energies Article
A Novel Robust Model Predictive Controller for Aerospace Three-Phase PWM Rectifiers Tao Lei * , Weiwei Tan, Guangsi Chen and Delin Kong Key laboratory of Aircraft Electric Propulsion Technology, Ministry of Industry and Information Technology of China, Northwestern Polytechnic University, Xi’an 710072, China;
[email protected] (W.T.);
[email protected] (G.C.);
[email protected] (D.K.) * Correspondence:
[email protected] Received: 24 August 2018; Accepted: 14 September 2018; Published: 19 September 2018
Abstract: This paper presents a novel Model Predictive Direct Power Control (MPDPC) approach for the pulse width modulation (PWM) rectifiers in the Aircraft Alternating Current Variable Frequency (ACVF) power system. The control performance of rectifiers may be largely affected by variations in the AC side impedance, especially for systems with limited power volume system. A novel idea for estimating the impedance variation based on the Bayesian estimation, using an algorithm embedded in MPDPC is presented in this paper. The input filter inductance and its equivalent series resistance (ESR) of PWM rectifiers are estimated in this algorithm by measuring the input current and input voltage in each cycle with the probability Bayesian estimation theory. This novel estimation method can overcome the shortcomings of traditional data based estimation methods such as least square estimation (LSE), which achieves poor estimation results with the small samples data set. In ACVF systems, the effect on the parameters estimation accuracy caused by the number of sampling points in one cycle is also analyzed in detail by simulation. The validity of this method is verified by the digital and Hard-in-loop simulation compared with other estimation methods such as the least square estimation method. The experimental testing results show that the proposed estimation algorithm can improve the robustness and the control performance of the MPDPC under the condition of the uncertainty of the AC side parameters of the three-phase PWM rectifiers in aircraft electrical power system. Keywords: Model Predictive Direct Power Control (MPDPC); Aircraft Electrical Power System; PWM rectifiers; Bayesian estimation methods
1. Introduction In the recent years, thanks to technological advances in microprocessors, model predictive control (MPC) has been proposed and studied as a promising alternative for the control of power converters and drives [1]. This control strategy has been used for inverter [2], rectifier [3,4], active power filter [5] and Uninterrupted Power Supply (UPS) [6]. In aircraft electrical power system, the pulse width modulation (PWM) rectifiers have more advantages than the traditional multi-pulse Transformer/Autotransformer-Rectifiers (TRU or ATRU) applied in the aircraft electrical power system, such as the regulated DC voltage and high power factor in the AC side [7]. In the aircraft electrical power system, the limitation about the total harmonic distribution (THD) of AC input current is very strict about less than 10% [8]. Based on a mathematical model of PWM rectifiers, and predicting the possible future controlling input variables, the Finite Control Set-Model Predictive Control (FCS-MPC) [9] achieves numerous similar performance including the high power factor and sinusoidal input currents with little harmonic pollution compared with other traditional control methods in the aircraft electrical power system. Furthermore, the advantages of MPC within PWM Energies 2018, 11, 2490; doi:10.3390/en11092490
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rectifiers are the abilities to handle constraints, multiple inputs and outputs and nonlinearities in a simple and intuitive manner. A commonly used MPC control algorithm for three-phase PWM rectifiers is Model Predictive Direct Power Control (MPDPC) [10–14]. The input active power and reactive power in DPC are controlled by measuring input AC current. The three-phase bridge leg corresponds to eight possible switching states and eight possible input current states. Finally, the cost function is used to evaluate the eight switching states and select the best switching vector in the next cycle. The control performance of the FSC-MPC depends on the accurate predictions of the future state. Mismatches between the parameters used in the model and the actual parameters of PWM rectifiers may result in inaccurate selection of the optimal control vector. Inaccurate system models can cause some degradation in control performance. This issue is very important for the MPC applied to the aerospace PWM rectifier, which makes the system potentially susceptible to the mismatches from the temperature and environmental effect. Recently, in order to solve the problem of uncertainty of model parameters, several estimation methods with MPC are suggested. These methods can be classified into the model based estimation methods and data based methods. Paper [15] proposed a parameter estimation method based on least square estimation (LSE). These methods belong to data based methods and the LSE methods just are applied to the 50 Hz power system in this paper. The least square method is applied to accurately estimate the inductance value but the response of this estimator will become slow with an increase of the program runtime. A MPDPC control with the inductance parameters compensation methods is applied to the single phase three-level rectifiers [16]. The research is focused on low switching frequency from 1–20 KHz, it is naturally data based methods. The MPC with the model based estimation methods also has been widely researched. Paper [17] proposed a kind of adaptive robust observer based on direct torque control for the field of AC motor drive but it is not prevalently studied in the three-phase PWM rectifier. In Reference [18], Based on a three-phase PWM rectifier, a robust model predictive control algorithm with disturbance observer is proposed. The influence of model parameter mismatch on the system is regarded as a perturbation and the Luenberger observer is used to eliminate system disturbances by feed-forward compensation and enhance the stability of the control system. Nevertheless, many parameters of observer need to be tuned in this algorithm and it is complicated in the process of designing Luenberger observer. A non-linear constrained control strategy based on robust model prediction and sliding mode control with the parameter estimation is proposed in Reference [19]. However, this method is not applicable to referring to discrete-time model of PWM rectifiers, which is only designed for continuous-time model. Paper [20] combined the MPDPC with the sliding mode Observer in PWM rectifiers under unbalanced grid conditions. The voltage sensor-less control can be realized by the observer and the error of parameters variation is compensated in a satisfied range. This method is only verified in 50 Hz power system. The extended Kalman-Filter with online grid impedance estimation was proposed for the control of grid connected converter in Reference [21], these control algorithms also belong to the model based methods, the ground grid impedance model was derived based on the rectifiers’ model. But the complexity of rectifier’s model with observer was only verified by simulation. The tuning of the covariance matrices of the extended kalman filter (EKF) is based on trial-and-error procedure. In additional, the grid-side online impedance estimation methods based on the power relationship and voltage vector relationship were presented respectively in Reference [22–26], these methods only consider the ground grid connected converter situation, the parameters of grid network changed largely and the control methods with parameter estimation is not limited to the MPC control. The Uncertainty of parameters of PWM rectifiers in ACVF electrical power system cannot be investigated, the characteristics of ac side impedance parameters changed with main AC generator frequency with the load profile. So, the research topic about the aerospace PWM rectifiers with MPDPC control under parameter uncertainty is very significant. Compared with the model based methods, the parameters estimation methods from the data based are easily accomplished by the MPC controller
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by measuring the input and output measurement data to estimating the AC side of the inductor parameters to make it sure that the rectifier model parameters and the actual circuit parameters are kept same in real time. So, the data based estimation method with the MPDPC is the topic of focus in this paper. The main frequency may suffer obvious variations as high as 100%, depending upon the power consumption in the aircraft electrical power system [27]. The main frequency can be changed from the 360–800 Hz and this further deteriorates control performance of MPDPC. Based on the aerospace three-phase PWM rectifier, the cause and influence of inaccurate input filter inductance’s variation are discussed in this paper. MPDPC with Bayesian estimation algorithm is proposed to estimate the actual value of inductance in limited samples data set. The shortcomings of slow convergence rate and instability for the estimated value are overcome by this method compared with the LSE methods proposed in Reference [15]. The realization of control algorithm does not require additional sensors and can be implemented directly in the digital microprocessor. The main contributions of this work include the following: 1.
2.
3.
4.
A MPDPC with the Bayesian estimation for PWM rectifiers in the aircraft ACVF power system is presented, which has not been fully investigated in existing literature for PWM rectifier’s parameter estimation. The probability distribution function of estimated parameters is presented through the analysis of variation for inductance’s value caused by different factors such as temperature. The Bayesian estimation algorithm integrated with MPDPC is derived in detail. The performance of the Bayesian estimation is compared with LSE according to the converging rate, stability of estimated parameters by digital simulation. For the Bayesian methods with MPDPC, the result of the estimation is also compared by using the input current signal with white noise. The digital simulation results validate the robust of Bayesian estimation to disturbance from the input signals. From the numbers of samples for the input measuring signal, The Bayesian estimation methods do not need more samples which would greatly reduce the computational complexity. The proposed Bayesian estimation method is compared with the LSE method and traditional MPDPC in experiments. The advantage of estimation methods is verified by digital simulation and experimental results. The several performance indexes are listed in following Tables to be discussed, the final conclusion can be obtained to test the feasibility of the proposed method. The Hard In Loop (HIL) and experimental testing rig was built to verify the theoretical analysis under the aircraft ACVF Power system.
