energies Article
Frequency-Adaptive Modified Comb-Filter-Based Phase-Locked Loop for a Doubly-Fed Adjustable-Speed Pumped-Storage Hydropower Plant under Distorted Grid Conditions Wei Luo *, Jianguo Jiang and He Liu Key Laboratory of Control of Power Transmission and Conversion, Ministry of Education (Department of Electrical Engineering, Shanghai Jiao Tong University), Shanghai 200030, China;
[email protected] (J.J.);
[email protected] (H.L.) * Correspondence:
[email protected]; Tel.: +86-021-62932648 Academic Editor: Ånund Killingtveit Received: 28 January 2017; Accepted: 19 May 2017; Published: 23 May 2017
Abstract: The control system of a doubly-fed adjustable-speed pumped-storage hydropower plant needs phase-locked loops (PLLs) to obtain the phase angle of grid voltage. The main drawback of a comb-filter-based phase-locked loop (CF-PLL) is the slow dynamic response. This paper presents a modified comb-filter-based phase-locked loop (MCF-PLL) by improving the pole-zero pattern of the comb filter, and gives the parameters’ setting method of the controller, based on the discrete model of MCF-PLL. In order to improve the disturbance resistibility of MCF-PLL when the power grid’s frequency changes, this paper proposes a frequency-adaptive modified, comb-filter-based, phase-locked loop (FAMCF-PLL) and its digital implementation scheme. Experimental results show that FAMCF-PLL has good steady-state and dynamic performance under distorted grid conditions. Furthermore, FAMCF-PLL can determine the phase angle of the grid voltage, which is locked when it is applied to a doubly-fed adjustable-speed pumped-storage hydropower experimental platform. Keywords: adjustable speed pumped storage hydropower plant; distorted grid conditions; FAMCF-PLL; versa module eurocard (VME) bus
1. Introduction Pumped-storage hydropower plants (PSHP) are the most widely-used energy storage technology for large-scale energy storage levels, as it has the characteristics of being rapid, effective, economical, and reliable [1,2]. In recent years, along with the rapid growth of intermittent renewable energy sources, demands for power supply quality and reliability have become stricter. As large-capacity, reversible, pump-turbine manufacturing technology, and high-power converter technology, advance, adjustable-speed pumped-storage technology has become the optimal scheme for pumped-storage power plants. The adjustable-speed, pumped-storage technology has more benefits than constant-speed pumped storage power stations, such as faster grid support in the generating mode and power/frequency regulating ability in the pumping mode. Generally, the installed capacity of pumped-storage plants can reach hundreds of megawatts. Simultaneously, the AC excitation converter is about 20% of the motor capacity and can reach tens of megawatts. Currently, the thyristor based cycloconverter is the most widely-used converter topology of doubly-fed adjustable-speed pumped storage plants, due to the development of the power electronic devices, control strategy, etc. [3–5]. The cycloconverter-based adjustable speed pumped-storage hydropower plants consist of a three-phase breaker, doubly-fed induction machine (DFIM), stator and rotor transformers, and cycloconverter. The DFIM’s stator side is connected directly to the stator transformer’s secondary Energies 2017, 10, 737; doi:10.3390/en10060737
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and rotor transformers, and cycloconverter. The DFIM’s stator side is connected directly to the stator transformer’s secondary winding via the three-phase breaker. In addition, the DFIM’s rotor side is connectedvia tothe thethree-phase rotor transformer’s winding via the cycloconverter. Furthermore, the winding breaker. secondary In addition, the DFIM’s rotor side is connected to the rotor three-phase grid voltage connects stator winding in order tothe build an AC magnetic field, transformer’s secondary winding the via DFIM’s the cycloconverter. Furthermore, three-phase grid voltage synchronized with thestator grid winding frequency, is build constant under a balanced, system. connects the DFIM’s in which order to an AC magnetic field,three-phase synchronized with Therefore, the rotor’s mechanical rotating frequency can be adjusted by means Therefore, of controlling slip the grid frequency, which is constant under a balanced, three-phase system. the the rotor’s frequency. Meanwhile, the adjustable-speed, pumped-storage hydropower can operate in a mechanical rotating frequency can be adjusted by means of controlling the slip plant frequency. Meanwhile, certain range of variable speeds while maintaining synchronized of range grid voltage. The the adjustable-speed, pumped-storage hydropowerthe plant can operateoperation in a certain of variable configuration of the cycloconverter-based adjustable-speed pumped-storage is speeds while maintaining the synchronized operation of grid voltage. The hydropower configurationplant of the shown in Figure 1. cycloconverter-based adjustable-speed pumped-storage hydropower plant is shown in Figure 1.
Figure 1. Schematic diagram of the adjustable-speed, pumped-storage hydropower plant.
