Tan-Sun Transformation-Based Phase-Locked Loop in Detection of the Grid Synchronous Signals under Distorted Grid Conditions

: When three-phase voltages are polluted with unbalance, DC offsets, or higher harmonics, it is a challenge to quickly detect their parameters such as phases, frequency, and amplitudes. This paper proposes a phase-locked loop (PLL) for the three-phase non-ideal voltages based on the decoupling network composed of two submodules. One submodule is used to detect the parameters of the fundamental and direct-current voltages based on Tan-Sun transformation, and the other is used to detect the parameters of the higher-harmonic voltages based on Clarke transformation. By selecting the proper decoupling vector by mapping Hilbert space to Euclidean space, the decoupling control for each estimated parameter can be realized. The settling time of the control law can be set the same for each estimated parameter to further improve the response speed of the whole PLL system. The system order equals the number of the estimated parameters in each submodule except that a low-pass ﬁlter is required to estimate the average amplitude of the fundamental voltages, so the whole PLL structure is very simple. The simulation and experimental results are provided in the end to validate the effectiveness of the proposed PLL technique in terms of the steady and transient performance. and X.S.; Resources, G.T. and X.S.; Software, G.T. and C.Z.; Supervision, G.T.; Validation, G.T. and C.Z.; Visualization, G.T. and C.Z.; Writing—original draft, C.Z.; Writing—review & editing, G.T. and X.S. All


Introduction
With the widespread application of renewable energy and distributed generation (DG), more and more grid-connected inverters have been employed in the power grid. However, the power grid will be subject to some adverse impacts regarding the increased penetration level of DG and the excessive usage of single-phase, asymmetric, and non-linear loads, such as voltage unbalance, voltage distortion, frequency jumping, and so on [1,2]. In addition, DC offsets may appear in the detected grid voltage signals due to the saturation of voltage transducer, A/D conversion error, or power grid fault [3,4]. All the above problems will make it difficult for the grid-connected inverter to synchronize with the grid voltages normally, and the operation performance of the grid-connected inverter may deteriorate. Therefore, when the above problems occur in the grid voltages, it is an important guarantee to acquire the grid synchronization signals such as phases, frequency, and amplitudes of the grid voltages quickly and accurately for the reliable operation of the grid-connected inverter.
Among power grid synchronization techniques, phase-locked loop (PLL) is one of the most effective solutions. PLL is a control system which can generate the output signal synchronized with the phase of input signal. The most widely used PLL technique in the three-phase power electronic system is the synchronous reference frame PLL (SRF-PLL) [5,6]. SRF-PLL utilizes proportional-integral for each estimated parameter is realized. Both the estimated phase of phase A voltage, which is a ramp signal, and its differential signal estimated grid frequency need to be designed as second-order system; however, all the other estimated parameters only need to be designed as first-order systems. The system order equals the number of the estimated parameters in each submodule except that the estimation for the average amplitude of the fundamental voltages needs a low-pass filter (LPF), so the control structure is very simple. In addition, the response speed of the whole PLL system can be further improved by setting the control law with the same settling time for each estimated parameter. Finally, the simulation and experiments of the proposed PLL technique are carried out to verify the effectiveness of the proposed PLL technique in terms of the steady and transient performance.