The remainder of this paper is organized as follows. The basic operation theory of MPDPC is introduced in Section 2. The realization of an online parameter estimation algorithm based on Bayesian estimation with the MPDPC is proposed in Section 3. Digital simulation and semi-physical experiment results are shown in Section 4 and the hardware prototype experiments is carried out in Section 5. The conclusion is discussed in Section 6 and the detail derivative process of Bayesian estimation method is elaborated in Appendix A. 2. Traditional Model Predictive Control Algorithm The three-phase PWM rectifier mainly consists of input filter inductor, Insulated Gate Bipolar Transistor (IGBT) Rectifier Bridge and output filter capacitor. The input power and output voltage are controlled by traditional MPC algorithm through controlling the switch state of IGBTs. A typical three-phase PWM rectifier topology is shown in Figure 1.
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idc
0
vsa vsb vsc
LS
RS ia (t )
LS
RS ib (t )
LS
RS ic (t )
T1
T3
T5
a
Vdc C
b
RL
c
T2
T4
T6
N
Figure 1. The topology of three-phase pulse width modulation (PWM) rectifier. Figure 1. The topology of three‐phase pulse width modulation (PWM) rectifier.
The three-phase input voltage vs , input current iis and PWM rectifiers bridge voltage v REC are The three‐phase input voltage vs , input current s and PWM rectifiers bridge voltage vREC expressed in αβ orthogonal coordinates as (1)–(3), respectively.Note that the rectifier bridge voltages are expressed in orthogonal coordinates as (1)–(3), respectively.Note that the rectifier bridge are expressed with the dc-link voltage and the switching functions: voltages are expressed with the dc‐link voltage and the switching functions: 2 v2s = (vsa + vsb e j(2π/3) + vsc e j(4π/3) ) (1) (1) v ( v 3 v e j ( 2 /3) v e j ( 4 / 3) ) s
sa
sb
sc
3 2 2 (i + ji( 2 e/j3)(2π/3) +j (i4 e/ 3) j(4π/3) ) is is = isb e sb isc e sc ) (isa sa 3 3 2 2 j(2π/3) j ( 4 /j3) (4π/3) v REC vREC (= s a 3(s ab + e j s( 2be/ 3) s c + e sc e )V dc )Vdc 3
(2) (2) (3) (3)
where s a , sb , sc are the switching states of the three-phase rectifier bridge upper switch take on the savalues where 、 sb 、ofsc“1” are the switching states of the three‐phase rectifier bridge upper switch take on binary and “0” in the closed state and open state, respectively. The relationship of the the binary values of “1” and “0” in the closed state and open state, respectively. The relationship of input and output of a three-phase PWM rectifier system is expressed in vector form as: the input and output of a three‐phase PWM rectifier system is expressed in vector form as: dis (4) dis Ls dt = vs − v REC − Rs is L v v Ri (4) s
dt
s
REC
s s
By approximating the derivative of the inductance currents on the basis of the forward Euler By approximating the derivative of the inductance currents on the basis of the forward Euler approximation with a sampling period Ts , the three-phase PWM rectifier system in the continuous-time model will be with represented in the discrete-time as: approximation a sampling period Ts , domain the three‐phase PWM rectifier system in the
continuous‐time model will be represented in the discrete‐time domain as: R T T is (k + 1) = (1 − S S )is (k) + S [vs (k) − v REC (k)] LS RS TS LS TS
(5)
inductance current is represented as: R T T is (k + 2) = (1 − S S )is (k + 1) + S [vs (k + 1) − v REC (k + 1)] LS RSTSLS TS
(6)
is (k 1) (1 )is (k) [vs (k) vREC (k)] (5) LS delay in the LS model prediction algorithm, two-step prediction In order to solve the computational is performed, shifting the inductance current in (5) by one step forward, so the two‐step inductance In order to which solve isthe computational delay in the model prediction algorithm, current is represented as: prediction is performed, which is shifting the inductance current in (5) by one step forward, so the is (k 2) (1 )is (k 1) [vs (k 1) vREC (k 1)] (6) LS k + 1 instant LS can be assumed to be equal to the sampling value The input voltage at the next time at The the kinput instant, basedat onthe thenext assumption the sampling frequency is much larger than the main voltage time k that 1 instant can be assumed to be equal to the sampling frequency of the aircraft electrical power system, According to the instantaneous power calculation value at the k instant, based on the assumption that the sampling frequency is much larger than methodin Reference [28], the future instantaneous input real and reactive powerare presented as: the main frequency of the aircraft electrical power system, According to the instantaneous power calculation methodin Reference [28], the future instantaneous input real and reactive powerare P(k + 2) = isα (k + 1)vsα (k + 1) + isβ (k + 1)vsβ (k + 1) (7) presented as: k +2i) =( kisα k v+ 1()kvsβ (kv+ (1k)vsα1)(k + 1) P ( kQ(2) (1) (1)k + i1s) (+k isβ1) s s s
(7)(8)
Q ( k 2) is ( k 1)vs ( k 1) is ( k 1)vs ( k 1)
(8)
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The common cost function of MPDPC is the error between the value of prediction and reference Energies 2018, 11, 2490 of active and reactive power.
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g DPC Pref P (k 1) Qref Q (k 1)
Active Power(W)
(9) The common cost function of MPDPC is the error between the value of prediction and reference of active and reactive power. 3. Online Parameter Estimation Algorithm As shown in (5) and (6), the rate of AC input current change is affected by the inductance value (9) gDPC = Pre f − P(k + 1) + Qre f − Q(k + 1) directly.Meanwhile, the uncertainty of model parameters will affect the accuracy of predicted active and reactive power, so the selection of the optimal switching states and the system performance also 3. Online Parameter Estimation Algorithm will be disturbed eventually. As shown in (5) and (6), the rate of AC input current change is affected by the inductance value As shown in Figure 2, assuming that the reference value of active power Pr e f is a constant, and directly. Meanwhile, the uncertainty of model parameters will affect the accuracy of predicted active represent the actual value and the predicted value of the active power, respectively. P and Pm t power, t , reactive so the selection of the optimal switching states and the system performance also will be disturbed eventually. According to the sampling progress, there is a deviation between the actual value of active power shown in Figure 2, assuming that the reference value of active power Pre f is a constant, tk Then, at the sampling time tk 1 , and the predicted value P t k As Pm t k at the sampling time and P(t), Pm (t) represent the actual value and the predicted value of the active power, respectively. the optimal to switching vector S3 is there selected on the basis of the active power value According the sampling progress, is a deviation between the predicted actual value of active power P(tk ) and the given reference value of active power . Thus, the actual value of the active P the predicted value P ( t ) at the sampling time t Then, at the sampling time t , the optimal Pmand t k +1 m k k k +1 ref switching vector S3 is selected on the basis of the predicted active power value Pm (tk+1 ) and the power obtained is P t k +2 , which has already deviate from the reference Pref . So, it should be S5 given reference value of active power Pre f . Thus, the actual value of the active power obtained is not optimal switching vector for in it model parameters to errors P (S tk3+ 2is ), the which has already deviate from thePreference Pre f . So, should be S5 not Slead optimal tk +1 . Errors 3 is the switching for P(and tk+1 )the . Errors in model parameters lead toand errors between the predicted andin the between the vector predicted reference value of the active reactive power and result reference value of the active and reactive power and result in inaccurate prediction results, meanwhile, inaccurate prediction results, meanwhile, affect the harmonic content of the AC side current, even affect the harmonic content of the AC side current, even reduce the performance of the three-phase reduce the performance of the three‐phase rectifier system. rectifier system.
P
P(tk)
P(tk+1)
P(t) Pm(tk)
Pm(t)
O
tk
Pm(tk+1)
P(tk+2) S0
Pm(tk+2) Pref S7
tk+1
tk+2
t
Time(s) Figure 2. Power prediction under the influence of model parameter errors. Figure 2. Power prediction under the influence of model parameter errors.