Phase-locked loops (PLLs) are the most widely-used synchronization technique in the field of Phase-locked loops (PLLs) are the most widely-used synchronization technique in the field of grid-connected power converters [6–12]. PLLs are typically composed of a phase detector (PD), loop grid-connected power converters [6–12]. PLLs are typically composed of a phase detector (PD), filter (LF), and a voltage-controlled oscillator (VCO) [13]. A major challenge of PLLs is how to loop filter (LF), and a voltage-controlled oscillator (VCO) [13]. A major challenge of PLLs is how to quickly and accurately estimate the phase angle and frequency when the grid voltage is distorted. quickly and accurately estimate the phase angle and frequency when the grid voltage is distorted. To overcome this challenge, combining filtering techniques into the structure of PLLs has been To overcome this challenge, combining filtering techniques into the structure of PLLs has been proposed in the literature [13–24]. Reference [14] used a double, or more synchronous reference proposed in the literature [13–24]. Reference [14] used a double, or more synchronous reference frame (DSRF/MSRF), to realize the decoupling between the fundamental negative sequence and the frame (DSRF/MSRF), to realize the decoupling between the fundamental negative sequence and positive sequence components, and then locking the phase angle by extracting the fundamental the positive sequence components, and then locking the phase angle by extracting the fundamental positive sequence component, which can realize a fast and dynamic response. However, this positive sequence component, which can realize a fast and dynamic response. However, this scheme scheme suffers from a high computational burden under harmonically-distorted grid conditions. suffers from a high computational burden under harmonically-distorted grid conditions. The delayed The delayed signal cancellation (DSC) based PLL can obtain good, steady-state performance, but it signal cancellation (DSC) based PLL can obtain good, steady-state performance, but it is difficult is difficult to balance the contradiction between dynamic performance and the amount of to balance the contradiction between dynamic performance and the amount of calculations [15]. calculations [15]. The second-order generalized integral (SOGI) based PLL has a strong, steady-state The second-order generalized integral (SOGI) based PLL has a strong, steady-state performance and a performance and a fast and dynamic response, but the digital implementation is complex [16,17]. fast and dynamic response, but the digital implementation is complex [16,17]. The first-order comb The first-order comb filter (CF) is a linear-phase, finite-impulse-response (FIR) filter that can realize filter (CF) is a linear-phase, finite-impulse-response (FIR) filter that can realize the ideal low-pass the ideal low-pass filter (LPF) characteristic, and is a widely-used technique in PLLs due to its simple filter (LPF) characteristic, and is a widely-used technique in PLLs due to its simple digital realization digital realization and low computational burden [18–25]; furthermore, the first-order CF is and low computational burden [18–25]; furthermore, the first-order CF is equivalent to a moving equivalent to a moving average filter (MAF). In Reference [22], small-signal modeling of an SRF-PLL average filter (MAF). In Reference [22], small-signal modeling of an SRF-PLL with an MAF-based with an MAF-based prefiltering was introduced so as to compensate for the phase and amplitude prefiltering was introduced so as to compensate for the phase and amplitude errors of MAF-PLL in the errors of MAF-PLL in the presence of frequency drifts. However, this scheme results in a presence of frequency drifts. However, this scheme results in a complicated design procedure and a complicated design procedure and a high digital implementation cost. As indicated in Reference high digital implementation cost. As indicated in Reference [24], the frequency-dependent attenuation [24], the frequency-dependent attenuation characteristic is a practical challenge of MAFs, and some characteristic is a practical challenge of MAFs, and some possible solutions to overcome this challenge possible solutions to overcome this challenge were mentioned; however, frequency adaptive digital were mentioned; however, frequency adaptive digital realization and hardware implementation were realization and hardware implementation were not considered in Reference [24]. not considered in Reference [24]. This paper presents a CF transfer function and schematic diagram for CF-PLL in Section 2; the This paper presents a CF transfer function and schematic diagram for CF-PLL in Section 2; major problem associated with CF-PLL is the slow dynamic response characteristic. Section 3 the major problem associated with CF-PLL is the slow dynamic response characteristic. Section 3
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proposes a modified comb filter (MCF) based PLL and its parameter setting method, based on the proposes a modified comb filter (MCF) based PLL and its parameter setting method, based on discrete model of MCF-PLL. Then, FAMCF-PLL and its digital implementation scheme are the discrete model of MCF-PLL. Then, FAMCF-PLL and its digital implementation scheme are proposed in order to enhance the disturbance resistibility when the power grid’s frequency proposed in order to enhance the disturbance resistibility when the power grid’s frequency changes. changes. The adjustable-speed, pumped-storage power plant’s excitation control system, based on The adjustable-speed, pumped-storage power plant’s excitation control system, based on the VME bus, the VME bus, and experimental results, are presented in Section 4 to validate the theoretical studies. and experimental results, are presented in Section 4 to validate the theoretical studies. Conclusions are Conclusions are drawn in the final section. drawn in the final section. 2. CF-PLL 2. CF-PLL The CF CF transfer transfer function function can can be be described described as as [18]: [18]: The
1 − e −Tw s GCF ( s) = 1 − e−Tw s GCF (s) = T s Tww s
(1) (1)
where TTw is the window length of CF. where w is the window length of CF. The transfer function(Equation (Equation(1)) (1))indicates indicatesthat thatthe theCF’s CF’s transient response time is equal to The transfer function transient response time is equal to its its window length. Assuming the window length the CF contains N samples of itssignal, input window length. Assuming thatthat the window length of theofCF contains N samples of its input w/Ts, Ts is the control system’s sampling time. The transfer function and the signal, and N = T and N = Tw /Ts , Ts is the control system’s sampling time. The transfer function and the pole-zero pole-zero expression ofz-domain CF in the can z-domain can beas: obtained as: expression of CF in the be obtained
1 1 − z − N 1 NNz − zk GCF ( z) =1 1 − z−−N = 1∏ z − zk (2) GCF (z) = N 1 − z−11 =N k =∏ (2) 1 z−p N 1−z N k=1 z −1 p1 By substituting z = ejω into Equation (2) By substituting z = ejω into Equation (2) j 2π ⋅ k N ⋅Ts ) 1 N e jω − e ( GCF ( e jω ) = N∏ (3) j 2π N ⋅Ts ) · Ts ) 1N k =1e jω e jω−−eej((2π ·k/N jω GCF (e ) = ∏ jω (3) N k=1 e − e j(2π/N ·Ts ) A schematic diagram of CF-PLL is shown in Figure 2. The PD is composed of a Clarke transformation anddiagram park transformation, three-phase voltage, ua, ub,ofuc,ainto DC A schematic of CF-PLL isconverting shown in the Figure 2. The grid PD is composed Clarke q under the synchronous rotating reference frame. In addition, ud contains the component ud and transformation andupark transformation, converting the three-phase grid voltage, ua , ub , uc , into DC q contains the input voltage’s phase angle error. Then, CF extracts input voltage’s amplitude, component ud and uq underand theusynchronous rotating reference frame. In addition, ud contains the d and uq, blocking the sinusoidal disturbances of integer multiples of the the DC components of u input voltage’s amplitude, and uq contains the input voltage’s phase angle error. Then, CF extracts changed into adisturbances per-unit value is transferred to the the disturbance frequency. uq is the the DC components of Following ud and uq , that, blocking sinusoidal of and integer multiples of LF. The estimated angular frequency consists of the grid voltage’s angular frequency and the disturbance frequency. Following that, uq is changed into a per-unit value and is transferred to estimate angular frequency (proportional Furthermore, the the LF. The estimated angularerror frequency consists ofintegral the gridregulator voltage’s output). angular frequency and the estimated phase angle is obtained by integrating the estimated angular frequency by means of the estimate angular frequency error (proportional integral regulator output). Furthermore, the estimated VCO. angle Finally, the estimated phase angle sent to the parkfrequency transformation as a of rotation transform phase is obtained by integrating the is estimated angular by means the VCO. Finally, angle. the estimated phase angle is sent to the park transformation as a rotation transform angle.