The Whole PLL Structure with Decoupling Network of Two Submodules
In the three-phase three-wire power electronic system, the non-ideal grid voltages e abc can be expressed as the sum of FCs (e abc1 ), DCs (e abc0 ), and HCs (e abch ) as follows In the first terms e abc0 as shown in Equation (1), E m is the average amplitude of e abc1 , θ a is the phase of e a , and ϕ ba and ϕ ca are the phase differences between e b and e a , and e c and e a , respectively. As the three-phase three-wire system under study is free from zero-sequence (ZS) components, the amplitudes of e abc1 are related to the phase differences ϕ ba and ϕ ca , and they are 2/ √ 3E m sin (ϕ ba − ϕ ca ), 2/ √ 3E m sin ϕ ca , and −2/ √ 3E m sin ϕ ba , respectively. In the second terms e abc0 as shown in Equation (1), the DC offsets of e a and e b can be set as E a0 and E b0 , respectively, and the DC offset of e c can be set as (−E a0 − E b0 ) as the system under study is free from ZS components. The third terms e abch as shown in Equation (1) are set to contain [n = (6k ± 1)]-th (k ∈ N + ) harmonic voltages, where (6k + 1)-th harmonics are set as PS, and (6k − 1)-th harmonics are set as NS. The amplitude of n-th harmonic is set as E nm , while the initial phase of n-th harmonic is set as ϕ n .
According to the expression form of the three-phase non-ideal voltages e abc as shown in Equation (1), this paper proposes two detection submodules (DSs), FC and DC detection submodule (FC&DC-DS) and HC detection submodule (HC-DS), to detect the above parameters contained in e abc1 , e abc0 , and e abch , respectively. In these two DSs, FC&DC-DS is used to detect the parameters of e abc1 and e abc0 successively through Tan-Sun and Park transformation, and HC-DS is used to detect the parameters of e abch successively through Clarke and Park transformation. Then, the fundamental voltages with DC offsets and the higher-harmonic voltages can be reconstructed respectively according to the detected parameters in these two DSs, and the input voltages of each DS can be decoupled by cross-subtraction of the voltages reconstructed by the other DS. The whole PLL structure composed of FC&DC-DS and HC-DS with decoupling network is shown in Figure 1, and these two DSs will be designed respectively in the following sections.

FC-DS in Detection of Fundamental-Voltage Parameters
The PLL technique for detecting the parameters of e abc1 in the FC&DC-DS will be briefly reviewed at first, whose detailed design process has been presented in [35]. e abc1 can be rewritten according to Equation (1) as where, the actual parameter vector is denoted as p 1 = θ a ϕ ba ϕ ca T . The expression form of the PD error e q1 can be calculated as the second element of the column matrix e dq1 derived successively though Tan-Sun and Park transformation of e abc1 as where T Park = cosθ a sinθ a − sinθ a cosθ a (5) And the estimated parameter vector is denoted asp 1 = θ aφbaφca T . Then, e q1 can be approximated as the following linear function about the estimated parameter errors ∆θ a , ∆ϕ ba , and ∆ϕ ca , the vector form of which is denoted as ∆p 1 = ∆θ a ∆ϕ ba ∆ϕ ca T , by Taylor expansion of it with respect to ∆p 1 about the point ∆p 1 = 0 as As e q1 in Equation (6) is the function about ∆p 1 with more than one parameter, Hilbert space should be built here to design the FC-DS structure. In the real space If the inner product formula is defined as then the function system 1, is an orthonormal basis in L 2 [0, 2π] [36]. The inner product defined in Equation (7) denotes the average value of the product of f (x) and g (x) in a fundamental period. If only the first three elements in Equation (8) are considered, the orthonormal basis in Hilbert space constructed by these three elements has a corresponding relationship with the space rectangular coordinate system in Euclidean space, and these three elements are corresponding to three unit vectors ı, , and k of three-dimensional Euclidean space. Based on the above relationship, if we define x = 2θ a , e q1 in Equation (6) can be expressed as the following vector form The control laws forω andp 1 can be expressed as the dot product of e q1 and certain vectors as where, K 1ω , K 1θ , K 2 , and K 3 are all greater than zero. In order to implement the decoupling control for each estimated parameter in the control laws Equation (10), v 1 should be perpendicular to b and c, v 2 should be perpendicular to c and a, and v 3 should be perpendicular to a and b, respectively. In order to simplify the design process, v 1 , v 2 , and v 3 should be set as unit vectors, and the signs of these unit vectors should be set to ensure the stability of the control laws Equation (10). Based on the above analysis, v 1 , v 2 , and v 3 can be calculated as By substituting Equation (11) into Equation (10), the control laws forp 1 can be rewritten as The control laws forp 1 should have the same settling time t s , and the damping ratio of second-order system is equal to its optimal value √ 2/2, so the gains K 1ω , K 1θ , K 2 , and K 3 can be approximately selected in the case of three-phase balanced voltages (ϕ ba = −2π/3, ϕ ca = 2π/3, and e d1 = 1.5E m ) as The control laws in Equation (10) can be rewritten as the following form by mapping the vectors v 1 , v 2 , and v 3 in Euclidean space to their corresponding scalar functions of Equation (8) in Hilbert space The real parameters p 1 in the control laws Equation (14) can all be replaced by their corresponding estimated parametersp 1 , and the initial values ofp 1 can be set aŝ According to the above design process, the FC-DS structure in the FC&DC-DS in detection of the fundamental-voltage parameters is shown in Figure 2.