In order to improving the steady state performance of three-phase PWM rectifier, realizing the In order to improving the steady state performance of three‐phase PWM rectifier, realizing the unity power factor and reducing the harmonic content of AC current, an online parameter estimation unity power factor and reducing the harmonic content of AC current, an online parameter algorithm based on Bayesian estimation is proposed with MPDPC. estimation algorithm based on Bayesian estimation is proposed with MPDPC The input current of (6) is transformed as: The input current of (6) is transformed as: v REC (k)] is (k 1)is (k + i1s)(= k )λis (k[) v+s (µk[v)s (k)v− REC ( k )] where where λ = 1−
Rs Ts Ls
(10) (10) (11)
RT (11) 1 s s T s L µ s= (12) Ls Ts of the input current i , input voltage v and rectifier (12) Based on the (10), the α-axis component s s Ls
bridge voltage v REC at stationary coordinate, can be represented as:
isα (k + 1) = λisα (k) + µ[vsα (k) − v RECα (k )] + ν + ε(k)
(13)
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where the v and ε(k) represent DC bias and error during measurement respectively. The (k − 1)th equations constructed with the (k − 1) successive sampled data points in (13) are used and rewritten in matrix form as:
isα (2) isα (1) .. .. = . . isα (k) isα (k − 1)
vsα − v RECα (1) .. . vsα (k − 1) − v RECα (k − 1)
1 ε (1) λ .. .. . . µ + v ε ( k − 1) 1
(14)
For simplicity, (14) is written in matrix notation as: Y = Φθ + ε
(15)
where
isα (2) .. Y= . isα (k)
vsα (1) − v RECα (1) .. . vsα (k − 1) − v RECα (k − 1) ε (1) λ .. θ = µ , ε = . ν ε ( k − 1)
isα (1) .. Φ= . isα (k − 1)
(16)
1 .. . 1
(17)
(18)
In (15), Y and Φ are known data that both can be measured. The coefficient θ is to be estimated parameter vector. The paper [15] gives the LSE methods but this method cannot achieve the good performance in small samples data set. The following Figure 3a gives the characteristic of power inductor value of inductance changed with load current and Figure 3b shows that the permeability of iron magnetic materials also changed with different temperature. Because of complexity for the variation of inductance value, So the relationship between variation of inductance and the current, temperature is nonlinear relationship. It is very difficult to describe the relationships with accurate analytical equation [29]. Through the data statistics processing procedure from the curve of inductance, the estimated parameters such as inductance value are assumed to follow the normal distribution around variation of inductance. The Figure 4a presents the histogram of inductance value which has been changed in different currents and temperatures. In the Figure 4a, the f k describing how often such result or values happened. So, Figure 4b presents the probability density function, which can describe the distribution of the inductance value. it is related to the histogram for infinite number of measurements for the inductance (continuous function approximation) while the cumulative distribution function is related to cumulative histogram.
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(a)
(b)
(a)
(b)
Figure 3. The characteristics of (a) inductance changed with the currents and (b) permeability of the iron magnetic material under different temperatures. Figure 3. The characteristics of (a) inductance changed with the currents and (b) permeability of the Figure 3. The characteristics of (a) inductance changed with the currents and (b) permeability of the
ironiron magnetic material under different temperatures. magnetic material under different temperatures.
(a)
(b)
Figure 4. The histogram of the inductance value (a) and probability density distribution function (b). Figure 4. The histogram of the inductance value (a) and probability density distribution function (b). (a) (b)
According to the above analysis, the priority distribution function of the estimation parameters Figure 4. The histogram of the inductance value (a) and probability density distribution function (b). According to the above analysis, the priority distribution function of the estimation parameters cancan be expressed by the Gaussian function. Based on Bayesian estimation theory in Reference [30], be expressed by the Gaussian function. Based on Bayesian estimation theory in Reference [30], estimation formula of coefficient is shown as Specific (19). Specific detailed formula derivation According to the above analysis, the priority distribution function of the estimation parameters the the estimation formula of coefficient θ isshown as (19). detailed formula derivation process is process is shown in the appendix. The following equation is same as the formula (17) in the can be expressed by the Gaussian function. Based on Bayesian estimation theory in Reference [30], shown in the appendix. The following equation is same as the formula (17) in the appendix. appendix. the estimation formula of coefficient is shown as (19). Specific detailed formula derivation −1 1 process is shown θˆin = the following as Φthe in the 0 is −same 0 ˆ appendix. ˆ)0Φ(E + Φ0 Φ))1(The (0 µ θˆ) =equation (E)1+(Φ Φ) (µ0 + Y) formula (17) (19) (19) Φ 0+ B B (E(E 0 Y) appendix. where where
ˆ) (19) (E R)sT1s( ˆB (E )1(0 0 Y) 1− λˆ B RsTs Ts Ls θˆB = µˆ B (20) where 1= Ls Ls ˆB νˆB v RsTs s ˆ ˆ 1T (20) Bayesian estimation formula ofthe B B resistance, Linductance L and DC bias are shown as: ˆ BˆB s ˆ s =v1− Rˆs,B Tµˆ BsλB ˆ ˆB (20) B Lˆ = Ts (21) ˆ s,B µLˆ Bs B νˆB = νˆBv Bayesian estimation formula of the resistance, inductance and DC bias are shown as:
actualcircuit parameterswill be guaranteed by The consistency of model parameters and the taking the estimated coefficient obtained at the end of each sampling time into the original model of Bayesian estimation formula of the resistance, inductance and DC bias are shown as: the rectifier by using the Equation (19), this can be expressed by Equation (22): ˆ s (k) + µˆ [vs (k ) − v REC (k)] is (k + 1) = λi
(22)
For convenience, θˆB can be easily calculated by introducing two dummy matrices A and B as the −1 θˆB = ( E + Φ0 Φ) (µ0 + Φ0 Y) = ( E + A)−1 (µ0 + B)
(23)
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where the E denotes the identity matrix. µ0 denotes the expectation of the coefficient θ.
a11 0 A = Φ Φ = a21 a31
a12 a22 a32
b1 a13 a23 , B = Φ0 Y = b2 . b3 a33
It is noted that regardless of the amount of measured data, A and B become 3-by-3 and 3-by-1 matrices, respectively, forthe three-phase AFE at all times. As a result, calculating θˆB , including the calculation of the inverse of A in (23), can be,without calculation burden, processed on the platform of a common DSP in real systems that are generally used for the three phase rectifiers.Using (19) and (23), all the elements of A and are calculated onthe basis of only the measurement of the input current and the input voltage. The block diagram of the control algorithm is shown in Figure 5. In this control block diagram, the input AC voltage and input AC current must be measured as the data set for estimation, the output dc voltage also need to be acquired to calculate the reference active power reference value. In this control block diagram, the Bayesian estimation algorithm can be integrated with MPDPC control methods. By using the MPDPC with the proposed Bayesian estimation methods, the PWM driving control signal can be selectedoptimally for the three phase PWM rectifiers. According to the control block diagram in Figure 5 and derived equation based on Bayesian estimation, the MPDPC methods with the Bayesian estimation algorithm can be expressed by following flow chart shown in Figure 6. Firstly, the AC side input voltage and input AC current of PWM rectifiers are measured in every time instants, then the optimal voltage vector from the last time instants can be applied to the rectifiers. Secondly, the parameters update algorithm based on the Bayesian estimation, the Ac side parameters such inductance and resistance can be updated in real time according to the operational conditions of rectifiers. Thirdly, the DPC control calculation can be carried out, the active power and reactive power can be calculated. Finally, the FCS-MPC control algorithm can be used to select optimal voltage vector from the 8 vectors. The process can be repeated in cycle time of input voltage. Energies 2018, 11, 9 of 23
Figure 5. Block diagram of the on-line parameter estimation algorithm with (model Figure 5. Block diagram of proposed the proposed on‐line parameter estimation algorithm with Predictive (model Direct Power Control) MPDPC. Predictive Direct Power Control) MPDPC.
is k vs k Vdc k
Vop
is k 2 is k 1 vs k Vop k
is k 2
Block diagram of the proposed on‐line parameter estimation algorithm with (model EnergiesFigure 2018, 11,5. 2490 9 of 22 Predictive Direct Power Control) MPDPC.
is k vs k Vdc k
is k 2 is k 1 vs k Vop k
is k 2
Vop
g DPC Q ref Q k 2 P ref P k 2
, if g DPC gop g op g DPC , xop x
is k 1 is k vs k Vop k x 8?
x x 1
V
x
vop k vxop k
Figure 6. The flow chart of MPDPC control algorithm with Bayesian estimation. Figure 6. The flow chart of MPDPC control algorithm with Bayesian estimation.
