APPLYING A NONLINEAR PID IN A SINGLE-PHASE PLL CONTROL

Luciane Agnoletti dos Santos1, Maurício dos Santos Kaster2, Sérgio Augusto Oliveira da Silva3

Federal University of Technology - Paraná Cornélio Procópio, Brazil

luciane.agnoletti@yahoo.com.br1, mkaster@utfpr.edu.br2, augus@utfpr.edu.br3

Abstract—This work presents an adaptive nonlinear PID control applied in a single-phase Phase-Locked Loop algorithm. In the design methodology adopted, the gains obtained by using conventional PID tuning techniques, are applied to a nonlinear function that determines dynamically the PID gains, which is afunction of the input signal, usually the output-to-reference error. This adjustment provides a more efficient PLL operation from the viewpoint of performance and robustness as compared to conventional PID. The response time for synchronism has been reduced by 2.3 times in relation to linear PI.L.

Keywords—nonlinear PI control, PLL

I. INTRODUCTION

The development of power electronics equipment dedicated to processing power has greatly evolved, improving performance, controllability and cost, while still allowing the execution of tasks that were previously not possible. However, most of these devices have the disadvantage of not working as line sources, providing non-sinusoidal currents, adding to the grid undesired harmonics.

Several reference institutions, such as IEEE and IEC have dealt with this problem and began to set limits for the generation of harmonics. Thus, electrical equipments used in several industrial segments are going increasingly stringent requirements aiming to minimize the problems of power quality[1].

The power quality is not only related to the quality provided by the utility but also to the quality of current drawn by connected equipment to utility power. However, you can add devices to the electrical grid that can compensate in some way these harmonic distortions and even be able to compensate reactive power [2].

These devices can be series and/or parallel active filters. One manner of generating the compensation of harmonic components and/or reactive is, for example, the SRF method [7]. This method requires a reference angle, in this case the angle of the grid, for the currents/voltages to be injected in synchronism and phase angle to the grid. A PLL is used to obtain such reference [3].

The PLL requires control in order to get in sync with the reference voltage. The control generally used for this algorithm is the PI control. Although effective, this control takes six or more cycles of the electric grid to achieve the exact sync. In

electrical systems, time is a factor for preventing the system to collapse.

The faster the devices come into sync, the faster the system respond in correcting disturbances. Thus, it is important trying to improve the dynamic controller. Much types of adaptive PI are proposed, employed in chemical processes [8], for servomotor control position [9] and for PLL control for three-phase systems [10].

The purpose of this paper is the application of nonlinear gain control of the PLL to improve the performance of the controller. The technique also allows increasing the robustness of the control system. The determination of the nonlinear function of the gains can be achieved by first designing the linear gains in the conventional manner using linear control theory, and then adjusting the parameters of the nonlinear function as needed to achieve the desired performance. This leaves room to the designer to choose the best way of inserting the curve such that these gains are applied according to the instantaneous value of the error. This approach allows obtaining a more efficient controller, decreasing the overshoot, settling time and rise time.

II. DESCRIPTION OF PLL AND NONLINEAR CONTROL

This section will present the studied PLL algorithm and the technique for obtaining the nonlinear gains.

A. A Single Phase PLL Algorithm (pPLL) The PLL compares the input signal phase to the phase

signal derived from oscillator output and adjusts its frequency to keep the combined phases. The PLL's algorithms are largely used in radio, telecommunications, computers and other electronics applications, with output frequencies from fractions of Hertz to several gigahertz.

The single-phase PLL system implemented in this work is based on the theory of three-phase instantaneous active power, discussed in [4]. In this work, a single-phase PLL, also known as pPLL, has implemented as the block diagram presented in Figure 1. The pPLL algorithm, which is described in detail in [5], is based on the theory of three-phase instantaneous active power.

Thus, a fictitious three-phase system must be created,which can be represented by a biphasic stationary axes system(αβ). The vα component (Fig. 1), which represents the measured voltage vs, is delayed in π/2 rad out of phase in

2012 IEEE International Conference on Power Electronics, Drives and Energy Systems December16-19, 2012, Bengaluru, India

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relation to vs, thus obtaining a fictitious quadrature component v'β.