FC&DC-DS in Detection of Fundamental Voltage and DC-Offset Voltage Parameters
Three-phase unbalanced voltages with only DC offsets, i.e., e abc10 , can be expressed according to Equation (1) as In order to detect the DC offsets in three-phase voltages, the estimated valuesÊ a0 andÊ b0 of E a0 and E b0 , respectively, can be constructed aŝ where, ∆E a0 and ∆E b0 are the estimated errors of E a0 and E b0 , respectively. The fundamental components of e abc10 in Equation (16) have been acquired by using FC-DS designed in Section 3, i.e.,θ a = θ a (18a) And only the control laws forÊ a0 andÊ b0 need to be designed in the FC&DC-DS. The actual parameter vector p 0 , the estimated parameter vectorp 0 , and the estimated parameter error vector ∆p 0 can be denoted respectively as The expression form of the PD error e q10 can be calculated as the second element of the column matrix e dq10 derived successively though Tan-Sun and Park transformation of the differences between e abc10 andÊ abc0 as e dq10 = T Park T Tan−Sun e abc10 −Ê abc0 (19) e q10 in Equation (19) can be approximated as the following linear function about ∆p 0 by Taylor expansion of it with respect to ∆p 0 about the point ∆p 0 = 0 as It can be derived according to Equation (20) that if ∆p 0 = 0, then e q10 = 0; on the other hand, if e q10 = 0, then As cos θ a and sin θ a are time-variant while the coefficients of them in Equation (21) are time-invariant, Equation (21) can be equivalent to The determinant of the matrix A can be calculated as In general, the difference between the instantaneous fundamental phases of any two voltages in e abc10 is neither 0 • nor ±180 • , so |A| in Equation (24) is unequal to 0, and it can be derived according to Equation (22) that ∆p 0 = 0. Through the above analysis, we can come to a conclusion that ∆p 0 = 0 is equivalent to e q10 = 0. Therefore, if e q10 is controlled to 0, the errors betweenp 0 and p 0 are eliminated as well.
If the second and third elements in Equation (8) are considered, the orthonormal basis in Hilbert space constructed by these two elements has a corresponding relationship with the plane rectangular coordinate system in Euclidean space, and these two elements are corresponding to two unit vectors l and m of two-dimensional Euclidean space. Based on the above relationship, if we define x = θ a , e q10 in Equation (20) can be expressed as the following vector form (25) The control laws forp 0 can be expressed as the dot products of e q10 and certain vectors as where, K 4 and K 5 are both greater than zero. In order to implement the decoupling control for each estimated parameter in the control laws Equation (26), v 4 should be perpendicular to e, and v 5 should be perpendicular to d. In order to simplify the design process, v 4 and v 5 should be set as unit vectors, and the signs of these unit vectors should be set to ensure the stability of the control laws Equation (26). Based on the above analysis, v 4 and v 5 can be calculated as By substituting Equation (27) into Equation (26), the control laws forp 0 can be rewritten as The control laws forp 0 should have the same settling time t s as that of the control laws forp 1 , so the gains K 4 and K 5 can be approximately selected in the case of three-phase balanced voltages (ϕ ba = −2π/3, ϕ ca = 2π/3) as The control laws in Equation (26) can be rewritten as the following form by mapping the vectors v 4 and v 5 in Euclidean space to their corresponding scalar functions of Equation (8) in Hilbert space The real parameters p 1 in the control laws Equation (30) can all be replaced by their corresponding estimated parametersp 1 . According to the above design process, the FC&DC-DS in detection of both the fundamental-voltage parameters and the DC-offset voltage parameters is shown in Figure 3, and the reconstruction structure of the unbalanced voltages containing DC offsets is shown in Figure 4.  In Figure 4, all the input variables are available through estimation except e d10 , which has a linear relationship with the average amplitude E m according to Equation (19). If e d10 is directly feed forward to the reconstructed voltages of the FC&DC-DS, the whole PLL system cannot be stable due to the direct connection from the inputs to the outputs of the FC&DC-DS, which are then fed back to the inputs of the HC-DS. Therefore, a low-pass filter (LPF) should be added in the path for estimating E m to avoid this direct connection, the transfer function of which is shown as where, ω f is the bandwidth of the LPF, which should be set much higher than that of the control laws designed for all the above estimated parameters, so as to alleviate the impact of the LPF to the transient performance of the whole PLL system.