4. Simulation and Experiment Results 4. Simulation and Experiment Results 4.1. Digital Simulation
4.1. Digital Simulation According to the on-line parameter estimation algorithm proposed in Section 3, the mathematic simulation model of PWM rectifier isestimation built in Matlab/Simulink (Matlab R2012, MathWorks, According to three-phase the on‐line parameter algorithm proposed in Section 3, the Natick, MA, USA). The algorithm MPDPC is written in S-function. The simulation parameters mathematic simulation model of of robust three‐phase PWM rectifier is built in Matlab/Simulink (Matlab are shown in Table 1. R2012, MathWorks, Natick, MA, USA). The algorithm of robust MPDPC is written in S‐function. The simulation parameters are shown in Table 1.
Table 1. Digital simulation parameters.
Parameters
Symbol
Value
Input voltage RMS/V Fundamental frequency/Hz Sampling period/µs Output voltage/V Input inductance/mH Input resistance/Ω Capacitance/µF DC load/Ω
vsRMS F Ts Vdc Ls Rs C RL
115 400 20 350 5 0.01 940 61.25
With the input inductance Ls changed from 5 mH to 2 mH, the specific simulation results are shown in Figure 7. The input AC current waveform of traditional MPDPC has become distortion and MPDPC with Bayesian estimation algorithm can still guarantee the current sinusoidal. To facilitate the observation of the current waveform, the current waveform has been amplified 10 times. The harmonic components of input current are shown in Figure 8. The total harmonic distribution (THD) of input AC current with traditional MPDPC has reached 23.95% when Ls = 2 mH, however the input current’s THD has been reduced to 10.57% with Bayesian estimation algorithm applied.
MPDPC with Bayesian estimation algorithm can still guarantee the current sinusoidal. To facilitate the MPDPC with Bayesian estimation algorithm can still guarantee the current sinusoidal. To facilitate the observation observation of of the the current current waveform, waveform, the the current current waveform waveform has has been been amplified amplified 10 10 times. times. The The harmonic components of input current are shown in Figure 8. The total harmonic distribution (THD) harmonic components of input current are shown in Figure 8. The total harmonic distribution (THD) of mH, of input input AC AC current current with with traditional traditional MPDPC MPDPC has has reached reached 23.95% 23.95% when when LL mH, however however the the s s22
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(b) (b)
(a) (a)
Figure 7. AC side voltage and current at inductance parameter changes (a) Traditional MPDPC; (b) Figure 7. AC side voltage and current at inductance parameter changes (a) Traditional MPDPC; Figure 7. AC side voltage and current at inductance parameter changes (a) Traditional MPDPC; (b) Bayesian estimation algorithm with MPDPC. (b) Bayesian estimation algorithm with MPDPC. Bayesian estimation algorithm with MPDPC.
(a) (a)
(b) (b)
Figure 8.8. The harmonic component of input current. (a) Traditional MPDPC; (b) MPDPC with Bayesian Figure harmonic component of (a) MPDPC; (b) MPDPC with Figure 8. The The harmonic component of input input current. current. (a) Traditional Traditional MPDPC; (b) MPDPC with Estimation Algorithm. Bayesian Estimation Algorithm. Bayesian Estimation Algorithm.
The estimated value of input inductance Ls and input resistance Rs at one sample cycle are shown in Figure 9, with two different parameter data based estimation algorithms: least square method and Bayesian estimation method. When Ls is being changed from the 5 mH to 2 mH quickly, the rate of change of Ls ’s estimation value in Bayesian estimation is more quickly than it in least square method and the estimated value of Ls keep the same size eventually, with an error about 0.22 mH compared to the simulation given value of Ls . As for the estimated value of Rs , Bayesian estimation indicates faster convergence rate and more stable performance than least square method in limited data samples set.
Bayesian estimation indicates faster convergence rate and more stable performance than least square least square method and the estimated value of Ls keep the same size eventually, with an error method in limited data samples set. about 0.22 mH compared to the simulation given value of Ls . As for the estimated value of Rs , Bayesian estimation indicates faster convergence rate and more stable performance than least square Energies 2018, 11, 2490 method in limited data samples set.
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(a)
(b)
Figure9. The estimated value of Ls and Rs from the different estimation methods. (a) Estimated
value of Ls ; (b) Estimated value of Rs . (a)
(b)
Figure 9. The The estimated value ofof Ls and Rs from the different estimation methods. (a) Estimated value Figure9. estimated the different estimation methods. (a) Estimated Ls and Rs from The application of value online parameter Bayesian estimation algorithm depends on the of Ls ; (b) Estimated value of Rs . ; (b) Estimated value of value of Lsdata of input voltage Rand measurement current. The k 1 samples of measuring data from the s.
power converter data are used in the matrix equation in (14). Figure 10 shows the estimated value of The application of online parameter Bayesian estimation algorithm depends on the measurement data of input voltage and k − 1 samples of measuring data from the power converter The of current. online The parameter Bayesian estimation algorithm depends on data the Ls and Rapplication s as different samples measurement data chosen with Bayesian estimation. In Figure 10, are used in the matrix equation in (14). Figure 10 shows the estimated value of L and R as different measurement data of input voltage and current. The samples of measuring data from the k 1 s s KHz and the 125 samples are the total samples of a full cycle as sampling frequency is 50 samples measurement data chosen with Bayesian estimation. In Figure 10, the 125 samples are the power converter data are used in the matrix equation in (14). Figure 10 shows the estimated value of fundamental frequency is 400 Hz. As the chosen samples are total samples of one or more full cycles, total samples of a full cycle as sampling frequency is 50 KHz and fundamental frequency is 400 Hz. Ls and Rs as different samples measurement data chosen with Bayesian estimation. In Figure 10, value of L and Rs can keepstable. From the Figure 10, it is can be seen that the the estimated As the chosen samples ares total samples of one or more full cycles, the estimated value of Ls and Rs the 125 samples are the total samples of a full cycle as sampling frequency is 50 KHz and more the sampling points is in one cycle, the more stable for the estimation value of impedance is, So can keepstable. From the Figure 10, it is can be seen that the more the sampling points is in one cycle, fundamental frequency is 400 Hz. As the chosen samples are total samples of one or more full cycles, in ACVF power system, the sampling frequency must be increased to 100 KHz properly by using the the more stable for the estimation value of impedance is, So in ACVF power system, the sampling R can keepstable. From the Figure 10, it is can be seen that the of Ls and the estimated Bayesian estimation methods. frequency mustvalue be increased to 100s KHz properly by using the Bayesian estimation methods. more the sampling points is in one cycle, the more stable for the estimation value of impedance is, So in ACVF power system, the sampling frequency must be increased to 100 KHz properly by using the Bayesian estimation methods.
(a)
(b)
Figure 10. The estimated value of and in different samples with Bayesian estimation method (Ls = 2 mH, Rs = 0.1 Ω) (a) Estimated value of L s ; (b) Estimated value of Rs .
(a) (b) Figure 11 shows the current harmonic content and power factor when the value of inductance changes. As the inductance parameter decreasing, the online parameter estimation algorithm and least square method can well suppress the rise of current harmonic content and keep the rectifiers operating at high power factor. In the Figure 11, the value of ∆L represents the corresponding value of inductance variation. For instance, ∆L = 0.6 means input inductance is 3 mH as the original inductance value is 5 mH. From the Figure 11, it is concluded that the control performance of PWM rectifiers with data based estimation methods is better than the traditional MPDPC. The Bayesian methods can achieve the same control performance as the LSE method.
value of inductance variation. instance, means input inductance is corresponding 3 mH as the L the 0.6value of L represents the operating at high power factor. For In the Figure 11, original inductance value is 5 mH. From the Figure 11, it is concluded that the control performance value of inductance variation. For instance, L 0.6 means input inductance is 3 mH as the of PWM rectifiers with data based estimation methods is better than the traditional MPDPC. The original inductance value is 5 mH. From the Figure 11, it is concluded that the control performance Bayesian methods can achieve the same control performance as the LSE method. of PWM rectifiers with data based estimation methods is better than the traditional MPDPC. The Bayesian methods can achieve the same control performance as the LSE method. 26 24 26 22 24 20 22 18 20 16 18 14 16 12 14 10 128 106 84 62 4 2
Bayesian Estimation Traditional MPC Least Square Bayesian Estimation Traditional MPC Least Square
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Bayesian Estimation Traditional MPC Least Square Bayesian Estimation Traditional MPC Least Square 0.8 1.0
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(b) Figure 11. Current (a) harmonic content and power factor as inductance parameter changes with different MPDPC methods. (a) Current harmonic content under the parameter’s variation; (b) Power Figure 11.11. Current harmonic content and power factor as inductance parameter changes with different Figure Current harmonic content and power factor as inductance parameter changes with factor under the inductance value’s variation. MPDPC methods. (a) Current harmonic content under the parameter’s variation; (b) Power factor different MPDPC methods. (a) Current harmonic content under the parameter’s variation; (b) Power under the inductance value’s variation. factor under the inductance value’s variation.
