# Adaptive fractional integral terminal sliding mode power control of UPFC in DFIG wind farm penetrated multimachine power system

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## Abstract

With an aim to improve the transient stability of a DFIG wind farm penetrated multimachine power system (MPN), an adaptive fractional integral terminal sliding mode power control (AFITSMPC) strategy has been proposed for the unified power flow controller (UPFC), which is compensating the MPN. The proposed AFITSMPC controls the *dq*- axis series injected voltage, which controls the admittance model (AM) of the UPFC. As a result the power output of the DFIG stabilizes which helps in maintaining the equilibrium between the electrical and mechanical power of the nearby generators. Subsequently the rotor angular deviation of the respective generators gets recovered, which significantly stabilizes the MPN. The proposed AFITSMPC for the admittance model of the UPFC has been validated in a DFIG wind farm penetrated 2 area 4 machine power system in the MATLAB environment. The robustness and efficacy of the proposed control strategy of the UPFC, in contrast to the conventional PI control is vindicated under a number of intrinsic operating conditions, and the results analyzed are satisfactory.

## Keywords

Adaptive fractional integral terminal sliding mode power control Doubly fed induction generator Multimachine power network Unified power flow controller## 1 Introduction

The increase in the penetration of wind power, especially from doubly fed induction generator based wind farms into the existing power grids, is although beneficial, but has significant pessimistic impacts [1] such as voltage and frequency control, power transfer capability, transient stability, etc. The employment of the power system stabilizers (PSS) is helpful in stabilizing such power systems, but they demonstrate an unreliable performance for the interarea oscillations between the generators of the multimachine power systems [2]. In the process, the application of the Flexible AC Transmission System (FACTS) devices, such as Unified Power Flow Controllers (UPFC) along with the PSS has illustrated excellent results, especially for improving the oscillations exhibited by the power system components [3]. Adding to it, they also control both the active and reactive power flows across the ac transmission lines [4, 5]. A number of configurations of the UPFC used for the ac transmission lines in the last few decades have been reviewed in [6]. Most of the models constitute a large number of parameters involved and hence are computationally more complex. On the contrary, a simple model which is easier for deriving the controls has been proposed in [7], where the voltage injected in the series portion of the UPFC is resolved into quadrature and phase components, with respect to the current flowing along the line. These components are further used to effectively control the reactive and active power flow through the transmission line, respectively.

A review on some of the control systems for the UPFC as well as their drawbacks has been discussed in [8]. The proportional-integral (PI) control is one of the effective conventional controller for the UPFC [9], but its performance is unreliable under some of the intermittent operating conditions (DFIG based wind farms) [10]. In the due course of time some of the non-linear controls such as sliding mode control [10] (SMC), Neuro-SMC techniques [11] etc., have been proposed for the control of the UPFC. The choice of an ideal hyper plane that assures the asymptotic stability of the non- linear systems is very much important for the controller design of the SMC. Adding to it, SMC is endowed with the well-known chattering phenomenon which makes it unreliable under certain operating conditions. Thus in order to overcome the above mentioned problems, a fractional integral terminal sliding mode power control (FITSMPC) has been investigated for the nonlinear and dynamic systems [12] that shows a very much promising result in terms of guaranteed finite time chatter free error convergence.

- (a)
A FITSMPC is proposed to control the dq- axis series injected voltage of the UPFC that subsequently controls the proposed admittance model of the UPFC.

- (b)
The controller gains of the proposed FITSMPC for the admittance model of the UPFC are made dynamic [13], such that they adjust with the intermittent operating conditions.

- (c)
This subsequently controls the active power injection at the DFIG wind farm terminal, which maintains the power balance of the nearby generators. As a result, the rotor angular oscillations between the generators in the multimachine system gets recovered, which will help in improving the power transfer capability of the associated transmission lines (though this later portion has not been investigated in the current paper, but is considered for future work).