ua ub uc
αβ abc
uα uβ
dq
αβ θˆ1+
ud
ud
uq
uq
k p s + ki
uq 2 d
2 q
u +u
Δωˆ g
s
ωn ωˆ g
θˆ1+
Figure 2. 2. Schematic diagram of of CF-PLL. CF-PLL. Figure Schematic diagram
Generally, the disturbance frequency of CF is equal to twice that of the fundamental grid Generally, the disturbance frequency of CF is equal to twice that of the fundamental grid voltage’s voltage’s frequency (Tw = 0.01 s). At the same time, assuming that the sampling frequency of the frequency (Tw = 0.01 s). At the same time, assuming that the sampling frequency of the control system control system is 10 kHz (Ts = 0.0001 s), then, selection of the parameters of the PI regulator is done is 10 kHz (Ts = 0.0001 s), then, selection of the parameters of the PI regulator is done according to the according to the symmetrical optimum method [23]: kp = 83.3 and ki = 2893.5. The bode plot of the symmetrical optimum method [23]: kp = 83.3 and ki = 2893.5. The bode plot of the open-loop transfer open-loop transfer function of the CF-PLL is shown in Figure 3. It can be seen that the CF-PLL has a unity gain at zero frequency and zero gain, at the notch frequencies 100 m (m = 1, 2, 3, ...) Hz.
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function of the CF-PLL is shown in Figure 3. It can be seen that the CF-PLL has a unity gain at zero frequency zero gain, the notch frequencies 100 mand (m =completely 1, 2, 3, ...) Hz. Therefore, CF-PLL can pass Therefore,and CF-PLL canatpass a DC component block the notch frequency acomponents. DC component and completely block can the notch components. In other words, multiples CF-PLL can In other words, CF-PLL block frequency the sinusoidal disturbances of integer of block the sinusoidal disturbances ofaddition, integer multiples the disturbance frequency 1/Twmargin . In addition, Figure 3ofshows that CF-PLL has a phase (PM) the disturbance frequency 1/Tw. In Figure shows thatmargin CF-PLL has a margin (PM)the of stability 43.3◦ and gain margin (GM) of 14.1 dB, of 43.3°3and a gain (GM) ofphase 14.1 dB, ensuring ofaCF-PLL. The cut-off frequency ensuring theisstability CF-PLL. The indicates cut-off frequency CF-PLL is 2π ·13.8 rad/s, which indicates of CF-PLL 2π·13.8ofrad/s, which that theofdynamic response is slow, which is the that the dynamic response is slow, which is the drawback of CF-PLL. drawback of CF-PLL.
100
101
102
103
Figure 3. 3. Bode Bode plot plot of of the the open-loop open-loop transfer transfer function function of of CF-PLL; CF-PLL; N N= = 100. Figure 100.
3. FAMCF-PLL 3. FAMCF-PLL 3.1. MCF-PLL
In order to improve the dynamic response of CF-PLL, this paper proposes MCF-PLL by introducing N poles and one zero when the sampling order of CF is N. The transfer function and pole-zero expression expression of of MCF MCF in in the the z-domain z-domain are are
K 1 − z − N 1 − α ⋅ z −1 K N +1Nz+−1 zk K z − zk GMCF ( z)K= 1 − z− N−1 1 − α · z−N1 = ∏ GMCF (z) = N 1− z N k =1 ∏ z − pk − (α ⋅ z −−1 )1 N = N 1 − z−1 11− N z − pk (α · z ) k =1
(4) (4)
where α is attenuation factor, 0 < α < 1. K is the gain factor, K = (1 − αN)/(1 − α). where α is attenuation factor, 0 < α < 1. K is the gain factor, K = (1 − αN )/(1 − α). By substituting z = ejω into Equation (4) By substituting z = ejω into Equation (4) 1 jω e − zk K NN++11 e jωjω− e j[2π( ·k/() Ns +1)·T ] K1 N +jω jω N+ Gjω e = = K ∏ e − ej 2π ⋅k ( N +1)⋅T s ( ) K e − (5) ∏ jω z k jω GMCF (e MCF) = (5) N k =jω − pk= N ∏ ∏ 1 e α ⋅ e j[2π ·k/( Ns +1)·Ts ] N k =1 e − p k N kk==11 ee jω−− α·e When the frequency is ωm (m = 1, 2, ..., N + 1), ejωm = zm = pm. MCF has N + 1 zeros and N + 1 poles When the frequency is ωm (m = 1, 2, ..., N + 1), ejωm = zm = pm. MCF has N + 1 zeros and N + 1 poles when the sampling order is N. Moreover, MCF has a unity gain at zero frequency, N − 1 zeros, and when the sampling order is N. Moreover, MCF has a unity gain at zero frequency, N − 1 zeros, N − 1 poles at notch frequency. Selecting N = 10, the pole-zero pattern of MCF is shown in Figure 4 and N − 1 poles at notch frequency. Selecting N = 10, the pole-zero pattern of MCF is shown in when α = 0.9, 0.95, and 0.99. It can be seen that CF has 10 zeros