (VSI). The inverter operates as a voltage source. The voltage amplitue E an the frequency ω an are irectly controlle. The active power P an reactive power Q are etermine by the interaction of the inverter with the gri. The inverter usually cannot operate in parallel to other gri forming inverters, since the frequency ω is constant. Typical applications are stanby UPS systems an isolate small networks. In power flow stuies: represente as a reference bus (slack bus), with a constant voltage amplitue E an angle δ =. Also calle Gri Following Inverters, or inverters with P-Q control. The inverter operates as a power source. The active power P an reactive power Q are irectly controlle. The frequency ω an voltage amplitue E are etermine by the interaction of the inverter with the gri. Suitable for parallel operation. Cannot operate in isolation. The system must inclue other generators (e.g., other inverters, synchronous machines) that control the voltage amplitue an frequency. Typical applications are renewable energy systems, an istribute generation systems. In power flow stuies: represente as a P-Q bus, with constant active power P an reactive power Q. Delivers power to the gri, while promoting stability an reliability. Regulates the frequency an voltage, an balances the active an reactive power generation. Implements a linear relationship between P an ω, an between Q an E. Suitable for parallel operation. Suitable for isolate operation. Combines well with energy storage systems, an online UPS systems. In power flow stuies: in general, cannot be represente as a stanar bus. However, if the frequency ω is known, an the voltage roop mechanism can be ignore, then the inverter is represente as a P-V bus with constant active power an voltage amplitue E, similarly to a synchronous machine.

4 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 4 Gri Forming Inverters A basic control scheme for gri forming inverters is shown Fig. 2. V c u V 1 c q q a b c PWM 3-phase inverter u c u b u a t L ω C To gri Figure 2: A basic control scheme for gri forming inverters. The main inverter stage often consists of three Buck converters connecte in parallel to a DC source, as shown in Fig. 3. Assuming these converters are lossless an store no energy, the average voltages u a (t), u b (t), u c (t) are given by u a (t) = V c a (t) u b (t) = V c b (t) u c (t) = V c c (t). (2) Note that in this lecture the uty cycles are efine in the range [ 1, 1] such that V c u x (t) V c. The LC filer at the output removes the switching harmonics, an elivers sinusoial voltages an s to the gri. V c Figure 3: A basic three-phase inverter stage.

5 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 5 Gri forming inverters are controlle as voltage sources with a constant frequency. A possible control law is = V 1 c u q = (3) =, where u is the esire voltage amplitue. In aition, the reference angle for the q transformation is (t) = ω t, (4) where ω is the esire frequency. The resulting output voltage is u = u u q = u = (5) with a reference angle ω t. Therefore, the gri forming inverter implements a voltage source with constant frequency an amplitue. In practical esigns, the inverter s shoul be controlle to improve the ynamic performance an efficiency, an to avoi over s. In aition, often there is a nee to precisely control the output voltage. An improve esign with aitional an voltage loops is shown in Fig. 4. The objective of the system is to regulate the output voltages such that v = v, v q =, v =. This is achieve by two loops: an inner loop that controls the s, an an outer loop that controls the voltages. V c ω t q a b c PWM 3-phase inverter u c u b u a q control i q q i L sensors i q voltage control v q q v C voltage sensors v v q = v = To gri Figure 4: A gri forming inverter with aitional an voltage loops. The inner loop regulates the s such that i (t) = i (t) an i q(t) = i q(t), by ajusting the inverter uty cycles. This loop is typically moele as follows. The average voltages u (t),

6 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 6 u q (t), u (t) are u = V c u q = V c q u = V c, an the inuctor equations in the q reference frame are (6) t i = ω i q 1 L (u v ), t i q = ω i 1 L (u q v q ). The cross terms ω i q an ω i that appear in these equations complicate the controller esign, since i epens on i q, an i q epens on i. A typical solution is to eliminate the cross terms, using the following control law (t) = V 1 c (v (t) ω Li q (t) k p (i (t) t ) i (t)) k i (i (τ) i (τ))τ q (t) = V 1 c (v q (t) ω Li (t) k p (i q(t) t ) i q (t)) k i (i q(τ) i q (τ))τ (t) =, where i (t) an i q(t) are the target s, an k p, k i are parameters of a PI controller. This control law is illustrate in Fig. 5. Substitution of (6) an (8) into (7) yiels (7) (8) L t t i = k p (i (t) i (t)) k i (i (τ) i (τ))τ, L t t i q = k p (i q(t) i q (t)) k i (i q(τ) i q (τ))τ. (9) k p v V 1 c i k i t ω Li q i k p v q V 1 c q i q k i t ω Li i q Figure 5: The inner loop: a typical control scheme that eliminates the cross terms ω i q an ω i. Since the cross terms are eliminate, the s are governe by two single-input singleoutput systems, as shown in Fig. 6. This enables straightforwar tuning of the loop parameters.