28 26 28 24 26 22 24 20 22 18 20 16 18 14 16 12 14 10 128 106 84 62 4 2
12 Bayesian Estimation Traditional MPC Least Square Bayesian Estimation Traditional MPC Least Square
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Bayesian Estimation Least Square Bayesian Estimation Estimation Before Noise Bayesian Least Square Square Before Noise Least Bayesian Estimation Before Noise Least Square Before Noise
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THD% THD%
It us inevitable that the input current which is measured from sensors and ADC (Analog‐to‐Digital Converter) contains small noises. Figure 12 shows the input current harmonic content with traditional ItIt us inevitable that the input current which is measured from sensors and ADC (Analog‐to‐Digital us inevitable that the input current which is measured from sensors and ADC (Analog-to-Digital MPDPC, Bayesian estimation and least square algorithms when adding white noise to input AC current Converter) contains small noises. Figure 12 shows the input current harmonic content with traditional Converter) contains small noises. Figure 12 shows the input current harmonic content with traditional signal artificially as inductance parameter changes. As the inductance parameter changes, the THD of MPDPC, Bayesian estimation and least square algorithms when adding white noise to input AC MPDPC, Bayesian estimation and least square algorithms when adding white noise to input AC current input current will increase a little after adding white noise with the measuring current value. Form the current signal artificially as inductance parameter changes. As the inductance parameter changes, signal artificially as inductance parameter changes. As the inductance parameter changes, the THD of Figure 12a, the THD content input current with adding Bayesian estimation and LSE with the MPDPC the THD of input current will of increase a little after white noise with the measuring currentis input current will increase a little after adding white noise with the measuring current value. Form the better Form than the Figure THD content content with the traditional MPDPC the inductance variation. From the value. 12a, theof THD content of with input currentunder with Bayesian estimation and with Figure 12a, the the THD input current Bayesian estimation and LSE with the LSE MPDPC is Figure 12b, the robust of control performance with the Bayesian estimation is better than the estimation the MPDPC is better than the THD content with the traditional MPDPC under the inductance variation. better than the THD content with the traditional MPDPC under the inductance variation. From the with LSE when the white noise is added to the input AC current. From the Figure 12b, the robust of control performance with the Bayesian estimation is better than the Figure 12b, the robust of control performance with the Bayesian estimation is better than the estimation estimation with LSE when the white noise is added to the input AC with LSE when the white noise is added to the input AC current. current.
2
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(a) when adding white noise as inductance parameter (b) changes. (a) THD of Figure 12. THD of current Figure 12. THD of current when adding white noise as inductance parameter changes. (a) THD of current after adding noise; (b) THD of current before and after adding noise. current after adding noise; (b) THD of current before and after adding noise. Figure 12. THD of current when adding white noise as inductance parameter changes. (a) THD of current after adding noise; (b) THD of current before and after adding noise.
To verify the dynamic performance of online parameter Bayesian estimation algorithm for load disturbances, an instantaneous load-unload simulation experiment was carried out. The Power reference Pre f is from 1000 W to 2000 W and DC load R L is from 122.5 Ω to 61.25 Ω. Simulation results are shown in Figure 13. Active power has doubled and reactive power is unchanged. The DC bus voltage drops or rises at the moment of loading and unloading but the controller can control the DC voltage back to 350 V. From the Figure 13, the transient dynamics response is achieved by the MPDPC with the Bayesian estimation method.
reference
Pref
is from 1000 W to 2000 W and DC load
RL is from 122.5 Ω to 61.25 Ω. Simulation
results are shown in Figure 13. Active power has doubled and reactive power is unchanged. The DC bus voltage drops or rises at the moment of loading and unloading but the controller can control the DC voltage back to 350 V. From the Figure 13, the transient dynamics response is achieved by the Energies 2018, 11, 2490 13 of 22 MPDPC with the Bayesian estimation method.
Figure 13. Instantaneous load-unload simulation results by MPDPC with Bayesian estimation method. Figure13.Instantaneous load‐unload simulation results by MPDPC with Bayesian estimation (a) active power; (b) reactive power; (c) DC bus voltage. method. (a) active power; (b) reactive power; (c) DC bus voltage.
According to the results of digital simulation, the controlling performance of MPDPC with LSE According to the results of digital simulation, the controlling performance of MPDPC with LSE and Bayesian estimation methods is not so different in power factor and THD. But their controlling and Bayesian estimation methods is not so different in power factor and THD. But their controlling performance is much better than that of traditional MPDPC’s under the variation of parameters of ac performance is much better than that of traditional MPDPC’s under the variation of parameters of ac side of rectifiers. side of rectifiers. 4.2. Hard-in-Loop Simulation 4.2. Hard‐in‐Loop Simulation At the same time, in order to validate the correctness of the robust MPDPC (RMPDPC) algorithm At the same time, platform, in order ato validate the of by the MPDPC (RMPDPC) for the digital simulation semi-physical testcorrectness is carried out therobust Typhoon-600 semi-physical algorithm for the digital simulation platform, a semi‐physical test is carried out by the Typhoon‐600 test platform [31], which is a new and high-fidelity model-based power-electronics testingsystem. semi‐physical test block platform [31], which is a and model‐based power‐electronics The testing system diagram is shown innew Figure 14.high‐fidelity The real-time semi-physical test system can testingsystem.The testing system block diagram is shown in Figure 14. The real‐time semi‐physical achieve simulation time step of 0.1–1 µs, which can provide a test development environment and is test system achieve simulation time step 0.1–1 μsPWM , which can provide a test 14a, development closer to the can practice operating situation for theof aerospace rectifiers. In the Figure the basic environment and is closer to the practice operating situation for the aerospace PWM rectifiers. In the theoretical diagram of HIL is given, The MPDPC control with the Bayesian estimation is programmed Figure 14a, the basic theoretical diagram of HIL parameters, is given, The control with the Bayesian in the upper computer. Using the same simulation theMPDPC control algorithm is downloadinto estimation is programmed in the upper computer. Using the same simulation parameters, the the DSP-28335 embedded processor by the Ethernet interface and the aerospace three-phase PWM control algorithm is downloadinto the DSP‐28335 embedded processor by the Ethernet interface and rectifier model is built on the Typhoon-600 virtual prototype to test control algorithm of RMPDPC for the aerospace three‐phase PWM rectifier model is built on the Typhoon‐600 virtual prototype to test the uncertainty of the inductance parameter. In the Figure 14b, the real hardware configuration with control algorithm of RMPDPC for the uncertainty of the inductance parameter. In the Figure 14b, the some HIL software are given, through the HIL simulation operation, the simulation waveform can be real hardware with package some HIL software are given, HIL simulation processed by theconfiguration analysis software in the upper computer. Inthrough Typhoonthe platform, the input operation, the simulation waveform can be processed by the analysis software package in the upper inductance parameters are easily changed, the controller performance test results shown in Figure 15. computer. In Typhoon platform, the input inductance parameters are easily changed, the controller From the Figure 15, it is can be seen that the HIL simulation result is very similar with the digital performance test The results shown in Figure 15. From the Figure 15, it is can be seen that than the HIL simulation result. MPDPC with the Bayesian estimation’s control performance is better the simulation result is very similar with the digital simulation result. The MPDPC with the Bayesian traditional MPDPC under the variation of input inductance.