and 1 pole. On the other hand, MCF Figure 4 when α = 0.9, 0.95, and 0.99. It can be seen that CF has 10 zeros and 1 pole. On the other hand, has 11 poles and 11 zeros by introducing 10 poles and 1 zero at the CF frequency. The distance MCF has 11 poles and 11 zeros by introducing 10 poles and 1 zero at the CF frequency. The distance between the pole-zero pattern of CF and MCF becomes smaller as the attenuation factor increases. If between the pole-zero pattern of CF and MCF becomes smaller as the attenuation factor increases. α = 1, MCF turns into an all-pass filter. A bode plot of the open-loop transfer function of MCF-PLL, If α = 1, MCF turns into an all-pass filter. A bode plot of the open-loop transfer function of MCF-PLL, when selecting different α values, is shown in Figure 5. The phase and gain margins of MCF-PLL when selecting different α values, is shown in Figure 5. The phase and gain margins of MCF-PLL become larger as the attenuation factor increases. This paper selects α = 0.99, and MCF-PLL has a become larger as the attenuation factor increases. This paper selects α = 0.99, and MCF-PLL has a phase margin of 45°, gain margin of 17.3 dB, and a cutoff frequency of 2π·36 rad/s. Compared with phase margin of 45◦ , gain margin of 17.3 dB, and a cutoff frequency of 2π·36 rad/s. Compared with CF-PLL, MCF-PLL improves the dynamic response on the premise of maintaining stability. CF-PLL, MCF-PLL improves the dynamic response on the premise of maintaining stability. j 2π ⋅ k N +1 ⋅T
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1 0.8 0.6
0.96
0.4
z33 α=0.99 ω33=6000π rad/s
0.2 0
α=0.95
-0.2 -0.4 -0.6 -0.8
0.85 -0.35
α=0.9 -0.25
-1 -1 -0.8 -0.6-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Axis
Phase(deg) Phase(deg)
Magnitude(dB) Magnitude(dB)
Figure 4. 4. The Thez-plane, z-plane, pole-zero pole-zero pattern pattern of of MCF, MCF, with 10. Figure with N N == 10.
1000
1011
1022
1033
Figure Figure 5. 5. Bode Bode plot plot of of the the open-loop open-loop transfer transfer function function of of MCF-PLL, MCF-PLL, with with N N == 100. 100.
3.2. Discrete Model of FAMCF-PLL 3.2. Discrete Model of FAMCF-PLL Assuming the three-phase grid voltage Assuming the three-phase grid voltage − Ui+i+ cosθ(+ = U+ θ ++ ) +U −−cos (θ −−)− ua =uaa ∑ i cos i ii + Uiii cos ii θi ii==1,5,7, i=1,5,7,... 1,5,7, − ub = ∑ Ui+ +cos θi++ −2π2π θ − + 2π (6) 3 +−− Ui cos 3 −− 2πi + + = Uii cos θii − +Uii cos θii + (6) iu=1,5,7,... u = bb ii==1,5,7, + + − − 2π 2π 3 3 Ui cos θi + 3 + Ui cos θi − 3 ∑ 1,5,7, c i=1,5,7,... uc = Ui++ cos θi++ + 2π +Ui−− cos θi−− − 2π i i i 3 sequence 3 c positive i amplitudes ii==1,5,7, 1,5,7, and negative represent the of the grid voltage’s k
where Ui+ and Ui− order harmonic, respectively. θ + and θi− represent the positive and negative sequence phase angles of and negative sequence amplitudes of the grid voltage’s where Ui+i+ and Ui−i− representi the positive the grid voltage’s k order harmonic, respectively. ++ k order harmonic, respectively. and θ i−− represent theusing positive and negative sequence phase Equation (6) can be rewrittenθ iin frame the Clarke transformation i the αβ ireference
angles of the grid voltage’s k order harmonic, respectively. Equation (6) can be rewritten in the αβ reference frame using the Clarke transformation
uα = Ui cos (θ + ) +Ui cos (θ − ) i i i=1,5,7, + − uβ = Ui sin (θi+ ) −Ui sin (θi− ) i=1,5,7,
Equation (7) Energies 2017, 10, 737
(7)
can be rewritten in the dq reference frame by using the park transformation
ud = Ui+ cos (θ + −θˆ+ ) +Ui− cos (θ − +θˆ+ ) i i 1 1 i =1,5,7, + + − − − + Ui (θcos +i− U = ∑ Ui+ sin sini (θcos uuαq = −θˆ1θ+i) −U +θˆθ1+i) i i =1,5,7,... i =i1,5,7, + Ui sin θi+ − Ui− sin θi− ∑ uβ = i =1,5,7,... When FAMCF-PLL completes the grid voltage phase locked, the phase + Equation (7) can be rewritten in the dq reference as: frame by using the park transformation ( δ = ωt + θ1 − θu ≈ 0 , Equation (8)) can be simplified + ˆ+ , 4+ U1+i cos cosδ +θi+fd− θω U, i6−ωcos θ)i− + θˆ1+ ∑ud =U 2 ω , ud = ( 1 g g g i =1,5,7,... ++ sin δ +θi+fq − ∑ ( 2ωθˆ1+g , 4−ωgU, 6i−ωsin ) i− + θˆ1+ g , θ uq =UU1 i sin uq = i =1,5,7,...