7 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 7 k p i k i t i t L 1 k p i q k i t i q t L 1 Figure 6: By eliminating the cross terms, the loop is moele by two single-input singleoutput systems. The outer voltage loop operates base on similar principles. As an example, assuming a balance resistive loa, the capacitor equations may be approximate as t v = ω v q 1 ( i v ), C R t v q = ω v 1 ( i q v ) q, C R an a possible control law that eliminates the cross terms is (1) i = v (t) R i q = v q(t) R ω Ci q (t) k p,v (v (t) v (t)) k i,v ω Ci (t) k p,v (v q(t) v q (t)) k i,v t t (v (τ) v (τ))τ, (v q(τ) v q (τ))τ, (11) where v an v q are the target voltages, an k p,v, k i,v are the parameters of a PI controller. Gri Feeing Inverters Gri feeing inverters are operate as power sources, an are mainly esigne to eliver power to an energize gri. They are also calle Gri Following Inverters, or inverters with P-Q control. This moe of operation is use often, since the tightly regulate output power enables robust an economical esigns. Gri feeing inverters are also use with renewable power sources, in which the active power must by equal to the power prouce by the source. In many applications the reactive power is set to zero, an the inverter operates with a power factor of unity. In this case the output is minimal, but the inverter oes not provie the reactive power that may be neee to support the gri. A basic control scheme for gri feeing inverters is shown in Fig. 7.

8 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 8 V c q a b c PWM 3-phase inverter q ω control i q q i sensors L i q P Q calculations v q q v voltage sensors ω PLL To gri Figure 7: A basic control scheme for gri feeing inverters. The esign consists of an inner control loop, an an aitional calculations block that generates the target s. These are given by i = i q = 2 v 2 v2 q 2 v 2 v2 q (P v Q v q ), (P v q Q v ), (12) where P, Q are the target active an reactive powers (for a single phase). As a result, the powers at steay state are given by P = 1 2 (v i v q i q ) = P, Q = 1 2 (v qi v i q ) = Q. (13) This calculation is base on the assumption that i = i an i q = i q at steay state. The reference frame for the q transformation is efine by the gri voltages. Since these voltages are not irectly controlle, there is a nee to extract the reference angle an the frequency ω from the voltage measurements. This is usually accomplishe by a Phase Locke Loop (PLL), as illustrate in Fig. 8. The PLL operates by controlling the reference angle in close loop, such that the quarature axis component v q is zeroe. If the loop is stable, then must be matche to the angle of the AC voltages v a, v b, v c. A gri feeing inverter connecte to a photovoltaic (PV) source is shown in Fig. 9. The challenge here is that the active power is etermine by the renewable source, an is not known in avance. This esign is similar to the one in Fig. 7, expect that there is an aitional loop that regulates the active power, in orer to match it to the source. The feeback is provie

9 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 9 k p ω s ω v a v b v c q v v q v k i t t PLL Figure 8: A basic implementation of a Phase Locke Loop (PLL). by the bus-capacitor voltage. If v c (t) > Vc set then P increases to ischarge the bus capacitor, an if v c (t) < Vc set then P is ecrease to charge the bus capacitor. At steay state, P pv 3P an v c (t) V set c. PV V set c v c (t) q a b c PWM DC/DC & MPPT v c 3-phase inverter P pv q active power control ω control i q q i sensors L i q P Q calculations v q q v voltage sensors ω PLL To gri Figure 9: Gri feeing inverter, connecte to a photovoltaic (PV) source. Droop Control As an introuction to the topic of gri supporting inverters, this section reviews basic principles of the roop control metho. Droop controllers are implemente locally in each generator, an form a istribute control system that stabilizes the gri. The communication between the controllers is minimal, an

