On-line self-tuning adaptive control of an inverter in a grid-tied micro-grid

Electric Power Systems Research 178 (2020) 106045

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On-line self-tuning adaptive control of an inverter in a grid-tied micro-grid Femina M. Shakeel , Om P. Malik ⁎

T

Department of Electrical and Computer Engineering, University of Calgary, Canada

ARTICLE INFO

ABSTRACT

Keywords: Grid connected inverter Self-tuning control Recursive least squares identification Pole shifting control Micro-grid PI control

Three phase grid connected inverters are commonly used in micro-grids as interphase between dc and ac systems. Traditionally, grid connected inverters (GCI) are controlled through standard decoupled d–q vector control mechanisms using proportional integral (PI) controllers. However, recent studies have indicated their limitations when applied to dynamic systems. For instance, PI controllers are unable to give optimal performance when the system operating conditions change. How such restrictions can be overcome using an adaptive self-tuning controller instead of a PI controller is investigated in this paper. A controller based on recursive least squares identification and pole shifting control is designed and implemented on the GCI in a micro-grid test system. Performance of the conventional and proposed controllers is compared and evaluated. The proposed controller outperformed the conventional controller in all test cases simulated, and showed robust and stable performance for a wide range of operating conditions.

1. Introduction Three phase voltage-source pulse-width-modulated converters are widely used in micro-grid applications as interphase between dc and ac systems. Among the various converter configurations, a grid connected inverter (GCI) with bi-directional power flow capability is used for interconnecting battery storage systems in micro-grids. A typical microgrid configuration with a GCI is shown in Fig. 1. Traditionally, GCIs are controlled using the standard d–q vector control approach in the synchronous reference frame using proportional integral (PI) controllers [1]. The control design is based on the linear model of the system at some specific operating point. Recent studies have indicated some inherent draw-backs of the conventional control structure. In Ref. [2], it was found that this approach is sensitive to model uncertainties. The controller parameters have to be re-tuned in situations where wide dispersion of the system parameters is expected, as concluded in Ref. [3]. The importance of estimating the grid system parameters such as grid filter impedance is emphasized in Ref. [4] in order to improve the performance of the conventional control method for micro-grid applications. Various studies have been conducted to overcome the short-comings of the standard vector control approach which uses fixed gain PI controllers. An adaptive control approach was proposed recently that employs a direct current control (DCC) strategy [1]. The method uses intelligent control principles to reduce the error between the desired and actual d- and q-axis currents through an adaptive tuning process. The

⁎

inner current loop PI controllers are retained in the system and are tuned based on intelligent control principles. The major challenge in this approach is that there exists no well-established systematic approach for tuning the controller PI gains and hence an optimal DCC is difficult to obtain. A predictive current control (PCC) is proposed in Ref. [5], but the controller becomes unstable if the programmed filter inductance becomes different from its actual value or if there is an inaccuracy in measurement. Since filter parameters vary along with inverter operation, it is difficult to achieve an optimal performance. An internal model control was employed recently [6] to overcome the drawbacks of PI and PR (proportional resonant) control methods. Although this method has a simple control structure, faster response and stronger robustness, it is quite sensitive to model deviations and is hard to realize in practice. A harmonic impedance reinforcement control method was proposed in Ref. [7] in-order to reduce the current harmonics induced by the non-linear loads at the point of common coupling. The main objective of this control scheme is to enhance the harmonic rejection ability of the droop-controlled inverter. The last decade has seen a growing interest in applying artificial intelligence (AI) techniques such as fuzzy logic (FL) and artificial neural networks (ANN) for on-line control of power systems. A neural network based vector control is proposed for GCI in Ref. [8]. With the ANN characteristic of a ‘black box’ like description, it is difficult for an outside user to identify the control process. The authors of Ref. [9] have proposed a fuzzy logic based control design for a three phase voltage source inverter (VSI). Here the inner current loop PI controllers are