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(8) (7)
error
(9) (8)
The disturbance frequency FAMCF is equal to twice that the fundamental grid When FAMCF-PLL completesofthe grid voltage phase locked, theof phase error (δ = ωt + θ1+ voltage’s − θu ≈ 0, w = 0.01 s. The grid voltages in the dq reference frame are obtained frequency, T Equation (8)) can be simplified as: (
ud =U1+ cos δ ud = U1+ cosδ + f d + 2ωg , 4ωg , 6ωg , . . . sin δ, 4ω , 6ω , . . . q =fU1 2ω uq = U1+ sinδu+ q g g g
(10) (9)
Then, converting the grid voltages’ q axis component into per-unit value expression The disturbance frequency of FAMCF is equal to twice that of the fundamental grid voltage’s frequency, Tw = 0.01 s. The grid voltages ∗in theudqq reference frame are obtained uq = = sin δ ≈ δ (11) ( u2 + u2 + d q ud = U1 cos δ (10) uq =discrete U1+ sin δmodel is shown in Figure 6. The transfer The schematic diagram of the MCF-PLL function ofconverting MCF in thethe z-domain is given by Equation (4).into Theper-unit transfervalue function of the PI controller Then, grid voltages’ q axis component expression in the z-domain is: uq =k sin (11) u∗q = q ⋅T δ ≈ δ 2 GPIu( z+ =uk p2 + i s ) (12) q d z −1 The diagram of the MCF-PLLofdiscrete model is shown 6. The coefficient transfer function where kp schematic is the proportionality coefficient the PI controller and kiinisFigure the integral of the of MCF in the z-domain is given by Equation (4). The transfer function of the PI controller in the PI controller. z-domain is: The transfer function of the integrator in the z-domain is: k · Ts GPI (z) = k p + i (12) Tsz − 1 GI ( z ) = (13) z −1 where kp is the proportionality coefficient of the PI controller and ki is the integral coefficient of the PI controller.
Δθ
+ 1
θ1+
G MCF ( z )
G PI ( z )
ωn Δωˆ g ωˆ g
GI ( z)
θˆ1+
θˆ1+ Figure 6. 6. Schematic diagram of of MCF-PLL MCF-PLL discrete discrete model. model. Figure Schematic diagram
Selecting a phase margin within the range of 30°~60°, to guarantee the stability of the control The transfer function of the integrator in the z-domain is: system, is recommended. This paper selected PM = 45°, ωc = 2π·36 rad/s. The discrete model of MCF-PLL is set up in MATLAB/Simulink (MathWorks: Ts Natick, MA, USA), based on the Equations GI (z) = (13) z−1 Selecting a phase margin within the range of 30◦ ~60◦ , to guarantee the stability of the control system, is recommended. This paper selected PM = 45◦ , ωc = 2π ·36 rad/s. The discrete model of MCF-PLL is set up in MATLAB/Simulink (MathWorks, Natick, MA, USA), based on the Equations (4), (12), and (13). Then, the values of the phase margin and the cut-off frequency in the
The MCF-PLL cannot completely block the disturbance components if the grid voltage frequency changes [23–25]. In such cases, MCF-PLL needs to adjust the sampling order, according to the variations in the grid voltage’s frequency. Then, the MCF sampling order, N, is given by means of the nearest integer approach Energies 2017, 10, 737
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N = round(Tw Ts )=round( π fs ωˆ g )
(14)
where Ts is the control system’s sampling period. ωˆ is the estimated angular frequency of proportional integral derivative (PID) Tuner were set. Thegparameters of the proportional integral (PI) MCF-PLL. regulator are obtained: kp = 194.15, ki = 28,008.07. Figure 7 shows the digital implementation schematic diagram of FAMCF-PLL. It can be seen 3.3. FAMCF-PLL that FAMCF-PLLRealization can be easily implemented. Equation (15) presents the digital implementation method of FAMCF-PLL The MCF-PLL cannot completely block the disturbance components if the grid voltage frequency k −1 k −1 the sampling order, according to the variations changes [23–25]. In such cases, MCF-PLL to adjust 1 needs = − y ( ) x ( ) x ( korder, − − i ) N, is given by means of the nearest k k i MCF sampling in the grid voltage’s frequency. Then,Nthe i=N i =0 (15) integer approach k−N k − N −1 xN N (14) ) =round K (αTw /T ⋅ ys()k = )− π α f⋅sy/(ωˆkg−) i ) − iround ( k= i =0 i =0 ˆ g is the estimated angular frequency of MCF-PLL. where Ts is the control system’s sampling period. ω where x(k) 7isshows the input variable of FAMCF-PLL at k sample time. y(k) is the output variable of Figure the digital implementation schematic diagram of FAMCF-PLL. It can be seen that FAMCF-PLL at k sample FAMCF-PLL can be easilytime. implemented. Equation (15) presents the digital implementation method The FAMCF-PLL digital implementation scheme in this paper consists of four steps: of FAMCF-PLL k −1 k −1 1 y(k)the voltages’ q axis component, x (k), to A (i − 1); then saving the (1) At sample time k, adding = grid x k − i − x k − i ( ) ( ) ∑ ∑ N i= N i =0 (15) result to array A (i). k− N k − N −1 N x ( kN K ∑ α · y(k(14), − i) − −(ik) − N )] N . ) =using Equation (2) Computing sample order y ( k )∑= [ Aα (·ky)(−k A i =0
i =0
(3) Adding y (k) to α N ⋅ B ( k − N ) , then saving the result to array B (i). where x(k) is the input variable of FAMCF-PLL at k sample time. y(k) is the output variable of (4) x ( k ) = [ Bat( kk)sample − α ⋅ B ( ktime. − 1)] K . FAMCF-PLL
1 y(k) N
x( k )
z −1
z−N
K z−N α N
α
x (k)
z −1
Figure Figure 7. 7. Digital Digital implementation implementation schematic schematic diagram diagram of of FAMCF-PLL. FAMCF-PLL.
