Small-Signal Modeling of Single-Phase PLLs Using Harmonic Signal-Flow Graphs Shahil Shah1 and Leila Parsa2 1Department of Electrical, Computer and Systems Engineering
Rensselaer Polytechnic Institute, Troy, NY 12180, USA Telephone: (518) 545-9024; Email:
[email protected] Abstract Small-signal modeling of converters interfacing with ac power systems is difficult as they involve time-periodic quantities in the steady-state and their linearized dynamics result in a linear time-periodic (LTP) system. This paper presents signal-flow graphs for LTP systems to complement the harmonic linearization method and simplify the modeling process. The proposed graphs visually describe the flow of small-signal perturbation through converters and frequency cross-coupling among different variables. Since complex PLL structures are most difficult to model in a converter control system, the paper demonstrates modeling using signal-flow graphs for two singlephase PLLs: SOGI-PLL and Park-PLL. The developed loop-gain models accurately predict the phase-lag introduced by orthogonal signal generator block in these PLLs, which limits the maximum bandwidth for which a PLL can be designed. Automated modeling using a signal-flow graph solver is also presented.
I. INTRODUCTION Increasing penetration of power electronics based renewable resources and the application of HVDC and FACTS devices in utility power systems has increased the importance of modeling of power converters. Modeling of converters interfacing with ac power systems is particularly difficult as they involve timeperiodic quantities in the steady-state; the linearized dynamics of such converters form a linear time-periodic system (LTP) instead of a linear time-invariant (LTI) system. In other words, when a converter is perturbed by injecting perturbation at an arbitrary frequency fp, multiple linear components may appear in the converter variables at frequencies fp±n·f1, where f1 is the fundamental frequency of the ac power system. The frequency-domain relationship between any two variables, for example between the terminal voltages and currents for the impedance modeling, can be described by an infinite-dimensional transfer matrix termed as harmonic transfer function (HTF) [1, 2]. The HTF relates the perturbation components of the two variables at fp±n·f1. Two fundamental methods exist for modeling such relationships: frequency-domain harmonic linearization method [3-5] and timedomain harmonic state-space (HSS) modeling method [6-8]. Modeling complexity increases exponentially in both of these methods for converters with rich circuit and control dynamics, such as modular multilevel converter (MMC) [9, 10]. Several approximations are usually applied to simplify the modeling process. For example, neglecting some control functions and dynamics of networks at the converter ports [13]. Another implicit approximation generally employed is to represent the
2Department of Electrical Engineering University of California, Santa Cruz, CA 95064, USA Email:
[email protected]
frequency-domain relationship between two variables by onedimensional transfer function, instead of the multi-dimensional HTF model required for the LTP systems [12]. This paper introduces signal-flow graphs for LTP systems to complement the harmonic linearization method and simplify the modeling process by visually describing the dynamics of advanced converters. Since complex PLL structures are most difficult to model in a converter control system, modeling using the proposed signal-flow graphs is presented in this paper for two PLL implementations. Rotating synchronous reference frame (SRF) based PLLs are among the most popular PLL structures. Modeling of single-phase SRF-PLLs is more complicated than three-phase SRF-PLLs as they require an additional signal, orthogonal to the input voltage, for transforming the input voltage to a dq reference frame. Different orthogonal signal generators (OSG) are presented in literature [14]. Among them, secondorder generalized integrator (SOGI) and back-to-back Park transformation are commonly used. Presence of multiple nonlinearities and time-periodic variables in these advanced PLL structures make their modeling difficult. Modeling has been achieved by linearizing the PLL dynamics in the time-domain [16, 17] while replacing the time-periodic dynamic variables by their steadystate values to simplify the modeling process. These approximations have resulted in models that are accurate only at very low frequencies, limiting the understanding of PLL instabilities experienced when they are designed for higher bandwidths. Additionally, the time-domain linearization and ensuing assumptions do not reveal the differences in the small-signal dynamics of the SOGI-PLL and Park-PLL [17]. This paper first reviews the harmonic linearization and harmonic state-space modeling methods, and introduces signalflow graphs for LTP systems. Next, it applies the signal-flow graphs for the modeling of single-phase SOGI-PLL and ParkPLL. Using the developed PLL models, the paper unravels interaction between the phase-locking and frequency-locking dynamics as the root cause of stability problems when the PLL bandwidth is increased. Automated modeling of PLL loop-gain using a signal-flow graph solver is also presented. The rest of the paper is organized as follows: Section II reviews the frequency-domain modeling methods and introduces signal-flow graphs for the modeling of LTP systems. Using the proposed signal-flow graphs, Section III develops loop-gain models for the single-phase SOGI-PLL and Park-PLL. Section IV concludes this paper.