10 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 1 information is share by means of the gri frequency an voltage. Droop controllers are use in power systems in orer to Regulate the frequency an active power Regulate the voltage an reactive power Promote fair sharing of active power among generators Promote fair sharing of reactive power among generators Allow generators of ifferent sizes to operate in parallel In essence, this is achieve by implementing an inverse relationship between active power an frequency, an between reactive power an voltage, as explaine next. Frequency roop control Consier a lossless generator with a single-phase active power P an a frequency ω. At steay state, the roop control law is where P ref is the reference power D is the amping constant ω s is the nominal gri frequency (a constant) 3P = 3P ref 1 D (ω ω s), (14) A primary objective of the frequency roop controller is to stabilize the gri frequency, by regulating the active power. Consier first a synchronous generator, in which the mechanical power is p m = 3P ref 1 D (ω ω s). (15) As the frequency increases, the mechanical power ecreases, an vice versa. This behavior regulates the generate power, an matches it to the actual loa, as escribe in Fig System is balance mechanical power = electrical power 4 Due to roop control the mechanical power increases The loa power increases 2 The rotor spee ω graually ecreases Figure 1: Conceptual operation of the frequency roop mechanism. 3

11 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 11 As an example, assume that a single synchronous generator is feeing a loa with active power P L (t). The angular acceleration of the rotor is given by where K is the swing equation constant. With no roop control (D = ), t ω = K(p m(t) 3P L (t)), (16) t ω = 3K(P ref P L (t)), (17) an since in general P ref P L (t), the frequency ω is unstable. On the other han, when the roop control mechanism is active (finite D), the resulting ynamics equation is t ω = K(3P ref 1 D (ω ω s) 3P L (t)) = K D ω K(3P ref 3P L (t) ω s ). (18) D The term (K/D)ω provies negative feeback, an the frequency ω is stabilize. The relationship between the active power an frequency at steay state is illustrate in Fig. 11. P D increases D D P ref 1 ω/ω s Figure 11: Droop characteristics in steay state: active power as a function of frequency. The controller operates as follows: In case D : The frequency ω is constant such that ω ω s. The generator provies active power as neee to stabilize the frequency. The active power varies. The generator operates like a gri forming inverter, or an infinite bus. In case D : The active power is constant, P P ref. The frequency ω varies. The generator operates as a power source, like a gri feeing inverter. Mile values of D:

12 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 12 Combine the properties of these two extreme cases. The active power is regulate, but is not constant. The generator provies variable active power to ajust the frequency. The frequency ω is regulate, but is not constant. As we shall see, the generator operates like a gri supporting inverter. An aitional objective of the controller is to share the active power among the generators. Assume a system with N generators, such that in steay state, Also assume that the total loa in the system is P L, such that 3P i = 3P ref,i 1 D i (ω ω s ). (19) N P i = P L. (2) i=1 These equations may be written as 3D 1 1 P 1 3D 1 P ref, D N 1 P N =. 3D N P ref,n, (21) 1 1 ω ω s P L an the solution is ω ω s = 3 ( N i=1 1 D i ) 1 ( P L P i = P ref,i 1 3D i (ω ω s ). As an example, for two generators in steay state: ) N P ref,i, i=1 (22) ω = ω s 3D 1D 2 D 1 D 2 (P L P ref,1 P ref,2 ) P 1 = P ref,1 1 3D 1 (ω ω s ) P 2 = P ref,2 1 3D 2 (ω ω s ). (23) If the generators are synchronous machines, an the amping constant of each machine is inversely proportional the rotor moment of inertia, such that J i D i = const, then ( ) P i = P ref,i J N i P L P ref,i, (24) J tot where J tot = N i=1 J i. Here, eviations of the loa power from the reference power are share among the generators, accoring to their size, such that larger generators provie more power. For this reason, we say that the roop control metho promotes fair sharing of active power among the generators. i=1