The proposed controllers for the admittance model of the UPFC is installed in a standard 2-area 4-machine system [7], which has been penetrated by a DFIG based wind farm [14]. The overall model with their controllers have been simulated in the MATLAB/Editor environment following the necessary requirements for multimachine simulation [15]. The size and location for installation of the UPFC and DFIG based wind farm in the multimachine power system has been followed as per the references [7, 16], respectively. Critical Clearing Time (*T*_{ CCL }) [1], one of the key indicators of the transient stability index, has been taken as the basis for comparison of the performance of the proposed controller with the conventional PI control of the admittance model of the UPFC installed in the DFIG wind farm penetrated multimachine power system, and which is subjected to three phase fault and the DFIG wind farm experiences a realistic wind profile [17]. It is observed that as compared to the conventional PI control, the proposed control strategy for the admittance model of the UPFC is very much significant and robust in improving the transient stability of the DFIG penetrated multimachine power system and exhibits the largest *T*_{ CCL } for all most all the cases simulated in this paper. These outputs as illustrated in the simulation and results section are satisfactory and vindicate the real time application of the proposed technique.

## 2 Proposed admittance model of UPFC

### 2.1 Basic model of the UPFC

*V*

_{ m }

*∟δ*

_{ m }and

*V*

_{ n }

*∟δ*

_{ n }, respectively). As a generalized concept of the UPFC [7], it comprises of a series transformer (

*T*

_{ sbt }), a shunt exciter transformer (

*T*

_{ shet }), a common dc link capacitor and two voltage source converters. In fact the bus ‘

*n*’ is very nearer to bus ‘

*m*’, as compared to bus ‘

*o*’ (with bus voltage magnitude

*V*

_{ o }

*∟δ*

_{ o }), as per the inequality relation “(

*x*

_{ SE }+

*x*

_{ Tsbt }) < <

*x*

_{ Tr. Line }”, where

*x*

_{ SE },

*x*

_{ Tsbt }and

*x*

_{ Tr. Line }are the reactance’s of the series portion of UPFC, transformer

*T*

_{ sbt }and the transmission line between bus ‘

*n*’ and ‘

*o*’, respectively.

*x*

_{ SH }represents the reactance’s across the shunt portions of the UPFC.

*ξ*

_{ SH }and

*ξ*

_{ SE }represent the susceptances of the shunt and series converter transformers of respectively. Similarly,

*υ*and

*υ*

_{ SH }represent the angle of the

*V*

_{ Tsbt }the UPFC, and

*V*

_{ Tshet }with respect to

*V*

_{ m }, respectively;

*δ*

_{ m }and

*δ*

_{ n }are the phase angle at buses ‘m’ and ‘n’, respectively.

_{α}|

*V*

_{ m }| and μ

_{β}|

*V*

_{ m }| be the induced voltage across

*T*

_{ sbt }and

*T*

_{ shet }, respectively, where

*μ*

_{ α }and

*μ*

_{ β }are the voltage proportion level of the series and shunt circuits of the UPFC, respectively. Thus expression for the injection of the active and reactive powers at buses ‘

*m*’ and ‘

*n*’ (

*P*

_{ m },

*Q*

_{ m },

*P*

_{ n }and

*Q*

_{ n }), respectively, can be defined as:

*υ*

_{ SH }≈ 0. Thus we derive:

*I*

_{ SH }corresponds to the real component of the current across the shunt converter.

### 2.2 Admittance model of the UPFC

*Y*

_{ m }and

*Y*

_{ n }) which can be expressed in terms of the active and reactive power across the two buses ‘

*m*’ and ‘

*n*’ (

*P*

_{ m },

*Q*

_{ m },

*P*

_{ n }and

*Q*

_{ n }), respectively, as shown in Fig. 3.

*Y*

_{ m }and

*Y*

_{ n }can be expressed mathematically as:

*P*

_{ m },

*Q*

_{ m },

*P*

_{ n }and

*Q*

_{ n }from Eqs. (4), and (1) in Eq. (5), we derive,

### 2.3 Design of μ_{α} and υ

*ϑ*

_{ ζD }and

*ϑ*

_{ ζQ }are the direct and quadrature axis voltage injected in the series circuit of the UPFC (

*ϑ*

_{ ζ }), i.e.

*ϑ*

_{ ζ }can be resolved (

*ϑ*

_{ ζP }and

*ϑ*

_{ ζQ }) across the series current

*I*

_{ ζ }such that:

Where *i*_{ ζD } and *i*_{ ζQ } re the dq- axis component of the series injected current *I*_{ ζ }.