The input signals used were: pure sine wave and sine wave amplitude variation. In the final version, other inputs, like harmonics disorders, will be tested.

Figure 1. Block diagram for the pPLL method

We chose to use the same gains obtained in [5] to compare them to the proposed nonlinear PI control. The gains used were

500 /kp rad Ws and 225000 /ki rad Ws .

B. Nonlinear PI Control By analyzing the influence of proportional and integral

gains in the system response, it is noticed that the higher the gain pk is, the system error is smaller when the system is stable compared to the reference signal or under the influence of constant disturbance. Thus, the proportional controller can be used satisfactorily in the low-order processes with open loop response well damped, where the optimum value would speed up the transient response.

However, if the system is of higher order, the high gains are able to reduce the steady-state error, but the system will lead to stability problems or transients with large oscillations. For a ikwith small value results in an accelerated transient response, however, in certain systems, the integrator can cause amplification of the natural oscillation, [6].

The nonlinear PI control is proposed based on the theory [6] but using Gaussian curves. The equation related to the proportional gain is presented in(1).

2.1 1 0(( .) ) e pk k kf (1)

where p is the control variable of the concavity of the

Gaussian curve of the gain, is the error in the reference, 1k

is the top limiting value and 0k is the minimum limiting value of gain. The parameter ,can be calculated by establishing areference value of ( ) and its correspondent gain as a percentage of the interval 1 − 0where:

1

1 0

2 2

max max

( )ln

ln(1 )

k fk k

p (2)

When 1 0k k , has the concavity of the Gaussian curve side

up, as presented in Fig. (2). When 1 0k k , has the concavity of the Gaussian curve side down, as presented in Fig. (3)

In this system, the maximum error value will be max 1and the value of 0,9 represent the test value chosen in order to obtain the value of the concavity p. Using(2), results:

2

ln(1 0,9)2,3026

1p (3)

Figure 2. Gain function representation (a) ( )p

k ; (b) ( )i

k .

The gain curves used are presented in Fig. 3.

Figure 3. Gain function representation (a) ( )pk e (b) ( )ik .

III. SIMULATION RESULTS

The first simulation was performed taken as input signal apure sine wave of unit amplitude, shown in Fig. 4(a). As presented in the theory of the single-phase PLL, the same figure shows the fictitious voltage vβ delayed on / 2 radcompared to the input voltage.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1400

600

800

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-line

ar K

p

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kp

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3x 10

4

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ki

It is observed that using the conventional PI, synchronism occurs at time of 0.07 seconds, presented in Fig. 4(b), while utilizing the nonlinear PI the time is reduced to 0.03 seconds, as presented in Fig. 4(c), reducing 2.3 times the conventional PI transient response. The Fig. 4(d) presents the error signal generated and the desired reference, proving the time values already mentioned above.

The power grid is subject to suffer some kinds of disturbances. Voltage sags and swells disturbances are very common and will be our first target. In order to evaluate the control performance under disturbances on the power grid, a sag with amplitude 30% was considered at time 0.2 seconds and removed at 0.25 seconds, as presented in Fig. 5(a).

The Fig. 5(b) presents the phase error using the linear PI and nonlinear PI, and it can be observed again that even under the influence of disturbances on the power grid amplitude, nonlinear PI has a better performance than linear PI.

Figure 4. (a) Input Voltage Vα (reference voltage) with Vβ delayed in π/2 rad(b) PLL synchronism response using PI (c) PLL synchronism response using

nonlinear PI (d) Phase error for both.

Figure 5. (a) Input voltage Vα under sag disturbance (30%); (b) Phaseerror relative to reference using PI Linear and nonlinear PI

Similarly to the previous disturbance, a swellwith 30% of the peak value at time 0.15 was applied for 3 cyclesand then removed, as presented in Fig. 6(a).