13 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 13 Voltage roop control A primary objective of the voltage roop controller is to regulate the reactive power an voltage. To see this, consier a generator that is represente in steay state as a voltage source behin a series reactance, as escribe in Fig. 12. jx E δ P, Q = V Figure 12: The generator is moele in steay state as a voltage source behin a series reactance. The voltage roop control law is where E ref is the reference voltage Q ref is the reference reactive power k q is a constant E = E ref k q (Q Q ref ), (25) In a synchronous machine this control law is implemente in practice by ajusting the in the fiel wining. The generator reactive power Q may be compute from the circuit in Fig. 12, an is given by Q = V ( E cos(δ) V ), (26) X an if δ an E V, then Q = E X ( E V ). (27) This relationship between the reactive power an voltage is plotte in Fig. 13. Q slope E 2 X 1 V / E Figure 13: Reactive power as a function of voltage. Variations in V may result in high an unpreictable reactive power flow. If the voltage E is fixe, variations in V may result in high an unpreictable reactive power flow. This is potentially angerous, since high reactive power causes loss, an may lea to stability problems.

14 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 14 To solve this, the roop mechanism regulates the reactive power by ajusting E in inverse proportion to Q. Substitution of (25) in (27) yiels which may be written as Q E X ( E V ) E ref X (E ref k q (Q Q ref ) V ), (28) Q = This equation is illustrate in Fig. 14. E ref X E ref k q (E ref k q Q ref V ). (29) Q k q increases k q = slope: E 2 ref /X Q ref K q Q Q ref 1 XQ ref E 2 ref V / E ref Figure 14: Droop characteristics in steay state: reactive power as a function of voltage. In steay state, the controller operates as follows: In case k q = (no roop control): The voltage V varies in a small range. The generator provies reactive power as neee to maintain a certain voltage amplitue. As V ecreases, the reactive power increases to compensate. The reactive power varies. The generator operates as a voltage source, or like a gri forming inverter. In case k q : The reactive power is regulate, Q Q ref. The voltage V varies. The generator operates as a power source, or like a gri feeing inverter. Mile values of k q : Combine the properties of these two extreme cases. The reactive power is regulate, but is not constant. The voltage V is somewhat regulate. maintain a stable voltage. The generator provies reactive power to As we shall see, the generator operates like a gri supporting inverter.

15 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 15 jx E 1 δ 1 Q 1 V jx Q 2 P L, Q L E 2 δ 2 loa Figure 15: Example: reactive power sharing between two generators. An aitional objective of the voltage roop controller is to share the reactive power among the generators. As an example consier the system in Fig. 15, which is escribe by the following set of equations: Q 1 E 1 X ( E 1 V ) Q 2 E 2 X ( E 2 V ) E 1 = E ref,1 k q,1 (Q 1 Q ref,1 ) E 2 = E ref,2 k q,2 (Q 2 Q ref,2 ) Q L = Q 1 Q 2. Assume first that there is no roop control (k q,1 = k q,2 = ), an Q L =. The resulting reactive powers in this case are (3) Q 1 = Q 2 = E 1 E 2 E 1 E 2 X E 1 E 2. (31) Here, even though Q L =, there may be significant reactive power flow between the two generators. Now assume that the roop controller is active, an k q,1 = k q,2 = k q E ref,1 = E ref,2 = E ref Q ref,1 = Q ref,2 = Q ref. (32) The resulting reactive powers in this case are Q 1 = Q 2 = Q L 2. (33) The controller prevents circulation of reactive power, an shares the reactive power evenly between the two generators. Gri Supporting Inverters Gri supporting inverters eliver power to the gri, while contributing to the stability an reliability of the system, similarly to synchronous generators. This is achieve by means of