II. FREQUENCY-DOMAIN MODELING OF CONVERTERS
1 – 2 cos 2f 1 t
A. Harmonic Linearization Method The harmonic linearization approach injects a singlesinusoidal perturbation in an input variable, say at fp, and analytically develops response at the perturbation frequency in an output variable for developing transfer function between the two variables [3]. Harmonic linearization is a versatile method as its application is not limited by the kind of nonlinearities in the converter dynamics. Additionally, the method is also amenable for the large-signal modeling for the evaluation of large-signal phenomena as sustained resonance [11]. However, the harmonic linearization method in its native form captures only the linear time-invariant (LTI) dynamics of a converter by modeling onedimensional transfer function based models [3-5] and ignoring linear responses at frequencies different from the input perturbation frequency. Nonetheless, the method can be easily extended by considering simultaneous perturbations at several frequencies to capture the frequency cross-coupling effects for developing multi-dimensional HTF based models [12, 13].
v(t)
C. Harmonic Signal-Flow Graphs It was noted before that the frequency-domain relationship between two variables in an LTP system must be represented by an infinite dimensional HTF. Instead of the harmonic linearization or harmonic state-space approach for obtaining such HTF model, an alternate modeling approach using infinite-dimensional signal-flow graphs is presented here to enable separate linearization of individual nonlinearities and preserve the control structure in the modeling process. Infinite dimensional graphs are termed “harmonic signal-flow graphs” as the frequencies of nodes in the graph differ by the harmonics of the fundamental frequency. Harmonic signal-flow graphs are introduced in the following by employing them for the modeling of Lossy Mathieu
x(t)
1 ---------------------s s + 2
k
a)
W s + j 1 V s + j 1
– 1
V s
G1 s
X s + j 1
G0 s
Xs
–k Ws
– W s – j 1 V s – j 1
B. Harmonic State-Space Method Harmonic state-space modeling achieves linearization in the time-domain by representing a small-signal disturbance in each state variable by a perturbation vector [18]. Elements of this vector are fourier components of the state variable at frequencies fp±n·f1 [2]. These elements can also be interpreted as smallsignal disturbance in the steady-state fourier coefficients of the state variable since the HSS modeling method is equivalent to the dynamics phasor modeling approach [15]. The HSS modeling has been applied for the modeling of single-phase voltage-source converters (VSC) [6, 7] and modular multilevel converters (MMC) [10]. HSS modeling is essentially a state-space approach and requires each state to be defined and dealt separately. Hence, individual control functions are not modeled independently from each other, as in the harmonic linearization method. Dealing with all the control functions simultaneously and the state-space representation with large number of states make the HSS modeling complicated. The final model, which is obtained in the form of state-space matrices, is also difficult to interpret and the effects of different control functions are not directly evident.