Corresponding author at: Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive NW, Calgary AB T2N 1N4, Canada. E-mail address: [email protected] (F.M. Shakeel).

https://doi.org/10.1016/j.epsr.2019.106045 Received 13 February 2019; Received in revised form 7 June 2019; Accepted 23 September 2019 Available online 14 October 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

feedback loop and the type number of the system is increased to ensure steady state correspondence with the reference signal. This method is adopted for the current study due to its simplicity and ease of implementation. The main contributions from this work are: (1) A novel adaptive control based on RLS system identification and PS control is proposed for a grid connected inverter. (2) The proposed control scheme accommodates the reference signal tracking problem by including an integrator in the feedback loop. The paper is structured as follows: Section 2 presents the GCI model in synchronous reference frame and summarizes the conventional vector control method. The proposed self-tuning control is explained in detail in Section 3. Performance of the ST controller is evaluated and compared with that of the conventional method in MATLAB/Simulink simulation platform and the results are presented in Section 4. Finally, the paper concludes with a summary of the main points. 2. GCI model and control in synchronous reference frame A schematic diagram of the GCI with a dc-link capacitor on one side and a three phase voltage source representing the voltage at the point of common coupling (PCC) on the other side is shown in Fig. 2. The three phase voltage balance equation across the inverter filter is given by:

Fig. 1. Micro-grid configuration with a bi-directional GCI.

replaced with FL based self-tuned PI controllers. However, the task of constructing the rule base, selection of membership functions and parameter tuning increases the complexity of control design. A self-tuning (ST) control, that addresses these deficiencies by exploiting the principles of indirect adaptive control theory, is presented in this paper. The proposed controller uses an on-line recursive least squares (RLS) algorithm for system identification and pole shifting (PS) control for control synthesis. The RLS algorithm is commonly used for system identification due to its simplicity. A variable forgetting factor is incorporated in the algorithm to discount the importance of older data to give emphasis on new information [10]. This enhances the ability of the identifier to better track the change in operating conditions of the actual system and improve numerical stability. The plant model is estimated at each sampling interval. A linear pole shift controller generates the control signal based on the identified plant parameters. Pole shift control needs to tune a single pole shift factor, , which shifts the open loop poles towards the center of the unit circle in z-plane. This increases the stability of the closed loop system. The self-tuning control technique based on pole shift control has previously been implemented in adaptive power system stabilizers (APSS) [10–12] and some flexible ac transmission systems (FACTS) devices [13,14]. However, this is the first time it is being developed for a GCI. In the previous studies, the self-tuning controller was used as a transient gain stabilizer to solve the regulator problem of reducing the disturbance upon the output. The steady state performance was maintained by an automatic voltage regulator in APSS and PI controller in FACTS devices. ST controller was used as an additional/supplementary controller in the system. The novelty of the control approach proposed here comes from the fact that the ST controller invariably replaces the two PI controllers in the inner current loop of the conventional vector control structure. The idea is to convert the regulator problem into a reference tracking problem. Some approaches have been reported in the literature for incorporating a reference input in the self-tuning algorithm. In Ref. [15], the regulator equation in pole shifting control is modified by incorporating a reference input. A direct classical control approach was implemented in Ref. [16] by including servo compensators into the control scheme by the set-point feed forward method. This method allows the tuning of both the regulator and servo properties of the closed loop system at the cost of increased computational effort. A self-optimizing pole shift control was proposed in Ref. [11] which combines the quintessence of pole assignment algorithm for stability and minimum variance algorithm for reference tracking. Another approach was proposed in Ref. [17] where digital integrators are incorporated in the

ia ia va vta d vb = R ib + L ib + vtb dt vc vtc ic ic

(1)

where va, vb, vc are the three phase grid voltages, ia, ib , ic are the three phase grid currents, vta, vtb, vtc are the terminal voltages of the GCI, and L and R are the effective inductance and resistance of the inverter filter. Using Clarke’s and Park’s transformation matrices, the three phase vectors can be transformed from abc reference frame to dq reference frame. In the dq reference frame (synchronous reference frame), (1) becomes (2) where is the angular frequency of the grid voltage.