16 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 16 V c u V 1 c q q a b c PWM 3-phase inverter u c u b u a 2 E t ω L E ref ω s k q D Q ref 3P ref 3P Q power calc. i q v q q q i v sensors voltage sensors To gri Figure 16: Gri supporting inverter operating as a voltage source (conceptual control scheme). a roop control mechanism, that maintains an inverse relationship between ω an P, an between E an Q. A basic esign is shown in Fig. 16. Gri supporting inverters may be escribe as gri forming inverters with an aitional roop control mechanism: ω = ω s 3D(P ref P ), u = E = E ref k q (Q ref Q), 2 (34) where ω s is the nominal gri frequency, ω s = 2π5 ra/s or ω s = 2π6 ra/s. Such inverters may be viewe as a combination of gri forming an gri feeing inverters, an the exact balance between these two moes of operation epens on the roop parameters D an k q. For instance: If D =, then the frequency is constant ω = ω s, as in gri forming inverters. If D, then the active power is constant P = P ref, as in gri feeing inverters. If k q =, then the voltage amplitue is constant E = E ref, as in gri forming inverters. If k q, then the reactive power is constant Q = Q ref, as in gri feeing inverters. Mile values: the inverter supports the gri by regulating the active power, reactive power, frequency an voltage. As an example, consier a gri supporting inverter connecte to an infinite bus with a frequency ω g, where ω g ω s. We will use this example to emonstrate the typical ynamics of gri supporting inverters, an specifically to explain how the inverter synchronizes to

17 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 17 the gri. To simplify, it is assume that k q = (no voltage roop), an that the infinite bus voltage is E = E ref (RMS). Intuitively, if ω > ω g the voltage angle at the inverter output graually increases, which increases the active power P, an ecreases ω. On the other han, if ω < ω g the voltage angle graually ecreases, which ecreases the active power P, an increases ω. If this ynamic process is stable, then in steay state ω = ω g, an the inverter is synchronize to the gri. This ynamic process is moele as follows. The voltages at the output of the inverter stage are given by ũ 2 E ũ q = ũ, (35) where the reference angle of these q signals is (t). In aition, the voltages of the infinite bus are v 2 E v q = v, (36) with a reference angle ω g t. To represent the signals in the same reference frame, the voltages in (35) are transforme to the reference frame of the infinite bus, as explaine in the Synchronous Machine lecture. The result is u cos( ω g t) sin( ω g t) 2 E u q = sin( ω g t) cos( ω g t) u 1 (37) 2 E cos( ωg t) 2 E cos(δ) = 2 E sin( ωg t) = 2 E sin(δ), where the angle δ is efine as δ = ω g t. To simplify the ynamic equations, we represent the system using time-varying phasors, an estimate the active power P base on the DC power flow approximation. The two phasors are U(t) = 1 (u ju q ) = E (cos(δ) j sin(δ)) = E δ, 2 V = 1 (38) (v jv q ) = E, 2 an base on the DC power flow approximation, the active power is The resulting ynamic moel is P = E 2 δ. (39) ω g L δ = ω g t, P = E 2 ω g L δ, t = ω = ω s 3D (P ref P ), an several substitutions yiels the ifferential equation t δ = ω s ω g 3D (P ref E 2 ω g L δ (4) ). (41)

18 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 18 Due to the negative feeback, this ifferential equation is stable for any D >. The steay state is compute by solving δ =, which yiels t δ = ω ( gl P E 2 ref ω ) s ω g. (42) 3D In aition, accoring to (4), δ = ω g t t δ = ω ω g = ω = ω g. (43) So in steay state ω = ω g, an the inverter is synchronize to the gri. Note that the system is not stable if D =, since in this case the inverter operates as a gri forming inverter, an cannot be connecte in parallel to an infinite bus. Control of Permanent Magnet Synchronous Motors (PMSM) This section presents a typical control scheme for Permanent Magnet Synchronous Motors (PMSM). These motors are synchronous machines, in which permanent magnets are embee in the rotor to create a constant magnetic fiel. As in all synchronous machines, at steay state the rotor spee is proportional to the frequency of s an voltages in the stator. For this reason, such motors are especially useful in applications that require precise spee or position control. This section focuses on three-phase permanent magnet synchronous motors that have sinusoial EMF (as oppose to motors with trapezoial EMF). Permanent magnet synchronous motors (with sinusoial EMF) are moele like synchronous generators, with three moifications: The stator s are efine positive when flowing into the machine. The term L af i f is replace with λ, which is the amplitue of the flux inuce in the stator phases by the permanent magnets on the rotor. The electromagnetic torque accelerates the rotor, an the mechanical torque ecelerates the rotor. The angular acceleration is efine as t ω m = 1 J (T e T m ). Using analysis similar to the one presente in the Synchronous Machine lecture, the resulting moel is t = pω m t ω m = 1 J (T e T m ) t i = 1 v R i L q pω m i q L L L (44) t i q = 1 v q R i q L pω m i λpω m L q L q L q L q t i = 1 v R i. L L The inputs of the moel are v, v q, v an T m, an several aitional outputs are T e = 3 2 p (λi q (L L q ) i i q ), p m = T m ω m, p e = T e ω m. The reference angle for the q transformation is the electrical angle. The symbols appearing in these equations are (45)