4. Experimental Results The FAMCF-PLL digital implementation scheme in this paper consists of four steps: The proposed FAMCF-PLL scheme for doubly-fed adjustable-speed pumped storage plants (1) At sample adding grid voltages’ q axis component, (k), toSystems, A (i − 1);General then saving the was carried outtime on k,the VMEthe bus based control system (RX7i xPAC Electric result to array A (i). Company: Fairfield, CT, USA). The control system consisted of a CPU board (VMIVME-7807, (2) Computing orderan N using Equation y(k)an= analog A(k − Nboard [ A(k) −output )]/N. (VMIVME-4140, General Electric sample Company), encoder signal (14), board, N · B(k − N ), then saving the result to array B (i). (3) Adding y (k) to α General Electric Company) and three current control boards. The current control board was implemented Texas Instruments Company: Dallas, TX, USA) and a (4) x(k) = [ Bon α · B(k(TMS320F28335, − 1)]/K. (k) a− DSP FPGA (EPF10K30A, Altera Company: San Jose, CA, USA). The current control board algorithms 4. Experimental Resultsclose loop control, linear trigger, AD samplings, and generated the firing completed the current pulses. virtual distorted scheme grid conditions, FAMCF-PLL, the Clarke transformation, TheThe proposed FAMCF-PLL for doubly-fed adjustable-speed pumped storage plantspark was transformation, and power calculation algorithms were developed in IOWorks using the carried out on the VME bus based control system (RX7i PAC Systems, General Electric Company, programming unit. Furthermore, the control algorithms were downloaded to the CPU board using Fairfield, CT, USA). The control system consisted of a CPU board (VMIVME-7807, General Electric Company), an encoder signal board, an analog output board (VMIVME-4140, General Electric Company) and three current control boards. The current control board was implemented on a DSP (TMS320F28335, Texas Instruments Company, Dallas, TX, USA) and a FPGA (EPF10K30A, Altera Company, San Jose, CA, USA). The current control board algorithms completed the current close loop control, linear trigger, AD samplings, and generated the firing pulses. The virtual distorted grid conditions, FAMCF-PLL, the Clarke transformation, park transformation, and power calculation algorithms were developed in IOWorks using the programming unit. Furthermore, the control algorithms were downloaded to the CPU board using an Ethernet connector. All results were obtained under 10-kHz sampling frequency experimental conditions. Finally, the internal variables
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an Ethernet connector. All results were obtained under 10-kHz sampling frequency experimental conditions. Finally, the internal variables were exported by the analog output board and were were exported by an theoscilloscope analog output board andTektronix were monitored using an oscilloscope (DPO4034B, monitored using (DPO4034B, Company, Beaverton, OR, USA). Figure 8 Tektronix Company, OR, USA). Figure shows images of the of experimental and shows images of theBeaverton, experimental platform and 8the control diagram the system.platform The DFIM’s the control diagram of in thethe system. The DFIM’s parameters are shown in the Appendix A. parameters are shown Appendix A.
(a)
(c)
u gR \ u gS \ u gT
usA \ usB \ usC isA \ isB \ isC
ira \ irb \ irc
ωr \ θ r
Tem
ira∗
Us ird+ , q
qs∗ ps∗
Δqs Δps
ωr
ird∗
ωm
urd∗ Δurd
irq∗
u
ura∗ ura∗ urb∗ urc∗
∗ rq
Δurq ird irq
ωs
irc∗
θl
ωsl
ps , qs
irb∗
urb∗ urc∗
θl
ira
irb
θ r θl
θs
irc
ωˆ g
θu
usA usB usC
Us
usd usq isd isq
qs ps
θl
isA isB isC
(d) Figure 8.8.Images Imagesofof experimental platform. (a) Cycloconverter power cabinet; (b) DIFM and Figure thethe experimental platform. (a) Cycloconverter power cabinet; (b) DIFM and prime prime motor; (c) Control system; (d) Control diagram of the doubly-fed, adjustable-speed, pumped motor; (c) Control system; (d) Control diagram of the doubly-fed, adjustable-speed, pumped storage storage hydropower hydropower plant. plant.
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The paper selects CF-PLL and MCF-PLL as the control groups to verify the steady-state and paper selects and MCF-PLL as the under controlthe groups to verify the steady-state and dynamic The performances of CF-PLL the proposed FAMCF-PLL following conditions:
• • • • •
dynamic performances of the proposed FAMCF-PLL under the following conditions:
Condition 1: 40 ms, the grid voltage undergoes a frequency step change of +5 Hz.
• Condition 1: 40 ms, the grid voltage undergoes a frequency step change of +5 Hz. Condition 2: 40 ms, the grid voltage undergoes a phase angle jump of +40◦ . • Condition 2: 40 ms, the grid voltage undergoes a phase angle jump of +40°. 3: 403:ms, voltage undergoes a harmonics injection of 20% fifthfifth andand 10%10% seventh •Condition Condition 40 the ms, grid the grid voltage undergoes a harmonics injection of 20% harmonic components. At 100 ms, the grid voltage undergoes a frequency step change of seventh harmonic components. At 100 ms, the grid voltage undergoes a frequency step change+5 Hz. of +5 Hz. Condition 4: 40 ms, the voltage amplitude of phase A sags by 50% and the voltage amplitude of •phase Condition 4: 40 ms,At the100 voltage amplitude of phase A sags bya 50% and thestep voltage amplitude B sags by 30%. ms, the grid voltage undergoes frequency change of +5 Hz. B sags by 30%. At 100 ms, the grid voltage undergoes a frequency step change of +5 of phase Condition 5: 40 ms, the frequency step change, phase angle jump, harmonics injection, and voltage Hz. sag occur at the same time. • Condition 5: 40 ms, the frequency step change, phase angle jump, harmonics injection, and voltage sag occur the same time. result when the grid voltage underwent a frequency step Figure 9a shows theatexperimental
change ofFigure +5 Hz. can be that the CF-PLL suffered from a slow transientaresponse. 9a Itshows theseen experimental result when the grid voltage underwent frequency However, step change of +5 Hz. It can be seenofthat CF-PLL suffered from a slowwere transient response. However, the steady-state characteristics thethe MCF-PLL and FAMCF-PLL as good as those achieved the steady-state characteristics of theFAMCF-PLL MCF-PLL andcould FAMCF-PLL were assteady-state good as thoseperformance achieved by and by CF-PLL. Moreover, the proposed obtain better CF-PLL. Moreover, the proposed FAMCF-PLL could obtain better steady-state performance and results a a faster transient response than CF-PLL and MCF-PLL. Figure 9b shows the experimental transient response than CF-PLL and MCF-PLL. Figure experimental results whenfaster the grid voltage underwent a phase angle jump of +40◦ .9bIt shows can bethe observed that the transient when the grid voltage underwent a phase angle jump of +40°. It can be observed that the transient response of MCF-PLL and FAMCF-PLL were much faster than that of CF-PLL, which was similar to response of MCF-PLL and FAMCF-PLL were much faster than that of CF-PLL, which was similar to the experimental results in Figure 9a. Detail analyses results are shown in Table 1. the experimental results in Figure 9a. Detail analyses results are shown in Table 1.