HC-DS in Detection of Higher-Harmonic Voltage Parameters
The higher harmonics in three-phase voltages can be expressed according to Equation (1) as In order to facilitate the subsequent analysis, Equation (32) can be rewritten as the following form Clarke transformation matrix is According to Equation (5), Equation (33), and Equation (35), the dq-axes voltages e dh and e qh can be expressed as where,θ a in Park transformation matrix is provided by the FC&DC-DS, which has been acquired by using FC-DS designed in Section 3, i.e.,θ a = θ a . It can be seen from Equation (36) that the (6k ± 1)-th harmonics in three-phase voltages Equation (32) are finally converted into 6k-th harmonics in the dq-axes voltages Equation (36) successively through Clarke and Park transformation. In order to simplify the expression form of e dh and e qh as shown in Equation (36), the relationships of the harmonic coefficients can be set as By substituting Equation (37) into Equation (36), e dh and e qh can be rewritten as The estimated valuesÊ d6kcm ,Ê q6kcm ,Ê d6ksm ,Ê q6ksm of the real parameters E d6kcm , E q6kcm , E d6ksm , E q6ksm , respectively, can be constructed aŝ where, ∆E d6kcm , ∆E q6kcm , ∆E d6ksm , ∆E q6ksm are the estimated parameter errors of E d6kcm , E q6kcm , E d6ksm , E q6ksm , respectively. The real parameter vector p h , the estimated parameter vectorp h , and the estimated parameter error vector ∆p h can be denoted respectively as Through the above analysis, we can come to a conclusion that ∆p h = 0 is equivalent to ∆e dh = 0 and ∆e qh = 0. Therefore, if ∆e dh and ∆e dh are controlled to 0, the errors betweenp h and p h are eliminated as well.
If we establish the relationship that cos 6kθ a and sin 6kθ a in Equation (8) are corresponding to  k and k k in Euclidean space, respectively, e dh and e qh in Equation (40) can be expressed as the following vector form As the expression form of e dh is the same as that of e qh , only the control laws forÊ d6kcm andÊ d6ksm need to be designed, and the control laws forÊ q6kcm andÊ q6ksm can be designed in the same way. The control laws forÊ d6kcm andÊ d6ksm can be expressed as the dot products of e dh and certain vectors respectively as where, K dkc and K dks are both greater than zero. For any k ∈ N + , if v dkc and v dks only contain the unit vectors  k and k k , then v dkc and v dks will be certainly perpendicular to both {  n |n ∈ N + , n = k} and { k n |n ∈ N + , n = k}. Therefore, in order to implement the decoupling control forÊ d6kcm andÊ d6ksm in the control laws Equation (43), v dkc is only required to be perpendicular to k k , and v dks is only required to be perpendicular to  k . In order to simplify the design process, v dkc and v dks should be set as unit vectors, and the signs of these unit vectors should be set to ensure the stability of the control laws Equation (43). Based on the above analysis, v dkc and v dks can be calculated as By substituting Equation (44) into Equation (43), the control laws forÊ d6kcm andÊ d6ksm can be rewritten as The control laws forÊ d6kcm andÊ d6ksm should have the same settling time t s as that of the control laws forp 1 andp 0 , so the gains K dkc and K dks can be selected as The control laws in Equation (43) can be rewritten as the following form by mapping the vectors v dkc and v dks in Euclidean space to their corresponding scalar functions of Equation (8)  According to the above design process, the HC-DS in detection of the higher-harmonic voltage parameters is shown in Figure 5, and the reconstruction structure of the higher-harmonic voltages is shown in Figure 6, where the sine and cosine coefficients of (6k ± 1)-th harmonic voltages can be calculated according to Equation (37) as shown in Figure 6.