vd id d id vq = R iq + L dt iq + L

iq vtd + v tq id

(2)

Here vdq, idq, and vtdq are the PCC voltage, grid current and inverter output voltage in dq reference frame, respectively. If the d-axis of the reference frame is aligned along the PCC voltage position, vd will be a constant dc variable and vq voltage vector will be zero. The instantaneous real and reactive power injected into or absorbed from the ac system is given by:

p=

3 3 (vd id + vq iq) = vd id 2 2

(3)

q=

3 (vq id 2

(4)

vd iq) =

3 vd iq 2

Eqs. (3) and (4) show directly the possibility of controlling the active and reactive powers by adjusting the dq grid currents id and iq , respectively. In other words, one can control the active power (i.e., regulate the dc bus voltage) by controlling the d-axis current (id ) and control the reactive power (i.e., regulate the voltage at PCC) by controlling the q-axis current (iq) . The conventional standard vector control using 4 PI controllers in a nested loop is shown in Fig. 3 [18]. The control structure consists of two

Fig. 2. Schematic diagram of a grid connected voltage source inverter. 2

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

B (z 1) = b1 z

1

+ b2 z

2

+

+ nb z

(7)

nb

where na and nb are the order of the polynomials A and B, respectively. The aim of the RLS identification is to find the coefficients of A and B polynomials. Re-writing (5) in a form suitable for identification, gives: T y (t ) = ˆ (t ) (t ) + e (t )

(8)

ana b1 b2 bnb ] is the parameter vector, where ˆ (t ) = [ a1 a2 y (t na ) u (t 1) u (t 2) u (t nb )]T and (t ) = [ y (t 1) y (t 2) is the measurement variable vector. The prediction error, (t ) is given by: T

Fig. 3. Conventional dq vector control structure using PI controllers.

outer voltage control loops and two inner current control loops. The daxis outer loop controls the dc bus voltage and inner loop controls the active ac current. Similarly, the q-axis outer loop regulates the ac voltage magnitude by adjusting the reactive current, which is controlled by the q-axis inner current loop. Also, dq decoupling terms L and feedforward voltage signals are added to improve the performance during transients.

ˆ (t ) = ˆ (t

na

P (t 1) (t ) (t ) + T (t ) P (t 1) (t )

P (t ) =

1 [1 (t )

(11)

1) (t )]

(12)

K T (t ) (t )] P (t

1)

(13)

0

(t

1) + (1

0 ),

2 (t )

1

0

}

(14)

1 and 0 is a positive constant. Eq. (14) can be exwhere 0 0 plained as follows: If the system is running in steady state, (t ) will be very small or close to zero. This forces to be near or equal to 1. Upon the occurrence of a disturbance, (t ) decreases as (t ) increases. This in turn is reflected on P (t )and K (t ) matrices, which improves the parameter tracking property of the identifier. 3.2. Pole shift controller

where y (t ), u (t )and e (t ) are the system output, input and white noise, respectively. z 1 is the delay operator. The polynomials A and B are defined as:

+ na z

ˆ (t

1) + K (t )[y (t )

K (t ) =

(t ) = min{

(5)

A (z 1) y (t ) = B (z 1) u (t ) + e (t )

+

(10)

Here (t ) is the time varying forgetting factor, P (t ) is the error covariance matrix and K (t ) is the modifying gain vector. The effect of forgetting factor in (13) is that P (t ) and K (t ) are kept large if < 1. This makes the algorithm to always remain alert to track the changes in system dynamics. A variable forgetting factor is obtained by using the following equation [11]:

Recursive least squares identification has the advantages of simple calculation and good convergence properties, and hence it is a desired technique for use in the design of self-tuning controllers for real time applications. To enhance the ability of the identifier to track the operating conditions of the actual system and to avoid the parameter burst-out, a forgetting factor is used to discount the importance of the older data [11]. The plant is assumed to be a discrete Auto Regressive Moving Average (ARMA) model of the form:

2

1 (t ) 2 N k=1

The system parameter vector ˆ (t ) can be calculated by the following set of recursive equations [11]:

3.1. System identification using RLS algorithm

+ a2 z

(t ) that

N

J (N ) =

The major drawback of the conventional vector control approach discussed in Section 2 is that it is solely based on PI controllers. PI controllers create the need for frequent tuning and hence are inefficient in providing optimal performance under varying operating conditions. The proposed controller is adaptive and self-tunes itself in real-time according to the operating conditions. The self-tuning property lies in its on-line identifier which is used to estimate the parameters of the plant. The control is obtained based on the estimated parameters. A schematic overview of the ST controller is shown in Fig. 4. The plant output y (t ) is the grid injected current idq (t ), and the plant input u (t ) is the control voltage vtdq (t ) fed into the pulse width modulated (PWM) switching circuit of the inverter.

1

(9)

The RLS algorithm determines the most likely value of minimizes the sum of squares of the prediction error:

3. Self-tuning (ST) control

A (z 1) = 1 + a1 z

ˆT (t ) (t ) = e (t )

yˆ (t ) = y (t )

( t ) = y (t )

The most important motivation for using the pole shift controller is its ability to increase the system stability margin. Under the pole shifting control strategy, the poles of the closed loop system are shifted towards the center of the unit circle in the z -domain by a factor which is less than 1. For a small pole shift (close to 1), a small control output may be required and a small increase in stability margin will result. Although a large pole shift can result in a large increase of stability margin, it often forces the control to hit its limits. The control algorithm finds the best value of which gives an acceptable increase in stability margin without violating the control constraints. The procedure for deriving the pole shift algorithm [10] is given below. For the system modeled in (5), assume the feedback loop has the form:

(6)

u (t ) = y (t )

G (z 1) F ( z 1)

(15)

where

F (z 1) = 1 + f1 z Fig. 4. Self-tuning controller for grid connected inverter.

G (z 1) = g0 + g1 z 3

1

+ f2 z 1

2

+ g2 z

+ 2

+

+ nf z

(16)

nf

+ng z

ng

(17)

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

and nf = nb 1 , ng = na 1. From (5) and (15), the characteristic equation of the closed loop system can be derived as: (18)

T (z 1) = A (z 1) F (z 1) + B (z 1) G (z 1)

The pole shifting algorithm makes T (z take the form of A (z 1), but the pole locations shifted by a factor . i.e., 1)

(19)

A (z 1) F (z 1) + B (z 1) G (z 1) = A ( z 1)

Rearranging (19) in matrix form and comparing the coefficients gives:

1 a1

0 1 a1

0 0

ana 0 ana 0 0

b1 b2

0 b1 b2

0 0 0 b1 b2

1 bnb a1 0 bnb 0

0

ana

0

f1

×

a1 ( a2 (

fn f g0

=

ana (

1) 1)

2

na

Fig. 5. Modified controller block diagram for reference tracking.

u (t ) = u (t ) u (t 1) as the control signal. Also, the regulator in (15) is replaced by (28):

1)

0

0

gn g

bnb

0

(20)

u (t ) =

or in matrix form: If parameters ai s and bi s are identified at every sampling period, and pole shift factor is known, the control parameters Z = [{fi }, {gi}] can be solved using (21). This, when substituted in (15), gives: (22)

u (t , ) = X T (t ) Z ( ) = X T (t ) M 1L ( ) where X (t ) = [ u (t

1)

u (t

2)

u (t

nf )

y (t )

y (t

1)

y (t

2)

Assuming the practical control constraints are given by:

u min

u (t )

y (t

4. Simulation results The designed self-tuning controller is introduced in the grid connected inverter of the micro-grid test system shown in Fig. 1. The model order to be estimated was taken as 3. The diagonal elements of the initial co-variance matrix P were taken as 10000, the initial pole shift factor 0.7 and the forgetting factor 1. Initially, the designed system was simulated until parameters converged and the steady state conditions were reached. Then the following performance evaluations were done and the results were compared with that of a conventional PI controller. PI controller parameters are given in Appendix A.