w(t)
b)
G –1 s
X s – j 1
1 where G n s = --------------------------------------------------------------- s + jn 1 s + jn 1 + 2
Fig. 1. Lossy Mathieu equation: a) block diagram and b) harmonic signalflow graph. Note: 1 = 2f1
Equation, a widely studied LTP system [18]: ·· x t + 2x· t + k 1 – 2 cos 2f 1 t x t = 0
(1)
Stability of the equation can be analyzed using the feedback loop in Fig. 1a). For obtaining the loop-gain, when a perturbation is injected in v at an arbitrary frequency fp, w will have response at frequencies fp and fp±f1. Frequencies of the components generated in x are the same as those in w. The perturbation components in x will generate additional frequency components in v due to the feedback with gain k; this will further generate additional frequency components in the other variables. The flow of the injected perturbation is described using a signal-flow graph in Fig. 1b). In this so-called harmonic signal-flow graph: 1) nodes representing different frequency components of a particular variable are kept on the same vertical line; 2) nodes representing different variables at a particular frequency are kept on the same horizontal line; and 3) vertical separation between any two nodes is indicative of the frequency-shift between them. Gain of the horizontal branches between the nodes of v and w is unity. Moreover, gain of the transverse branches between the nodes separated by the fundamental frequency is . This is followed from the block-diagram in Fig. 1a). Gains of branches between the nodes of other variables can be similarly obtained. It is to be noted that even when there is only one physical loop in Fig. 1b), the LTP nature results in multiple loops in Fig. 1a). Hence, the loop-gain is a transfer matrix, describing the
¦ X s + j2 1 X s + j 1 Xs X s – j 1
=
X s – j2 1
¦ G2 s
¦ – G 2 s
¦ 0
– G 1 s
G1 s
0
– G 0 s
0
0
0
0
0
¦
¦
¦
¦
vac
_
v a)
0
G0 s
– G 0 s
¦ V s + j2 1
0
V s + j 1
0
Vs V s – j 1
V s – j2 1
– G –1 s G –1 s – G – 1 s – G –2 s G –2 s ¦
v v
vd dq v q
1 HPLL(s)
v v
dq
PLL
PLL
SRF-PLL vd
dq
(2)
¦
¦
PLL
vac
OSG
– G 1 s
OSG
b)
¦ 0
_
k
¦ 0
1 HPLL(s)
vq
PLL
PLL
vd0 vq0
SRF-PLL
Fig. 2. SRF based single-phase PLLs: a) SOGI-PLL and b) Park-PLL.
frequency-domain relationship between v and x. It is obtained in (2) using the harmonic signal-flow graph in Fig. 1b). The same relationship is developed in [18] using less intuitive HSS method. III. SMALL-SIGNAL MODELING OF SINGLE-PHASE PLLS Fig. 2 shows implementation of SOGI-PLL and Park-PLL. Loop-gain modeling for the SRF-based PLLs, including those in Fig. 2, can be achieved by obtaining the linear response in vq when a sinusoidal perturbation is injected in PLL. Due to the LTP nature of PLL dynamics, frequencies of the linear components under perturbation can be represented in general as fp±n·f1. Moreover, by tracing a sinusoidal perturbation at an arbitrary frequency through a PLL, analytically or using simulations, it can be shown that the perturbation components in the ac and dc variables appear at frequencies with the alternate values of integer n. This essentially means that the nodes for the ac and dc variables in the harmonic signal-flow graph of a PLL will be located at alternate horizontal levels. A. SOGI-PLL For small-signal analysis, the Park’s transformation block in the SOGI-PLL in Fig. 2a) can be resolved into two rotation matrices, as shown in Fig. 3. The first matrix captures the trans-
v v
cos 1 sin 1 – sin 1 cos 1
vdv vqv
vd 1 – 1
vq
Fig. 3. Park’s transformation in SOGI- PLL.
formation due to the grid-voltage angle 1. Whereas, the second matrix captures the transformation due to the phase estimation error = PLL – 1 . In Fig. 3, vdv and vqv are the grid voltages in the dq-frame aligned with 1. Under small-signal perturbation, V q s = – V 1 PLL s + V qv s .