The Fig. 6(b) presents the phase error using the linear PI and nonlinear PI. It can be observed that nonlinear PI has abetter performance than conventional PI.

Figure 6. (a) Input voltage Vα under swell disturvance (30%); (b) Phaseerror using PI Linear and nonlinear PI.

When the system presents harmonics disturbances, the quality of the extraction angle is compromised. Considering an input voltage with THD = 35%, as presented in Fig.7 (a),results in the response of phase error presented the Fig. 7(b) . It can be observed that for this kind of disturbance, the nonlinear PI presents a better dynamic response, but the amplitude of the phase error is slightly bigger.

Figure 7. (a) Input voltage Vα under THD=35%; (b) Phaseerror using PI Linear and nonlinear PI.

The last disturbance tested is the jumping phase. The jump phase is characterized by a variation in the instantaneous phase angle of the grid, which now assumes the new angle value.

The Fig. 8(a) presents the input voltage signal suffering a jumping phase of 40 degrees at the time 0.308 seconds. The Fig. 8(b) presents the phase error for conventional PI and nonlinear PI. It can be observed that the nonlinear PI had a

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cos( )V

cos( )V

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better response, decreasing the response time to a null error in 2.4 grid cycles.

Figure 8. (a) Input voltage Vα under THD=35%; (b) Phaseerror using PI Linear and nonlinear PI.

IV. CONCLUSIONS

This paper presented a comparison between single-phase PLL algorithm (pPLL) controlled by a conventional PI and aproposed adaptive nonlinear PI. It was observed that the nonlinear adaptive PI controller obtained a transient response2.3 times faster than the conventional PI.

In all cases, the nonlinear PI presents a better performance. In the case of harmonics disturbances, an alternative way to obtain a smaller phase error can be filtering the input signal,ensuring a signal with less or without harmonics disturbances.

In future works, a detailed analysis of nonlinear PI will be developed. Experimental results will also be compared tosimulation results.

REFERENCES

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[3] C. Cai; L. Wang and G. Yin; "A three-phase active power filter based on park transformation," Computer Science & Education, 2009. ICCSE '09. 4th International Conference on, pp.1221-1224, 25-28 July 2009

[4] H. Akagi, E. H. Watanabe and M. Aredes, Instantaneous Power Theory and Applications, 2007.

[5] S. A. O. Silva, R. Novochadlo and R. A. Modesto."Single-phase PLL structure using modified p-q theory for utility connected systems,"Power Electronics Specialists Conference, 2008. PESC 2008, pp. 4706-4711, 15-19 June 2008.

[6] M. Kaster, G. Cunha, I. Burgardt and C. Nascimento."ANÁLISE DO USO DE CONTROLE PID NÃO LINEAR APLICADO A CONVERSOR BUCK," 10 Conferência Brasileira de Dinâmica, Controle e Aplicações, 28 agosto 2011.

[7] P. Rodriguez, J. Pou, J. Bergas, J.I. Candela, R.P. Burgos, D. Boroyevich."Decoupled Double Synchronous Reference Frame PLL for Power Converters Control," Power Electronics, IEEE Transactions on , vol.22, no.2, pp.584-592, March 2007.

[8] R.A Wright, N. Kazantzis, C. Kravaris. "Optimization of nonlinear PI controllers for chemical processes," American Control Conference, 1998. Proceedings of the 1998 , vol.1, no., pp.328-332 vol.1, 21-26 Jun 1998.

[9] R. Shahnazi, H.M. Shanechi, N. Pariz. "Position Control of Induction and DC Servomotors: A Novel Adaptive Fuzzy PI Sliding Mode Control," Energy Conversion, IEEE Transactions on , vol.23, no.1, pp.138-147, March 2008.

[10] X. Qiu; J. Xiao; C. Wu. "Research of Variable Gain Nonlinear PI Controller Based Three-Phase Phase-Locked-Loop System," Electrical and Control Engineering (ICECE), 2010 International Conference on , pp.4050-4053, 25-27 June 2010.

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Jumping Phase