Simulation Analysis
This section will validate the correctness and effectiveness of the proposed whole PLL system, which is based on the combination of FC&DC-DS and HC-DS with decoupling network in MATLAB/Simulink. In the simulation, three-phase voltages are set unbalanced with DC offsets and higher harmonics, and the parameters of the three-phase non-ideal voltages are set as shown in Table 1. In Table 1, the fundamental voltage unbalance factor can be calculated as 14.58% according to [37]. In order to simulate the actual three-phase distorted grid voltages, the three-phase voltages mainly contain the 5th, 7th, 11th, and 13th harmonics, and the 5th and 11th harmonic voltages are NS (−5 and −11) while the 7th and 13th harmonic voltages are PS (+7 and +13). The amplitude of each order higher-harmonic voltage decreases as the harmonic order increase. The amplitudes of the 5th and 7th harmonic voltages, as well as those of the 11th and 13th harmonic voltages, are both nearly the same. The selection of the DC offsets in the three-phase voltages can refer to [3,16]. Amplitude of 13th harmonic voltage 3.8 V In the simulation, the settling time t s of the control law for each estimated parameter is set as 35 ms, and the bandwidth ω f of the LPF for estimatingÊ m is set as 2000 rad/s. The simulation for the proposed whole PLL system is performed under the following three different transient conditions of the three-phase non-ideal voltages, so as to validate the steady performance and response speed of the whole PLL system, respectively.

Three-Phase Unbalanced Voltages Being Injected with DC Offsets and Higher Harmonics
At t = 0.3 s, the three-phase unbalanced voltages are suddenly injected with DC offsets and higher harmonics, and the simulation results are shown in Figure 7. It can be seen from Figure 7 that the detection time for each parameter is less than 50 ms. The estimated error of each fundamental-voltage phase is almost zero, and the maximum estimated error of the grid frequency is less than 0.9 Hz. (g) Figure 7. Simulation results when three-phase unbalanced voltages are suddenly injected with DC offsets and higher harmonics. The waveforms of (a) e abc ; (b)ê abc1 . The actual and estimated parameters of (c) E a0 , E b0 ; (d) E 5m , E 7m , E 11m , E 13m ; (e) f ; (f) θ a ; (g) ϕ ba , ϕ ca .

The Phase Difference between Phase B and Phase A Voltages Having a Step Change
At t = 0.8 s, the phase difference between phase B and phase A voltages has a step change from −130 • to −150 • , and the simulation results are shown in Figure 8. It can be seen from Figure 8  (g) Figure 8. Simulation results when ϕ ba has a step change from −130 • to −150 • . The waveforms of (a) e abc ; (b)ê abc1 . The actual and estimated parameters of (c) E a0 , E b0 ; (d) E 5m , E 7m , E 11m , E 13m ; (e) f ; (f) θ a ; (g) ϕ ba , ϕ ca .

The Grid Frequency Having a Step Change
At t = 0.8 s, the grid frequency f has a step change from 50 Hz to 45 Hz, and the simulation results are shown in Figure 9. It can be seen from Figure 9 that the detection time for each parameter is about 60 ms. The estimated error of each fundamental-voltage phase is very little, and the maximum estimated error of the grid frequency is less than 1 Hz. (g) Figure 9. Simulation results when the grid frequency f has a step change from 50 Hz to 45 Hz. The waveforms of (a) e abc ; (b)ê abc1 . The actual and estimated parameters of (c) E a0 , It can be concluded according to the simulation results that the proposed whole PLL system can successfully separate the FCs, DCs, and HCs of the three-phase non-ideal voltages, respectively, and detect all the parameters in less than 60ms under various types of transient conditions. Except for the big overshoots in detecting the DC offsets and higher harmonics of the three-phase voltages, the overshoots in detecting the other parameters are relatively very little. It is worth to note from the simulation results that the proposed whole PLL system has good steady and transient performance.

Experimental Results
The performance of the proposed whole PLL system has been evaluated through simulation in Section 6. This section will validate the correctness and effectiveness of the whole PLL system on the digital experimental platform. The experimental platform, as shown in Figure 10, is composed of a waveform generation board and a waveform detection board which both take the digital signal controller TMS320F28335 as the core. The waveform generation board is used to simulate the three-phase voltages sampled from the voltage transducers, and the parameters of the three-phase voltages are set as the same as those used in the simulation ( Table 1). The waveform detection board is used to receive the three-phase voltages produced by the waveform generation board and detect the parameters of the received voltages. The settling time of each control law and the bandwidth of the LPF programmed in the waveform detection board are set also the same as those used in the simulation. This section will first validate the steady performance of the whole PLL system when both the DC offsets and higher harmonics are contained in the three-phase unbalanced voltages. And then, the transient performance of the whole PLL system will be validated under three different transient conditions, which are set as the same conditions as in the simulation.