(3)
For a three-phase SRF-PLL, vqv does not depend on the PLL variables (PLL or PLL). Hence, the loop-gain of a three-phase SRF-PLL is simply V 1 s [19], where V1 is the small-signal gain from PLL to vq and the integrator represents gain from PLL to PLL. For an SOGI-PLL, however, the small-signal response in vqv depends on the perturbation in PLL because of the frequency feedback to the SOGI block. Transfer function from PLL to vqv can be obtained using harmonic signal-flow graph for the SOGIPLL, which is drawn in Fig. 4 based on the implementation in
TABLE I SOGI-PLL PARAMETERS FOR SIMULATION
V ac s + j3 1 V s + j3 1
Parameter PLL compensator, HPLL(s)
PLL s + j2 1
SOGI gain, k
Value 0.52+65.68/s 1.414
PLL s + j2 1
V qv s + j2 1 V s + j 1
V1 V1 V s + j 1 = j ------------- G s + j 1 – j ----------------------------------------k 1 2 1 PLL s
V ac s + j 1
V1 V1 V s – j 1 --------------------------- = – j ------------- G s – j 1 + j -------------k 1 2 1 PLL s
PLL s V qv s
PLL s
(5)
where
V ac s – j 1
2
k 1 s k 1 G s = ----------------------------------- , G s = -----------------------------------. 2 2 2 2 s + k 1 s + 1 s + k 1 s + 1
V s – j 1 PLL s – j2 1 V qv s – j2 1
PLL s – j2 1
V ac s – j3 1 V s – j3 1
V s + j 1
Fig. 4. Harmonic signal-flow graph of SOGI-PLL.
Fig. 2a). For obtaining gains of the branches from PLL(s) to V(s±j1), the corresponding time-domain variables including steady-state and small-signal perturbations can be represented as: PLL t = 1 + ˆ s cos 2f p t + s
Vˆ n cos 2 f p – f 1 t + n
Vˆ n cos 2 f p – f 1 t + n
(4)
“Hat” in (4) signifies small-signal components. Eq. (4) considers perturbations only corresponding to the solid lines in the signalflow graph in Fig. 4. Using the harmonic linearization method [4] and the SOGI block implementation from Fig. 2a), relationships between the small-signal perturbation in PLL and vcan be obtained as: V1 V s + j 1 ---------------------------- = j ------------- G s + j 1 k 1 PLL s V1 V s – j 1 ---------------------------= – j ------------- G s – j 1 k 1 PLL s
V s – j 1 V qv s = – --j- --j- 1 --- 1 --2 2 2 2 V s + j 1
(6)
V s – j 1 Using (5) and (6) in (3), we get the response in vq due to the perturbation in PLL as: V 1 G s + j 1 + G s – j 1 Vq s ------------------- = ------ ----------------------------------------------------------------s 2 PLL s
v t = V 1 cos 2f 1 t + Vˆ p cos 2 f p + f 1 t + p +
v t = V 1 sin 2f 1 t + Vˆ p cos 2 f p + f 1 t + p +
Harmonic signal-flow graph in Fig. 4 shows that perturbation in v at fp±f1 will produce response in vqv at fp and fp±2·f1. Response in vqv at fp due to the perturbed v in (4) can be obtained using the harmonic linearization method as:
(7)
Eq. (7) gives the SOGI-PLL loop-gain excluding the compensator HPLL(s). Fig. 4 compares the loop-gain response obtained using the developed model and point-by-point simulations. It also shows response of the model ignoring the SOGI block dynamics. This is equivalent to assuming G(s) to be unity in (7) and the resulting loop-gain (=V 1 s ) is the same as the loop-gain of a three-phase SRF-PLL. It is evident from Fig. 4 that the SOGI block introduces significant phase-lag, limiting the maximum bandwidth of an SOGI-PLL to couple of tens of Hertz. The model in (7) shows that the loop-gain depends on the SOGI block design in a sophisticated manner than as interpreted in [17] by a first-order transfer function. Table I shows parameters used for the simulation of SOGI-PLL. It is to be noted that the model in (7) considers only the solid branches of the harmonic signal-flow graph in Fig. 4. Model accuracy can be improved by considering additional branches. Gains of the additional branches can be obtained from (5) and (6) using symmetry properties of harmonic signal-flow graphs [12].