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estimation’s control performance is better than the traditional MPDPC under the variation of input inductance. estimation’s control performance is better than the traditional MPDPC under the variation of input estimation’s control performance is better than the traditional MPDPC under the variation of input Energies 2018, 11, 2490 14 of 22 inductance. inductance. Analog
AO1 AO2
ia、ib、ic
AI1 AI2 AI3
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Input IO Output AO3 IO ia、ib、ic AI1 Analog AO1AO1 ia、ib、ic Extension AI2 AI1 Analog Interface Analog AO2 AI2 Analog AI3 S1、S2、S3、S4 Output AO3AO2 DI1 AI3 Input DO1 Output AO3 Digital DI2 IOIO Board IOIO Circuit InputDO2 Digital DO3 Input DI3 DI4 DO4 Interface Extension Output Extension Interface JTAG S1、S2、S3、S4 DI1 DO1 JTAG S1、S2、S3、S4 Board DI1 Circuit DO2 DO1Digital Interface Board Digital Digital DI2DI2 Circuit DO3 DO2 Digital DI3 DSP28335 Interface Input Input
Output Output
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DI4 DI3 DI4
DI
DI DI
AO
Development Board
DSP28335 DSP28335 Development DevelopmentBoard Board
AO AO
JTAG Interface Converter Model
Emulator Emulator Emulator USB
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USB USB
Ethernet FCS-MPC Controller Typhoon 600 Upper Typhoon 600 Computer Control Sortware Target Machine FCS-MPC Controller Ethernet FCS-MPC Controller Ethernet Typhoon Upper Typhoon Typhoon600 600 Upper Typhoon600 600 Computer Target Computer Control ControlSortware Sortware TargetMachine Machine
(a)
Converter Model Typhoon600 Converter Model Target Machine
Control Software
Upper Computer
DSP28335 Development Board
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DSP28335
DSP28335 (b) Development Board Development Board
(a) (b) (a) semi‐physical test platform diagram and practicality. (b) (a) The basic theoretical Figure 14. Typhoon diagram for the HIL system; (b) The hardware configuration for the HIL‐platform. Figure 14. semi‐physical platform practicality. basic theoretical Figure Typhoon test diagram and (a) The Figure 14. 14. Typhoon Typhoon semi-physical semi‐physical test test platform platform diagram diagram and and practicality. practicality. (a) (a) The The basic basic theoretical theoretical diagram for the HIL system; (b) The hardware configuration for the HIL‐platform. diagram for the HIL system; (b) The hardware configuration for the HIL-platform. diagram for the HIL system; (b) The hardware configuration for the HIL‐platform.
(a)
(b)
(b) (a) (b) (a) Traditional MPC (b) Figure 15. AC side (a) voltage and current at inductance parameter changes Online parameter estimation algorithm of MPDPC. Figure side voltage and current at inductance parameter changes (a) Traditional MPC (b) Online Figure 15. AC side voltage and current at parameter changes (a) MPC (b) Figure 15. 15. AC AC side voltage and current at inductance inductance parameter changes (a) Traditional Traditional MPC (b) parameter estimation algorithm of MPDPC. Online parameter estimation algorithm of MPDPC. Online parameter estimation algorithm of MPDPC.
The HIL simulation result of input current’s harmonic distortion content and the power factor are shown in Figure 16 when the inductance parameter is changed. In the case of load disturbance, The HIL simulation result of input current’s harmonic distortion content and the power factor The HIL simulation result of input current’s harmonic distortion content and the power factor The HIL simulation result of input current’s harmonic distortion content and the power factor the dynamic performance of the online parameter Bayesian estimation algorithm is verified and the are shown in Figure 16 when the inductance parameter is changed. In the case of load disturbance, are shown in Figure 16 when the inductance parameter is changed. In the case of load disturbance, are shown in Figure 16 when the inductance parameter is changed. In the case of load disturbance, specific HIL experiment results are shown in Figure 17. Both active and reactive power can quickly the dynamic performance of the online parameter Bayesian estimation algorithm is verified and the the dynamic performance of the online parameter Bayesian estimation algorithm is verified and the the dynamic performance of the online parameter Bayesian estimation algorithm is verified and the track the reference value and the voltage of DC bus voltage fluctuates at the moment of loading and specific HIL experiment results are shown in Figure 17. Both active and reactive power can quickly specific HIL experiment results are shown in Figure 17. Both active and reactive power can quickly specific HIL experiment results are shown in Figure 17. Both active and reactive power can quickly unloading but it can be the kept constant at 350 within a short time. HIL of simulation results track the reference value and voltage of DC bus V voltage fluctuates at the The moment loading and track the reference value and the voltage of DC bus voltage fluctuates at the moment of loading and track the reference value and the voltage of DC bus voltage fluctuates at the moment of loading and verified the control effect of MPDPC with the Bayesian estimation methods. unloading but itit can be kept constant at 350 V within a short time. The HIL simulation results verified unloading unloading but but it can can be be kept kept constant constant at at 350 350 V V within within a a short short time. time. The The HIL HIL simulation simulation results results the control effect of MPDPC with the Bayesian estimation methods. verified the control effect of MPDPC with the Bayesian estimation methods. verified the control effect of MPDPC with the Bayesian estimation methods. 30
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15
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10
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Bayesian Estimation Traditional MPC Bayesian BayesianEstimation Estimation Traditional TraditionalMPC MPC
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2525
THD%
25
THD% THD%
3030
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1010
5
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Bayesian Estimation Traditional MPC Bayesian BayesianEstimation Estimation 0.6 0.8 Traditional MPC Traditional MPC 1.0 0.8 0.8
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ΔL ΔL ΔL (a) ΔL (b) platform under the Figure 16. Current(a) harmonic content and power factor in semi-physical(b) test (a) (b) variation of inductance value. (a) AC Current harmonic content; (b) AC side Power factor.
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Figure 16. Current harmonic content and power factor in semi‐physical test platform under the Figure 16. Current harmonic content and power factor in semi‐physical test platform under the variation of inductance value. (a) AC Current harmonic content; (b) AC side Power factor. Energies 2018, 11, 2490 15 of 22 variation of inductance value. (a) AC Current harmonic content; (b) AC side Power factor.
Figure 17. Instantaneous load‐unload experiment results. (a) Active power; (b) Reactive power; (c) Figure 17. Instantaneous load-unload experiment results. (a) Active power; (b) Reactive power; (c) DC Figure 17. Instantaneous load‐unload experiment results. (a) Active power; (b) Reactive power; (c) DC bus voltage. bus voltage. DC bus voltage.
5. Experimental Verification 5. Experimental Verification 5. Experimental Verification The The Performance of the designed control system is evaluated on a laboratory test‐bench, which Performance of the designed control system is evaluated on a laboratory test-bench, which is The Performance of the designed control system is evaluated on a laboratory test‐bench, which is shown in Figure 18. In all test, the Grid voltages and currents are directly measured by probes, shown in Figure 18. In all test, the Grid voltages and currents are directly measured by probes, is shown in Figure 18. In all test, the Grid voltages and currents are directly measured by probes, while the estimated of inductance are by obtained by a TMS F28335 DSP with sampling while the estimated values values of inductance are obtained a TMS F28335 DSP with sampling frequency while the estimated values of inductance are obtained by a TMS F28335 DSP with sampling frequency 100 KHz. A programmable ac source is used to simulate the aircraft grid conditions. The 100 KHz. A programmable ac source is used to simulate the aircraft grid conditions. The proposed frequency 100 KHz. A programmable ac source is used to simulate the aircraft grid conditions. The proposed scheme is tested in the aircraft electrical power system conditions, which the control schemecontrol is tested in the aircraft electrical power system conditions, in which the inputin voltage proposed control scheme is tested in the aircraft electrical power system conditions, in which the input voltage and frequency of PWM Rectifiers is 115 Vrms/360–800 Hz. and frequency of PWM Rectifiers is 115 Vrms/360–800 Hz. input voltage and frequency of PWM Rectifiers is 115 Vrms/360–800 Hz.
Figure 18. The hardware prototype system. Figure 18. the hardware prototype system. Figure 18. the hardware prototype system.
The prototype AC/DC converter is operating under the normal load for 1 h interval. After that, The prototype AC/DC converter is operating under the normal load for 1 h interval. After that, the values of inductance and resistance can be changed greatly. The Figure 19a, bhighlights the The prototype AC/DC converter is operating under the normal load for 1 h interval. After that, the values of inductance and resistance can be changed greatly. The Figure 19a, bhighlights the three-phase PWM rectifier’s input voltagecan (red) inputgreatly. current The (green) in phase A by using the the three‐phase PWM rectifier’s input voltage (red) and input current (green) in phase A by using the values of inductance and resistance be and changed Figure 19a, bhighlights the Bayesian estimation methods under the variable frequency conditions from the 360 Hz to 800 Hz. three‐phase PWM rectifier’s input voltage (red) and input current (green) in phase A by using the Bayesian estimation methods under the variable frequency conditions from the 360 Hz to 800 Hz. The dc-link voltage (blue) can be maintained at the 350 V. So, the controlling performance of PWM Bayesian estimation methods under the variable frequency conditions from the 360 Hz to 800 Hz. The dc‐link voltage (blue) can be maintained at the 350 V. So, the controlling performance of PWM rectifiers with the proposed controller can be verified. The dc‐link voltage (blue) can be maintained at the 350 V. So, the controlling performance of PWM rectifiers with the proposed controller can be verified. rectifiers with the proposed controller can be verified.