Analysis of the Steady Experimental Results
When the three-phase unbalanced voltages contain DC offsets and higher harmonics, the steady experimental results are shown in Figure 11. It can be seen from Figure 11 that the whole PLL system can successfully separate the FCs, DCs, and HCs of the three-phase non-ideal voltages, respectively, and detect all the parameters without steady errors, which can demonstrate that the whole PLL system has good steady performance. (d) (e) Figure 11. Experimental steady results when three-phase unbalanced voltages contain DC offsets and higher harmonics. The waveforms of (a) e abc ; (b)f ,φ ba , andφ ca ; (c)ê abc andθ a ; (d)Ê abc0 ; (e)Ê 5m ,Ê 7m , E 11m , andÊ 13m .

Condition I
When the three-phase unbalanced voltages are suddenly injected with DC offsets and higher harmonics, the experimental results are shown in Figure 12. It can be seen from Figure 12 that the estimated errors of the grid frequency and the phase differences between phase B&C and phase A voltages are almost zeros. The detection time for the FCs, DCs, and the harmonic voltage amplitudes is less than 40 ms with little overshoots. (d) (e) Figure 12. Experimental results when three-phase unbalanced voltages are suddenly injected with DC offsets and higher harmonics. The waveforms of (a) e abc ; (b)f ,φ ba , andφ ca ; (c)ê abc andθ a ; (d)Ê abc0 ; (e)Ê 5m ,Ê 7m ,Ê 11m , andÊ 13m .

Condition II
When the phase difference between phase B and phase A voltages has a step change from −130 • to −150 • , the experimental results are shown in Figure 13. It can be seen from Figure 13 that the estimated errors of the grid frequency and the phase differences between phase B&C and phase A voltages are very little. The detection time for the FCs, DCs, and the harmonic voltage amplitudes is less than 25 ms, while the estimated DC offsets have large fluctuations and the estimated higher harmonic voltages have little fluctuations. (d) (e) Figure 13. Experimental results when ϕ ba has a step change from −130 • to −150 • . The waveforms of (a) e abc ; (b)f ,φ ba , andφ ca ; (c)ê abc andθ a ; (d)Ê abc0 ; (e)Ê 5m ,Ê 7m ,Ê 11m , andÊ 13m .

Condition III
When the grid frequency f has a step change from 50 Hz to 45 Hz, the experimental results are shown in Figure 14. It can be seen from Figure 14 that the estimated errors of the grid frequency and the phase differences between phase B&C and phase A voltages are very little. The detection time for the FCs, DCs, and the harmonic voltage amplitudes is less than 60 ms, while the estimated DC offsets have large fluctuations and the estimated higher harmonic voltages have relatively less fluctuations.  Figure 14. Experimental results when the grid frequency f has a step change from 50 Hz to 45 Hz. The waveforms of (a) e abc ; (b)f ,φ ba , andφ ca ; (c)ê abc andθ a ; (d)Ê abc0 ; (e)Ê 5m ,Ê 7m ,Ê 11m , andÊ 13m .
It can be concluded according to the experimental results that the proposed whole PLL system can successfully separate the FCs, DCs, and HCs of the three-phase non-ideal voltages, respectively, and detect all the parameters in less than 60ms under various types of transient conditions. Except for the large fluctuations of the estimated errors in detecting the DC offsets of the three-phase voltages, the overshoots or fluctuations in detecting the other parameters are little. Therefore, the experimental results can demonstrate that the proposed whole PLL system has good steady and transient performance.

Conclusions
This paper has proposed a PLL technique based on the combination of FC&DC-DS and HC-DS with decoupling network, which can realize the rapid detection of the parameters of FCs, DCs, and HCs of the three-phase non-ideal voltages, respectively. The control law for each estimated parameter can be designed as the first-order or second-order system (grid frequency and instantaneous phase of phase A voltage are the latter case) which is related to only the corresponding estimated parameter error, so the decoupling control is realized for each estimated parameter. The response speed of the whole PLL system can be further improved by setting the control law for each estimated parameter with the same settling time. The proposed PLL technique can directly detect the instantaneous phases of three-phase fundamental voltages other than those of the three-phase PS voltages. It can be concluded from the simulation and experimental results that the proposed PLL technique has good steady performance, and the detection time for each parameter does not exceed three grid cycles in various types of transient conditions, which can demonstrate that the PLL technique also has good transient performance and high stability. Future work will focus on the practical application of the proposed PLL technique to the three-phase AC microgrid.