TABLE II PARK-PLL PARAMETERS FOR SIMULATION Magnitude (dB)
20
Parameter PLL compensator, HPLL(s)
0
0.52+65.68/s
Low-pass filter corner frequency, f
- 20
533 rad/s
- 40 - 60 1 Hz
10 Hz
100 Hz
1 kHz
10Hz
100 Hz
1 kHz
Fig. 7 compares the Park-PLL loop-gain response obtained using the model in (8) [long-dashed lines] and point-by-point simulations [circles]. Table II shows the parameters used for simulations. Fig. 7 also shows response of the model ignoring the OSG block dynamics [short-dashed lines]; this is equivalent to neglecting the term in the brackets in (8). Clearly, the developed model better captures the Park-PLL dynamics. However, the model in (8) exhibits significant errors near 100 Hz. The model accuracy can be improved by considering additional branches drawn using dashed lines in Fig. 6b). Solid lines in Fig. 7 show response of the updated model. It can be seen that the model accuracy has significantly improved. The updated model is also
- 80
Phase (DEG.)
Value
- 100 - 120 - 140 - 160 - 180 1Hz
Fig. 5. SOGI-PLL loop-gain excluding the compensator HPLL(s). Solid lines: model including the SOGI block dynamics from (7); dashed-lines: model ignoring the SOGI block dynamics; circles: loop-gain obtained using point-by-point simulations.
vac
cos 1 sin 1 – sin 1 cos 1
vdv vqv
vd 1 – 1
vq
B. Park-PLL Unlike SOGI-PLL, which uses frequency feedback to the OSG block, the Park-PLL in Fig. 2b) uses feedback of the phase angle PLL for locking the OSG block with the grid voltage. Hence, for obtaining the Park-PLL loop-gain, we need to first obtain the small-signal gain from PLL to vq. Same as SOGI-PLL, each of the Park’s transformations in Fig. 2b) are resolved into two rotation matrices in Fig. 6a). Harmonic signal-flow graph for the Park-PLL OSG block is plotted in Fig. 6b) by following a single-sinusoidal-input perturbation in PLL to other variables. First we will consider only the solid branches in the graph. This is equivalent to considering only the fp component in the dc variables and fp±f1 components in the ac variables when a perturbation is injected in PLL at fp. Table III lists gains of the solid branches, which are developed through direct observation or the harmonic linearization method. For the sake of brevity, gains of the branches with a zero gain are not listed. Subscripts of G’s in Table III indicate source and sink nodes of branches depending on the numbering in Fig. 6b). It is to be noted that f is the corner frequency of the first-order low-pass filters in the Park-PLL OSG block (ref. Fig. 2b). Transfer function from PLL(s) to Vq(s) is obtained by solving the signal-flow graph using Mason’s rule solver from [20]. Using this transfer function, loop-gain of the Park-PLL is obtained as: 1 + s f V1 Vq s ------------------- = ------ --------------------------------s 1 + s f 2 PLL s
(8)
v v a)
cos 1 – sin 1 sin 1 cos 1
vdv0 vqv0
vd0 1 – 1
vq0
V s + j3 1 V dq0 s + j2 1 V dq s + j2 1
V dqv s + j2 1
V dqv0 s + j2 1
V s + j 1 8 1
PLL s
2, 3
V dq s
4, 5
V dq0 s
10, 11
6, 7
V dqv s
V dqv0 s 9
V s – j 1 V dq0 s – j2 1 V dq s – j2 1
b)
V dqv0 s – j2 1
V dqv s – j2 1
V s + j3 1
Fig. 6. Back-to-back Park’s transformation based OSG in Park-PLL: a) block diagram and b) harmonic signal-flow graph.