*V*

_{ m }’ and the series injected voltage ‘

*ϑ*

_{ ζ }’ can be expressed in dq axis as:

_{α}and

*υ*are designed as:

Thus UPFC has been modeled as controllable admittance loads as specified in Eqs. (6) and (7), respectively. It is to be noted that the admittance across the loads *Y*_{ m } and *Y*_{ n }, can be controlled by *μ*_{ α } and υ which are again controlled by *ϑ*_{ ζP } and *ϑ*_{ ζQ }, respectively, which depends upon δ_{ξ}, *ϑ*_{ ζD } and *ϑ*_{ ζD }, respectively. Thus the control of the UPFC as controllable loads is achieved by controlling the series injected voltage, as shown in Eq. (22), which is the main contribution of this manuscript. Modelling of the dc link voltage is referred to [7].

## 3 Methods

### 3.1 The non- linear dynamic model of the UPFC

*γ*

_{ Q }and

*γ*

_{ P }are the control terms associated with the non-linear dynamic Eq. (17), which are defined as:

*ϑ*

_{ xD }and

*ϑ*

_{ xQ }are defined as:

Thus *γ*_{ Q } and *γ*_{ P } are the target control terms which are used to control the final dq- axis series injected voltages (Eq. (34)), which subsequently control the proposed AM of the UPFC, respectively.

### 3.2 Proposed adaptive fractional integral terminal sliding mode power control (AFITSMPC) of UPFC

Where \( {p}_n^{\ast } \) and \( {q}_n^{\ast } \) are the reference value of the active and reactive powers, which are evaluated in the initial solution.

With *α*_{ 1 }, *α*_{ 2 } > 0, β_{1}, β_{2} are fractional numbers satisfying the relation 0 < [β_{1}, β_{2}] < 1.

*β*

_{ 1 },

*β*

_{ 2 }) are the fractional powers of the tracking errors (

*e*

_{ r1 },

*e*

_{ r2 }) with initial values (−

*e*

_{ r1 }(0)/

*α*

_{ 1 }, −

*e*

_{ r1 }(0)/

*α*

_{ 1 }), respectively.

**Theorem 1**

*The tracking error functions as defined in*Eq. (25)

*will converge to zero in a finite amount of time, and the system will remain robust and stable if the FITSS are chosen as in*Eq. (32),

*and the control is designed as follows*:

*γ*

_{ Pnom }and

*γ*

_{ Qnom }are the nominal controls, whereas

*γ*

_{ Prob }and

*γ*

_{ Qrob }are the robust controls, respectively, as introduced by the terminal sliding mode concept and can be derived as follows:

In the above Eqs. (34)–(37), {*β*_{ ProbV }, *β*_{ QrobV }, *β*_{ ProbA }, *β*_{ QrobA }} > 0, are the gains of the controller, whose initial values are mentioned in the Appendix. Eq. (37), which makes the controller adaptive i.e., adjusts the controller gains according to the varying operating conditions, is defined as the fractional integral terminal sliding mode adaptive law for the admittance model of the UPFC.

*Proof of convergence*On the surface \( {\dot{\sigma}}_{r1}=0 \) and \( {\dot{\sigma}}_{r2}=0 \) we have,

Equation (39) guarantees a finite time convergence of the tracking error functions [12] as defined in Eq. (25).

*Proof of stability*Let us consider the following Lyapunov function:

From Eq. (43), it is proved that as the value of \( {\dot{V}}_L\le 0 \) for all *σ*_{ r1 } ≠ 0 and *σ*_{ r2 } ≠ 0, which guarantees the system stability. This completes the proof.

## 4 DFIG wind farm model

*dq*reference frame based dynamic model of the DFIG wind turbine is developed from its flux linkage model [16], and is represented as:

*ω*

_{ rr },

*ω*

_{ tt }and

*ω*

_{ ss }are the rotor, turbine and synchronous speeds of the DFIG, respectively.

*E*

_{ sd }and

*E*

_{ sq }denotes the voltage behind the transient reactance of the DFIG. θ

_{tw}represents the shaft twist angle.