the control margin is defined as:

u=

umax u , u 0 u u min , u < 0

(24)

If u < 0 , it means that one of the control limits has been hit and the pole shifting factor has to be increased. If u > 0, it means that the control limits have not been reached and the pole shifting factor can still decrease if (t ) > 0. Modification of α (t) can be formulated as follows: The control at time t , u (t ) , is an implicit function of the pole shift factor (t ) at that instant of time as shown in (22). The sensitivity function can hence be computed as:

u

= XT

Z

= X TM

1

L

4.1. Step change in dc bus voltage reference A step change in dc bus voltage reference was applied at 3 s simulation time. The results are shown in Figs. 7 and 8. Although both PI and ST controllers provide quick and well damped response in tracking the changed voltage reference, performance of the ST controller is slightly better. It can be also seen that when the disturbance is applied at 3 s, there is ringing of the PI controller output (modulation index of the PWM inverter). However, there is no such problem in the case of the proposed self-tuning controller as there is a smooth variation of controller output in this case. The control signal along with the change in pole shift factor is shown in Fig. 8.

(25)

For the control margin u calculated in (24), the modification of pole shift factor is given by [10]:

=

K

u

1

u

(26)

where K is a positive constant chosen to avoid excessive variation of (t ). The variable pole shift factor, (t ), is then given by:

(t + 1) =

(t ) +

(28)

ng )]T

(23)

umax

yref (t ))

The modified block diagram for reference tracking as applied in the d-axis current control loop of the GCI is shown in Fig. 5. The q-axis current control loop has a similar structure. The inner current loop PI controllers in the conventional control scheme are replaced by two distinct self-tuning controllers and the overall control diagram is shown in Fig. 6. GPWM (s ) is the PWM gain containing the computation and modulation delay.

(21)

MZ ( ) = L ( )

G (y (t ) F

(27)

4.2. Three phase line to ground short circuit fault test

The variable pole shifting control algorithm described above has been normally used to solve the regulator problem given by (15). The control objective in a grid connected inverter is to solve a servo problem by tracking the reference grid current set by the outer voltage control loop (explained in Section 2). A reference input can be included in the control algorithm by incorporating integrators in the feedback loop. The placement of integrators in the feedback loop is explained in detail in Ref. [17]. To extend the self-tuning regulators to the case with constant and slowly changing reference values, it is assumed that the system contains an integrator or that one is introduced by using

A three phase short circuit fault was applied to one of the transmission lines at the PCC end of the inverter. The fault occurs at 3 s simulation time and is cleared at 3.1 s. Variation in dc bus voltage along with the change in modulation index of the inverter is shown in Fig. 9 and the variation in grid current for phase A of the three phase PCC voltage is shown in Fig. 10. Variation of both the dc bus voltage and grid current shows that the self-tuning controller is able to give better performance under transient conditions, when compared with that of the PI controller. 4

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

Fig. 6. Overall self-tuning control structure for GCI.

Fig. 7. Reference tracking for a step change in dc bus voltage reference and corresponding variation in modulation index.

Fig. 9. Variation in dc bus voltage and modulation index during three phase fault.

Fig. 8. Variation of control signal and pole shift factor.

Fig. 10. Variation in grid injected current during three phase fault.

5

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

Fig. 14. Variation in grid injected current during system parameter changes.

Fig. 11. Variation of grid injected current during line switching test.

Fig. 12. Variation of ‘a’ parameters of the d-axis current control loop.

Fig. 15. Response to step change in ac bus voltage reference.

Fig. 13. Variation of forgetting factor.

Fig. 16. Variation in grid injected current during line switching test.