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(a)
(b)
Figure The input input Figure 19.The voltage and current waveform during variable frequency conditions. (a) The Figure 19. 19.The input voltage voltage and and current current waveform waveform during during variable variable frequency frequency conditions. conditions. (a) (a) The The waveform of rectifiers operated at 360 Hz; (b) The waveform of rectifiers operated at 800 Hz. waveform of rectifiers operated at 360 Hz; (b) The waveform of rectifiers operated at 800 Hz. waveform of rectifiers operated at 360 Hz; (b) The waveform of rectifiers operated at 800 Hz.
The The Figures Figures 20 20 and and 21 21 give give the the experimental experimental result result of of three-phase three‐phase PWM PWM rectifiers rectifiers under under the the proposal estimation algorithm based on Bayesian methods. According to the experimental result, proposal estimation algorithm based on Bayesian methods. According to the experimental result, the the output voltage can be regulated to the reference voltage 350 V and input voltage is in same phase output voltage can be regulated to the reference voltage 350 V and input voltage is in same phase with the input current and unity power factor can be realized when the load transient variation and with the input current and unity power factor can be realized when the load transient variation and input input voltage voltage disturbance disturbance happened happened in in aircraft aircraft power power electrical electrical power power system. system. Figure Figure 20a 20a shows shows responses for the voltage drop to steady state operation from the 115 Vrms to 80 Vrms and Figure 20 responses for the voltage drop to steady state operation from the 115 Vrms to 80 Vrms and Figure 20 bgives the result of the voltage swell from the 115 Vrms to the 150 Vrms. It is seen that the input current bgives the result of the voltage swell from the 115 Vrms to the 150 Vrms. It is seen that the input (green) and voltage (red) is in same phase and dc-link voltage (blue) can be kept in constant value at current (green) and voltage (red) is in same phase and dc‐link voltage (blue) can be kept in constant 350 V. These disturbances the load and power sourceand canpower make the variable power value at 350 V. These from disturbances from the load source can parameters make the ofvariable circuit such asof the input circuit inductance serious but Bayesian very estimation algorithm parameters power such and as resistance the input very inductance and the resistance serious but the with MPDPC can maintain the better performance. Figure 21a, bprovide the transient waveform for Bayesian estimation algorithm with MPDPC can maintain the better performance. Figure 21a, the input voltage (red), current (green) dc-linkvoltage voltage(red), (blue)current with the load power changed. The bprovide the transient waveform for and the input (green) and dc‐link voltage results verify the validity of proposed control schemes. (blue) with the load power changed. The results verify the validity of proposed control schemes.
(a)
(b)
Figure TheThe output dc voltage, input voltage currentand waveform underthe input voltagetransient Figure output dc input voltage current waveform underthe Figure 20.20. 20. The output dc voltage, voltage, input and voltage and current waveform underthe input input variation by the MPDPC with (a) The waveform input voltageduring drop; voltagetransient variation by the MPDPC with estimation. (a) waveform voltagetransient variation by the the Bayesian MPDPC estimation. with the the Bayesian Bayesian estimation. during (a) The The waveform during (b) The waveform during input voltage swell. input voltage drop; (b) The waveform during input voltage swell. input voltage drop; (b) The waveform during input voltage swell.
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Figure 21. The transient waveforms under the dynamic load power variation. (a) The transient waveform of load power from 2.07 KW to 1.08 KW; (b) The transient waveform of load power from 1.38 KW to 3.45 KW.
(a) (b) According to the different kinds of MPDPC control scheme, the experimental comparison was Figure variation. Figure 21. 21. The The transient transient waveforms waveforms under under the the dynamic dynamic load load power power variation. (a) (a) The The transient transient carried out. From the result of the THD content of AC input current in Figure 22a, we can infer that waveform of load power from 2.07 KW to 1.08 KW; (b) The transient waveform of load power from waveform of load power from 2.07 KW to 1.08 KW; (b) The transient waveform of load power from the MPDPC controller based on the Bayesian estimation has better performance than other two 1.38 KW to 3.45 KW. 1.38 KW to 3.45 KW. methods. The variation of inductance can be compensated by the proposal schemes. In Figure 22b, the power factor to can be different calculated from the input voltage and input current, it is comparison seen that the According the kinds of MPDPC control scheme, the experimental was According to the different kinds of MPDPC control scheme, the experimental comparison was MPDPC with the Bayesian achieve the high power infactor the inductance’s carried out. From the resultestimation of the THDcan content of AC input current Figurewhen 22a, we can infer that the carried out. From the result of the THD content of AC input current in Figure 22a, we can infer that value changed. MPDPC controller based on the Bayesian estimation has better performance than other two methods. the MPDPC controller based on the Bayesian estimation has better performance than other two Therefore, the experimental results show the feasibility and the good performance for the The variation of inductance can be compensated by the proposal schemes. In Figure 22b, the power methods. The variation of inductance can be compensated by the proposal schemes. In Figure 22b, proposedrobust model control scheme with fixed sampling frequency in is the aircraft factor can befactor calculated from the input voltage input current, it input is seen that the it MPDPC with the power can predictive be calculated from the and input voltage and current, seen that the the electrical power system. Bayesian estimation can achieve the high power factor when the inductance’s value changed. MPDPC with the Bayesian estimation can achieve the high power factor when the inductance’s value changed. Therefore, the experimental results show the feasibility and the good performance for the proposedrobust model predictive control scheme with fixed sampling frequency in the aircraft electrical power system.
(a)
(b)
Figure 22. The Bar diagram of performance index under three kinds of MPC control scheme. (a) The Figure 22. The Bar diagram of performance index under three kinds of MPC control scheme. (a) The THD of input current; (b) the power factor of rectifiers. THD of input current; (b) the power factor of rectifiers.
Therefore, the experimental results show the feasibility and the good From the above experimental results and theoretical analysis, the performance comparison of (a) (b) performance for the proposedrobust modelin predictive control scheme with fixedcan sampling frequency inin theTable aircraft three methods applied three phase rectifier converters be demonstrated 2, electrical which Figure 22. The Bar diagram of performance index under three kinds of MPC control scheme. (a) The power system. shows the excellent performance of the proposed methods in the aspects: algorithm complexity, THD of input current; (b) the power factor of rectifiers. analysis, the performance comparison of From the above experimental results and theoretical steady state and estimation value dynamics characteristics. three methods applied in three phase rectifier converters can be demonstrated in Table 2, which shows From the above experimental results and theoretical analysis, the performance comparison of the excellent performance of the proposed methods in the aspects: algorithm complexity, steady state three methods applied in three phase rectifier converters can be demonstrated in Table 2, which and estimation value dynamics characteristics. shows the excellent performance of the proposed methods in the aspects: algorithm complexity, steady state and estimation value dynamics characteristics.