(a)
(b)
(c)
(d)
Figure 8. Cont.
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(e) Figure 9. Experimental results of PLLs under distorted grid conditions. (a) Condition 1; (b)
Figure 9. Experimental results of PLLs under distorted grid conditions. (a) Condition 1; (b) Condition Condition 2; (c) Condition 3; (d) Condition 4; (e) Condition 5. 2; (c) Condition 3; (d) Condition 4; (e) Condition 5.
In order to analyze the effects of the FAMCF-PLL’s grid voltage’s frequency variation rejection Table 1. Detailed experimental results of the PLLs. capability, the experimental results of the PLLs under comprehensive distorted grid conditions are shown in Figure 9c–e. Figure 9c shows that MCF-PLL and FAMCF-PLL had faster transient Condition MCF-PLL FAMCF-PLL responses than CF-PLL under the distorted grid voltage, CF-PLL with fifth- and seventh-order harmonic components set at 20% and 10%, respectively. CF-PLL and 33.4 FAMCF-PLL 24.0 obtained a frequency 5% setting time/msAt 100 ms, 66.8 much better rejection capability than MCF-PLL38.1 in experimental when the frequency overshoot/% 28.0operations33.8 1 disturbance peakchanged. phase error/deg 19.3 that, both 7.7 6.3and the grid voltage’s frequency As a result, it can be observed the phase errors frequency deviations of CF-PLL andtime/ms FAMCF-PLL were much MCF-PLL. phase 5% setting 64.6 smaller than 32.6 those of 23.6 Figure 9d2 shows that the phase errors and the frequency deviations of CF-PLL phaseboth overshoot/% 34.8 23.0 27.8 and FAMCF-PLL were peak muchfrequency smaller than those of MCF-PLL10.9 under the unbalanced grid error/Hz 14.3 18.3voltage amplitude sags and frequency changes, which was similar to the experimental results of Figure 9c. peak-to-peak phase error/deg 0.100 3.10 0 3 the transient response of MCF-PLL and FAMCF-PLL was much faster than that of In addition, peak-to-peak frequency error/Hz 1.00 12.0 0.500 CF-PLL. Figure 9e demonstrates that the transient response of CF-PLL was too slow to realize the peak-to-peak phase error/deg 0.6 3.60 0 phase lock. 4 Meanwhile, MCF-PLL showed an excellent filtering capability when the grid voltage’s peak-to-peak frequency error/Hz 0.800 6.70 0.200 frequency was at its nominal value. However, the phase errors and frequency deviations increased phase error/deg 0.800 FAMCF-PLL 5.00 could obtain 0 good if the grid5 voltage’speak-to-peak frequency changed from its nominal value. peak-to-peak frequency error/Hz 1.20 even when 17.2 the grid0.700 steady-state performance and disturbance rejection capability, voltage’s frequency changed, owing to adaptive frequency realization. The detailed results can be found in Table 1. to analyze the effects of the FAMCF-PLL’s grid voltage’s frequency variation rejection In order
capability, the experimental results of the PLLs under comprehensive distorted grid conditions are Table 1. Detailed experimental results of the PLLs. shown in Figure 9c–e. Figure 9c shows that MCF-PLL and FAMCF-PLL had faster transient responses MCF-PLL harmonic FAMCF-PLL than CF-PLLCondition under the distorted grid voltage, with fifth-CF-PLL and seventh-order components set 66.8 33.4 24.0 frequency 5% setting time/ms at 20% and 10%, respectively. At 100 ms, CF-PLL and FAMCF-PLL obtained a much better disturbance 1 frequency overshoot/% 38.1 28.0 33.8 rejection capability than MCF-PLL in experimental operations when the grid voltage’s frequency peak phase error/deg 19.3 7.7 6.3 changed. As a result, it phase can be5% observed that, both the phase and the frequency deviations of setting time/ms 64.6 errors 32.6 23.6 CF-PLL and FAMCF-PLL were much smaller than those of MCF-PLL. shows that both the phase overshoot/% 34.8 23.0Figure 9d 27.8 2 phase errors and the frequency deviations of CF-PLL and FAMCF-PLL smaller than those peak frequency error/Hz 10.9 14.3were much 18.3 peak-to-peak grid phasevoltage error/deg 0.100 3.10 0 of MCF-PLL under the unbalanced amplitude sags and frequency changes, which was 3 peak-to-peak frequency error/Hz 1.00 12.0 0.500 similar to the experimental results of Figure 9c. In addition, the transient response of MCF-PLL peak-to-peak phase error/deg 0.6 3.60 0 and FAMCF-PLL 4 was much faster than that of CF-PLL. Figure 9e demonstrates that the transient peak-to-peak frequency error/Hz 0.800 6.70 0.200 response of CF-PLL was too slow to realize the phase lock. Meanwhile, MCF-PLL showed an excellent peak-to-peak phase error/deg 0.800 5.00 0 5 when the grid voltage’s frequency was at its nominal value. However, the phase filtering capability peak-to-peak frequency error/Hz 1.20 17.2 0.700 errors and frequency deviations increased if the grid voltage’s frequency changed from its nominal value. FAMCF-PLL could obtain good steady-state performance and disturbance rejection capability,