TABLE III GAINS OF BRANCHES IN THE HARMONIC SIGNAL-FLOW GRAPH OF PARK-PLL OSG BLOCK
Vq s - = –V1 G 1 3 = ------------------ PLL s
V d0 s 1 = --------------------G 2 4 = --------------1 + s f Vd s
V q0 s 1 = --------------------G 3 5 = --------------1 + s f Vq s
V dv0 s - = 1 G 4 6 = ----------------V d0 s
V qv0 s - = 1 G 5 7 = ----------------V q0 s
V qv0 s - = V1 G 1 7 = ------------------ PLL s
V s + j 1 j = – --G 6,8 = ---------------------------2 V dv0 s
V s – j 1 j - = --G 6,9 = --------------------------2 V dv0 s
V s + j 1 1 = --G 7,8 = ---------------------------2 V qv0 s
V s – j 1 1 - = --G 7,9 = --------------------------2 V qv0 s
V dv s j = --G 8 10 = ---------------------------2 V s + j 1
V dv s j - = – --G 9 10 = --------------------------2 V s – j 1
V qv s 1 = --G 8 11 = ---------------------------2 V s + j 1
V qv s 1 - = --G 9 11 = --------------------------2 V s – j 1
Vd s = 1 G 10 2 = --------------V dv s
Vq s = 1 G 11 3 = --------------V qv s
IV. CONCLUSIONS
Magnitude (dB)
10 0 - 10 - 20 - 30
Phase (DEG.)
- 40 10 Hz - 60
100 Hz
1 kHz
100 Hz
1 kHz
- 80 - 100 - 120 - 140 - 160 10Hz
Fig. 7. Park-PLL loop-gain excluding the compensator HPLL(s). Solid lines: model using the harmonic signal-flow graph shown in Fig. 6b); long-dashedlines: model in (8), considering only the solid branches of the harmonic signalflow graph; and short-dashed lines: model ignoring the OSG block dynamics; circles: loop-gain obtained using point-by-point simulation.
obtained using the Mason’s rule solver from [20]. The input to the solver is a text file describing the signal-flow graph with branch gains. Gains of the additional branches are obtained from Table III using the symmetry properties of harmonic signal-flow graphs [12]. Phase-lag introduced by the OSG dynamics above few tens of hertz in the loop-gains of the SOGI-PLL (ref. Fig. 6) and ParkPLL (Fig. 7) limits the maximum bandwidth for which a PLL can be designed without losing stability.
With the increasing control and circuit complexity of advanced grid-connected converters, frequency-domain smallsignal modeling using the two fundamental methods harmonic linearization method and harmonic state-space method is becoming more and more mathematically intractable. The harmonic signal-flow graphs presented in this paper for LTP systems simplify the modeling process by visually describing linearization of advance converter dynamics. Since addition of any new converter dynamic in the modeling is equivalent to adding new nodes and branches without affecting the existing graph, the harmonic signal-flow graphs avoid increase in the modeling complexity and develop intuition on how different control functions affect the final model. The paper demonstrated modeling using the harmonic signal-flow graphs for single-phase SOGI-PLL and Park-PLL. The developed loop-gain models accurately predict the significant phase-lag introduced by the orthogonal signal generator blocks in these PLLs, which consequently limits the maximum bandwidth for which either of the PLLs can be designed. Finally, the automated modeling using a signal-flow graph solver can be used to quickly analyze effects of different control functions. REFERENCES [1] E. Mollerstedt, “Dynamic analysis of harmonics in electrical systems,” Ph.D. dissertation, Lund Inst. Technology, Sweden, 2000. [2] S. R. Hall and N. M. Wereley, “Generalized nyquist stability criterion for linear time periodic systems,” in Proc. 1990 Amer. Control Conf., pp. 1518-1525, San Diego, CA. [3] J. Sun and K. J. Karimi, “Small-signal input impedance modeling of line-frequency rectifiers,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 4, pp. 1489-1497, Oct. 2008. [4] M. Cespedes and J. Sun, “Impedance modeling and analysis of gridconnected voltage-source converters,” IEEE Trans. Power Electron., vol. 29, no. 3, pp. 1254-1261, Mar. 2014. [5] Z. Bing, K. J. Karimi, and J. Sun, “Input impedance modeling and analysis of line-commutated rectifiers,” IEEE Trans. Power Electron., vol. 24, no. 10, pp. 2338-2346, Oct. 2009. [6] V. Salis, A. Costabeber, S. M. Cox, and P. Zanchetta, “Stability
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