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Table 2. Performance Comparison of Three MPC methods under the inductance variation. Schemes Performance Operation time of the algorithm THD of line current Power factor Converging rate for the estimated value Stability for the estimated value
Traditional MPDPC
MPDPC with LSE
MPDPC with Bayesian Estimation
10.8 uS 8.6% 0.92 no no
15.2 uS 6.7% 0.98 low low
16 uS 6.8% 0.99 fast high
In order to clarify the significance of this research, the experimental results from LSE proposed in the paper [15] are used to be benchmarks for comparison of application in aircraft electricalpower system. The main results and parameters are listed in Table 3. From the comparison in Table 3, it is inferred that the proposed Bayesianmethods also can be applied to the industrial scenario based on the quantitative reasoning. The data set demanding in one cycle based on Bayesian estimation is less than the LSE methods in paper [15]. Table 3. Results and parameters forModel Predictive Direct Power Control (MPDPC) with Bayesian estimation. Value Parameters
Results from the Paper [15]
Results from This Paper
20 KHz 60 Hz 333 260 V 680 W 2.34% 10 ms
100 KHz 360–800 Hz 125–270 350 V 1.2 KW 4.68% 6.7 ms
Sampling frequency Main AC Frequency Number of sampling points in one cycle DC link voltage Rated Power THD of AC current Converging time to the true value
6. Conclusions This paper presents a novel online parameter estimation algorithm for the aerospacethree-phase PWM rectifiers based on Bayesian estimation with MPDPC. In view of the problem that the uncertainty model parameters, the model parameter can be updated every cycle with the proposed algorithm by sampling input Ac current and voltage waveform.With the estimated value of input inductance in each sampling period, the future value of input current, active power and reactive power can be accurately predicted. The problem of controller performance degradation resulted from the uncertainty parameters can be overcome.The successful results of the proposed schemeswere verified by using simulation and experimental test results. It is concluded that the proposed algorithm can reduce the current harmonic contents and improve power factor of PWM rectifiers without additional sensors.Compared with the industrial frequency 50–60 Hz ground power system application, it us also applicable to the aircraft electrical power system. The robust MPDPC based on Bayesian estimation methods can achieved better performance than the LSE-based MPC in small samples data set situation. Using limited small samples data set, The Bayesian methods can estimate the parameters with faster convergence rate and stable value than the result from LSE methods. In the future research, the different priority probability distribution function such as unity distribution or student distribution function can be used to estimate the parameters of AC side for PWM rectifiers. The different results including the accuracy and transient performance of PWM rectifiers can be compared in the Aircraft electrical power system. Author Contributions: T.L. and W.T. proposed the Bayesian estimation MPC control strategy; W.T. build the simulation model for the three-phase PWM rectifiers converter in aircraft electrical power system; G.C., D.K. and T.L. performed theexperiments for rectifiers converter; W.T. and T.L. analyzed the simulation and experimental data; T.L., W.T. and G.C. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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Nomenclature
is
Parameters matrix dimension 3 × 3. Parameters matrix dimension (k − 1) × 1. Parameters matrix dimension (k − 1) × 3. Estimated parameters matrix (3 × 1) and error matrix [(k − 1) × 1]. Estimated parameters. h iT Grid voltage vector vsa vsb vsc . h iT Grid current vector isa isb isc .
v REC Ls Rs vsα , vsβ isα , isβ vdc P, Q Pre f , Qre f s a , sb , sc Ts AC ACVF ATRU ESR EKF FCS-MPC IGBT LSE TRU THD MPC MPDPC PWM UPS
Voltage vector of PWM rectifiers. Grid filter inductance. Grid filter resistance. Grid α, β axis voltage. Grid α, β axis current. rectifiers dc link voltage. Grid active and reactive powers. Grid active and reactive reference powers Switching state of the upper switch at each leg. Sampling time. Alternating Current Alternating Current Variable Frequency Auto-Transformer Rectifiers Unit Equivalent Series Resistance Extended Kalman Filter Finite Control Set-Model Predictive Control Insulated Gate Bipolar Transistor Least Square Estimation Transformer rectifiers Unit Total Harmonic Distortion Model Predictive Control Model Predictive Direct Power Control Pulse Width Modulation Uninterruptable Power Supply
A,B Y Φ θ, ε λ, µ, ν vs
Appendix 1. Bayesian Theorem and Multivariate Distribution Bayesian estimation is a method of mathematically formalizing and using a priori information as a random variable with a known distribution π (θ ), which is usually called the prior distribution of π (θ ). Assuming that the distribution density of population X is p( x |θ ), since the prior distribution of θ is known, the distribution density of population X can be regarded as the conditional distribution density of X at a given θ. The distribution density of population X is changed to p( x |θ ). Assuming that X = ( X1 , · · · · · · Xn ) T is a sample taken from the X. When a given sample value= ( X1 , · · · · · · Xn )T , the joint density (or the likelihood function) is n
q( x |θ ) =
∏ p ( xi | θ )
(A1)
i =1
Thus, the joint probability distribution of sample X and parameter θ is f ( x, θ ) = q( x |θ )π (θ )
(A2)
By the total probability formula, (2) is transformed to the following equation. f ( x, θ ) = q( x |θ )π (θ ) = m( x )h(θ | x )
(A3)
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q( x |θ )π (θ ) h(θ x ) = m( x )
(A4)
At a given sample X = x, h(θ | x ) is called the posterior distribution of θ, which is also the conditional distribution of θ. Them( x ) in (4) is expressed as a prior distribution of the sample X. Because m( x ) does not depend on theta, if m( x ) is omitted, the Bayesian formula can be written as the following equivalent form. h( θ | x ) ∝ q ( x | θ ) π ( θ )
(A5)
where the symbol “∝” indicates that there is only one constant factor independent of θ on both sides. The right-hand side of (5) is not a normal distribution density function but it is the main part of the posterior distribution h(θ | x ), called the kernel of h(θ | x ). For the multivariate variable Z = (z1 , · · · · · · , zn ), if its distribution density is 1/2 1 f (z1 , . . . . . . , zn ) = (2π )−n/2 Σ−1 exp{− [z − µ]0 Σ−1 [z − µ]} 2
(A6)
Then Z is subject to multivariate normal distribution and is written as Z ∼ Nn (µ, Σ). 2. Bayesian estimation with the MPDPC for PWM rectifiers In Bayesian estimation, there is a distribution of probabilities for any event. In (15), the sampling Ts frequency is known. Suppose Rs , Ls and v are normal distributions and independent of each other. The Bayesian estimation method is used to estimate the actual values of Rs , Ls and v at time instant k. Rs ∼ N (r, Σr ) Ls ∼ N ( L, Σ L ) ν ∼ N (0, Σν )
(A7)
It is assumed that the measured interferences ε 1 , · · · · · · , ε k−1 follow the multivariate normal distribution and are independent and identically distributed. ε ( j) ∼ Nm (0, Σε )
(A8)
For (8), as the j row of the measurement error vector, ε is denoted by ε ( j) , Y( j) can be expressed as: Y( j+1) ∼ Nm (Φθ )( j) , Σ
(A9)
Therefore, Y(2) , · · · · · · , Y(k) are also independent of each other and obey multivariate normal distribution. For (15), the output matrix Y likelihood function is derived as shown below: (k−1)/2 1 L(Y, Φ|θ, Σ) = (2π )−(k−1)/2 Σ−1 • exp − tr [Y − (Φθ )]0 [Y − (Φθ )]Σ−1 2 where Σ denotes the covariance matrix of the measurement error. mathematical equation. L(Y, Φ|θ, Σ) =
−1 3/2 Σ
(2π )−(k−1)/2
The (10) is transformed through
(k−4)/2 1 1 exp − trSΣ−1 exp − tr (θ − θˆ)0 Φ0 Φ(θ − θˆ)Σ−1 · Σ−1 2 2
where
(A10)
−1 θˆ = (Φ0 Φ) Φ0 Y 0 ˆ S = [Y − (Φθ )] [Y − (Φθˆ)]
(A11)
(A12)
According to the assumption that Rs , Ls and v are normal and independent, the a priori distribution of the coefficient θ obeys the multivariate normal distribution as follows: h(θ |Σ
−1
1 −1 3/2 0 −1 ) ∝ Σ exp − tr (θ − µ0 ) E(θ − µ0 )Σ 2
where µ0 denotes the expectation of the coefficient θ and E denotes the identity matrix.
(A13)
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According to the Bayesian theorem, the posterior distribution of θ can be expressed as the product of the likelihood function and the prior distribution. (k+2)/2 ∗ ∗ 1 h(θ, Σ−1 |Φ, Y) ∝ h(θ Σ−1 )• L(Φ, Y θ, Σ−1 ) ∝ Σ−1 exp − tr (θ − θ )0 (E + Φ0 Φ)(θ − θ ) Σ−1 (A14) 2 where
∗
θ = ( A + Φ0 Φ)
−1
−1 ( Eµ0 + Φ0 Φθˆ) = ( E + Φ0 Φ) (µ0 + Φ0 Φθˆ)
(A15)
(15) shows the posterior distribution of θ, which indicates that the posterior distribution of θ also obeys multivariate normal distribution, that is ∗
h(θ |Φ, Y) ∼ MN3×1 [θ, Σ ⊗ (E + Φ0 Φ)
−1
]
(A16)
∗
Thus, the expectation of the posterior distribution of θ is θ, which is the Bayesian estimate of θ. ∗
−1 −1 θˆB = θ = (E + Φ0 Φ) (µ0 + Φ0 Φθˆ) = (E + Φ0 Φ) (µ0 + Φ0 Y)
(A17)
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