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even when the grid voltage’s frequency changed, owing to adaptive frequency realization. The detailed Energies 2017, 737 in Table 1. 11 of 13 results can be10,found Figure 10 shows the experimental results of the doubly-fed, adjustable-speed, pumped storage Figure 10 shows the experimental results of the doubly-fed, adjustable-speed, pumped storage hydropower experimental platform under normal grid conditions. During the process of rotor hydropower experimental platform under normal grid conditions. During the process of rotor acceleration, from 1425 rpm (0.9 5p.u.) to 1575 rpm (1.05 p.u.), and rotor deceleration, from 1575 rpm acceleration, from 1425 rpm (0.9 5p.u.) to 1575 rpm (1.05 p.u.), and rotor deceleration, from 1575 (1.05 p.u.) to 1425 rpm (0.95 p.u.), the proposed strategy obtained smooth rotor currents and phase rpm (1.05 p.u.) to 1425 rpm (0.95 p.u.), the proposed strategy obtained smooth rotor currents and sequence changes changes during the entire dynamic process,process, as shown Figurein10a,b, In a word, phase sequence during the entire dynamic asin shown Figurerespectively. 10a,b, respectively. theInresult showed thatshowed the doubly-fed, adjustable-speed system had a good performance, both under a word, the result that the doubly-fed, adjustable-speed system had a good performance, sub-synchronous, and super-synchronous, states using FAMCF-PLL. both under sub-synchronous, and super-synchronous, states using FAMCF-PLL.
(a)
(b)
Figure 10. Experimental results of the doubly-fed, adjustable-speed, pumped storage hydropower Figure 10. Experimental results of the doubly-fed, adjustable-speed, pumped storage hydropower experimental platform. (a) Sub-synchronous to super-synchronous; (b) Super-synchronous to experimental platform. (a) Sub-synchronous to super-synchronous; (b) Super-synchronous sub-synchronous. to sub-synchronous.
5. Conclusions 5. Conclusions This paper proposed a FAMCF-PLL for doubly-fed adjustable-speed pumped storage plants, This paper proposed a FAMCF-PLL doubly-fed plants, especially when grid voltage is not ideal.forThe proposed adjustable-speed scheme obtained pumped excellent storage disturbance especially when grid voltage is not ideal. The proposed scheme obtained excellent disturbance rejection rejection ability, by blocking the sinusoidal disturbances of integer multiples of the disturbance ability, by blocking the sinusoidal disturbances of integer multipleswith of the disturbance frequency, frequency, even when the grid voltage was adverse. Compared CF-PLL and MCF-PLL, even when theimproved grid voltage was adverse. Compared and MCF-PLL, FAMCF-PLL FAMCF-PLL the disturbance-rejection abilitywith whenCF-PLL the frequency of the grid voltage improved disturbance-rejection when the frequency of the gridpower voltage changed. Finally, changed.the Finally, the control systemability of adjustable-speed, pumped-storage plants was built, thebased control system adjustable-speed, pumped-storage power wasand built, based on the VME on the VMEofbus, and the experimental results validated theplants feasibility effectiveness of the proposed bus, and thealgorithms. experimental results validated the feasibility and effectiveness of the proposed algorithms. Acknowledgments: The authors appreciate the financial supports provided by the National High Technology Acknowledgments: The authors appreciate the financial supports provided by the National High Technology Research and DevelopmentProgram Programof ofChina China (863 (863 Program), and thethe National Basic Research and Development Program),Grant GrantNo. No.2014AA052602; 2014AA052602; and National Basic Research Program China(973 (973Program), Program),Grant Grant No. No. 2014CB046306. 2014CB046306. Research Program ofofChina Author Contributions:Wei WeiLuo Luoand andJianguo Jianguo Jiang Jiang developed thethe present research andand Author Contributions: developedthe theessential essentialidea ideabehind behind present research thethe system models; the test test bench benchand andcarried carriedout outthe theexperiments. experiments. system models;Wei WeiLuo Luoand andHe HeLiu Liu established established the WeiWei LuoLuo carried outout thethe design of of thethe FAMCF-based control strategies and the analysis ofofthe experimental carried design FAMCF-based control strategies and the analysis the experimentalresults. results.Wei WeiLuo completed the manuscript and the final manuscript correction was done by Jianguo Jiang. Luo completed the manuscript and the final manuscript correction was done by Jianguo Jiang. Conflicts of of Interest: Conflicts Interest:The Theauthors authorsdeclare declareno noconflict conflictsofofinterest. interest.
Appendix AA Appendix DFIM: Urr==380 380V,V,f =f 50 = 50 Hz, 2 pole pairs, Rr = 0.0345 s = 0.0728 DFIM:SnSn==110 110kVA, kVA, U Uss == U Hz, np =np2 =pole pairs, Rs = R 0.0728 Ω, Rr Ω, = 0.0345 Ω, Ls = Ω, Ls 6.57 = 6.57 mH, L = 3.27 mH, L = 43.93 mH. r m mH, Lr = 3.27 mH, Lm = 43.93 mH. References References 1. 1. Bose, B.K. of power powerelectronics electronicsinin21st 21stcentury. century.IEEE IEEE Trans. Appl. Bose, B.K.Global Globalenergy energyscenario scenario and and impact impact of Trans. Ind.Ind. Appl. 2013, 2638–2651.[CrossRef] 2013, 60,60, 2638–2651. 2. Hadjipaschalis, I.; Poullikkas, A.; Efthimiou, V. Overview of current and future energy storage technologies for electric power applications. Renew. Sustain. Energy Rev. 2009, 13, 1513–1522.
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