19 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 19 is the rotor electrical angle, measure with respect to a fixe point on the stator; p = poles/2 is the number of pole pairs; L, L q, L are the irect axis, quarature axis, an zero sequence inuctances; R is the resistance of the stator winings; i, i q, i are the stator s (positive when flowing into the machine); v, v q, v are the stator voltages; ω m is the angular velocity of the rotor; λ is the amplitue of the flux inuce in the stator phases by the permanent magnets on the rotor; J is the rotor moment of inertia; T m, T e are the mechanical an electromagnetic torques; p m, p e are the mechanical an electromagnetic powers. Similarly to synchronous generators, the moel may be simplifie by assuming a roun rotor (no saliency effects), such that L = L q = L s. In this case, the motor may be escribe by the equivalent circuit in Fig. 17. e a v a i a e b v b i b v c i c L s R e c energy storage energy losses energy conversion Figure 17: Equivalent circuit for a permanent magnet synchronous motor with a roun rotor (assuming L = L q = L s ). The inuce EMF is given by e e q = λpω m, (46) e where [ e e q e ] T is the q transformation of [ ea e b e c ] T. In aition, for L = L q = L s, the electromagnetic torque is given by T e = 3 2 pλi q. (47) Therefore, the inuce EMF is proportional to the angular velocity, while the electric torque is proportional to the quarature axis. The heart of the energy conversion process is escribe by the inuce EMF source, which converts electrical energy to mechanical energy. From a mechanical perspective, the electromagnetic power is p e = T e ω m = 3 2 pλω mi q, (48)

20 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 2 an from an electrical perspective, p e = 3 2 (e i e q i q 2e i ) = 3 2 ( i λpω m i q 2 ) (49) Both expressions are ientical. = 3 2 pλω mi q. A basic control scheme is shown in Fig. 18. The esign consists of two loops: an inner loop, an an outer spee loop. The inner loop regulates the s such that i i an i q i q, by ajusting the inverter uty cycles. The objective of the outer loop is to regulate the spee, such that in steay state ω m ωm. This is implemente by controlling i q, base on the approximate relation between torque an T e = 3pλi 2 q. If the spee ω m is too low, then i q increases to prouce more torque, an to accelerate the rotor. If ω m is too high, then i q ecreases to prouce less torque, an to ecelerate the rotor. Two sensors are place on the motor to measure the spee ω m an the electrical angle. The latter is use as a reference angle for the q transformation. V c q q a b c PWM 3-phase inverter c b a controller i q q i sensors i i q motor spee controller ω m ω m spee sensor position sensor Figure 18: A basic control scheme for a permanent magnet synchronous motor (PMSM).

21 Lecture 6: Power System Dynamics: Control of Three-Phase Inverters 21 References [1] J. Schiffer, D. Zonetti, R. Ortega, A. M. Stanković, T. Sezi, an J. Raisch, A survey on moeling of microgris From funamental physics to phasors an voltage sources, Automatica, vol. 74, pp , Dec [2] D. Baimel, J. Belikov, J. M. Guerrero, an Y. Levron, Dynamic moeling of networks, microgris, an renewable sources in the q reference frame: A survey, IEEE Access, vol. 5, pp , 217. [3] J. Rocabert, A. Luna, F. Blaabjerg, an P. Roríguez, Control of Power Converters in AC Microgris, IEEE Transactions on Power Electronics, vol. 27, no. 11, pp [4] A. M. Bouzi, J. M. Guerrero, A. Cheriti, M. Bouhamia, P. Sicar, M. Benghanem, A survey on control of electric power istribute generation systems for microgri applications, Renewable an Sustainable Energy Reviews, vol. 44, pp , 215. [5] R. Teoorescu, M. Liserre, an P. Roríguez, Gri converters for photovoltaic an win power systems, John Wiley & Sons, 211.

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