4.3. Line switching test

for the controlled variable to settle down to its steady state value when a PI controller is used. The grid current oscillations are damped out quickly when ST controller is used. The ‘a’ parameters shown in Fig. 12 are the identified plant parameters. During steady state operation, there is not much variation in these parameters. When there is a change in line impedance or other operating conditions of the actual physical system, the model parameters change and converge to a new value to reflect this change in the system. Similarly, the forgetting factor

A line switching test was simulated by switching on a 23 km distribution line between the inverter output terminals and PCC, at 2 s simulation time. The response curve of grid injected current is shown in Fig. 11. The variation of ‘a’ parameters of the d-axis current control loop and the corresponding forgetting factor is shown in Figs. 12 and 13, respectively. The grid current response in Fig. 11 shows that it takes more time 6

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

found to be deviating from its nominal value and gradually decreasing (Fig. 17). The physical causes for unstable operation of a GCI are the changes in system parameters and unbalanced/distorted ac system conditions [8]. From the tests 4.5 and 4.6, the superior performance of a selftuning controller to maintain stability, over fixed gain controllers is quite evident. The ST controller was able to avoid an unstable operation of the system under varying system conditions. This can be attributed to the online system identification based control which updates the system model parameters as the operating conditions change. Also, the selfoptimizing pole-shift control guarantees the closed loop stability by restricting the closed loop poles within the unit circle in the z-domain. 5. Conclusions An adaptive self-tuning controller for grid connected inverter based on on-line system identification and variable pole shift control is presented in this paper. The two PI controllers in the inner current control loop of the conventional vector control structure are replaced by two self-tuning controllers. The pole shift control algorithm previously used as a regulator has now been modified into a reference tracking controller. The designed controller is evaluated by performing different simulation studies on a micro-grid test system. Since the models as well as the controls are tuned online, the self-tuning controller shows better performance when compared with the conventional PI based vector control. There is significant improvement in system stability, transient response and robustness with the proposed ST controller. A well-tuned PI controller can give the same performance as a selftuning controller just for the specific power system configuration and the range of operating conditions for which it is designed off-line. A self-tuning controller can perform robustly over a wide range of operating conditions as testified by the simulation results. Being self-tuning, it determines its parameters within a few milliseconds at start and requires no off-line design of parameters. The self-tuning controller can be further verified in a real time, hardware-in-the loop simulation platform. Also, the possibility of using a Multiple Input Multiple Output (MIMO) structure instead of two individual control loops can be thought of as a future research topic.

Fig. 17. Variation in frequency during line switching test.

changes when disturbance is applied, in order to track the changed parameters as shown in Fig. 13. It settles back to 1 when steady state operating conditions are reached. 4.4. Change in grid filter inductance The modeling inductance was varied to test the robustness of the controller during system parameter changes. The modeling inductance was gradually increased until noticeable variations were found in the response of the system. It can be observed from Fig. 14 that the PI controller performance deteriorates after 3 s whereas the ST controller is able to maintain the optimal performance. 4.5. Unstable operation during step change in ac bus reference voltage The designed system was tested for a step change in the ac bus reference voltage at 4 s simulation time. The system response curves are shown in Fig. 15. The PI controller was unable to track the change in reference voltage as can be seen from Fig. 15. The ST controller showed smooth tracking of the reference voltage.

Conflict of interest

4.6. Unstable operation during line switching test

None.

A 25 km long distribution line was switched on at 2 s simulation time. The system responses are shown in Figs. 16 and 17. When a selftuning controller is used, the grid current settles to its steady state value after an initial transient as can be seen from Fig. 16. When a PI controller is used, the grid currents start oscillating at high amplitudes showing an unstable operation. The frequency of the grid voltage is

Acknowledgements This work was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada.

Appendix A Table A1

Table A1 PI controller parameters. VDC regulator gains

Current controller gains

KP

KI

KP

KI

0.5

500

1

50

7

Electric Power Systems Research 178 (2020) 106045

F.M. Shakeel and O.P. Malik

References

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