*C*

_{ dc }and

*V*

_{ dc }are the capacitance and voltage components across the dc terminal of the DFIG, respectively. The constants

*H*

_{ gg }

*, H*

_{ tt }are represents the inertia constants of generator and turbine, respectively. Similarly,

*L*

_{ ss },

*L*

^{ / }

_{ ss },

*T*

_{ rr }represent the synchronous inductance, transient inductance and time constant of DFIG respectively. Mathematically,

*L*

^{ / }

_{ ss }=

*ω*

_{ ss }[

*L*

^{ 2 }

_{ mm }/

*L*

_{ rr }] and

*T*

_{ rr }=

*L*

_{ rr }/

*r*

_{ rr }, where

*L*

_{ rr }and

*r*

_{ rr }are the rotor self-inductance and resistance, respectively, and

*L*

_{ mm }is the mutual inductance between the stator and the rotor terminals of the DFIG.

*t*

_{ mm },

*t*

_{ sh }and

*t*

_{ ee }are the mechanical, shaft and electromechanical torque of the DFIG respectively.

*I*

_{ rd },

*I*

_{ rq },

*I*

_{ sd },

*I*

_{ sq },

*I*

_{ id },

*I*

_{ iq },

*V*

_{ rd },

*V*

_{ rq },

*V*

_{ sd },

*V*

_{ sq },

*V*

_{ id }, and

*V*

_{ iq }, are the dq-axis current and voltage quantities of the RSC, stator, and GSC Terminals, respectively. Similarly,

*P*

_{ rr },

*P*

_{ ss },

*P*

_{ ii },

*P*

_{ df },

*Q*

_{ rr },

*Q*

_{ ss },

*Q*

_{ ii }, and

*Q*

_{ df }are the active and reactive power flows across the RSC, stator, GSC and DFIG terminals respectively.

The GSC operation has been restricted to unity power factor and hence, results in zero reactive power at the GSC terminal.

## 5 Results

*AFITSMPC*for the AM of the UPFC. The gains of the PI controllers for the DFIG as well as AM of UPFC are tuned through ITAE criterion [20], with possible maximum wind power penetration. The detailed values of the tuned gains for the PI controller for the DFIG as well as AM of UPFC are mentioned in the Appendix.

A three phase short-circuit fault is considered as the disturbance which has been simulated on one of the load bus (bus ‘7’ in Fig. 5) for certain duration of time ‘*t*_{ f }’ in sec. The highest value of *t*_{ f } in the post fault region within which synchronism between the relative rotor angles of the generators in a power system is maintained is defined as the critical clearing time denoted as *T*_{ CCL } [1]. Thus it is very much significant that, for a given operating condition, the *T*_{ CCL } provides an exact clue of the transient stability margin of the power system.

### 5.1 Performance of the control strategies of the UPFC in the DFIG wind farm penetrated two area four machine system (Fig. 5) with the DFIG wind farm subjected to fixed wind speed

In this particular case, a three phase fault is initiated on bus 7 of the test system (Fig. 5) at the timing instant *t*_{ s } = 2.11 s, where the DFIG wind farm is subjected to fixed wind speed of 6.29 m/s. The performance of the controllers has been tested for both the lower as well as higher level of penetration of the DFIG wind farm, which are illustrated in Figs. 7 and 8, respectively. In order to evaluate the *T*_{ CCL }, the performance of the proposed as well as conventional controls of the AM of the UPFC is observed by repetitive simulations by increasing the duration of fault ‘*t*_{ f }’. It is observed that, the *T*_{ CCL } for the conventional PI control for this particular case is 249 ms and 161 ms for lower and higher penetration of DFIG wind farm, respectively, at which the generators in the system losses synchronism (subplots (d)). On the contrary, for the same duration of fault (*T*_{ CCL } for PI control), the proposed control of the AM of the UPFC is very much significant in improving the ‘*Y*_{ m }’ and ‘*Y*_{ n }’ placed between the buses ‘m’ and ‘n’ as shown in subfigures (a) and (b), respectively. Subsequently, the power at the DFIG wind farm terminal (*P*_{ dg }) is improved (subplot (c)), which is responsible for restraining the electrical and mechanical power equilibrium of the nearby generators. This minimizes the rotor angular deviation of the generators and hence stabilizes the MPN, which is reflected in the improvement on the interarea oscillation between generators 1 and 4, (DW 1–4 (Rad/s)) as illustrated in subplot (d), in these figures, respectively. Further, the *T*_{ CCL } for the proposed control strategies is found out to be 266 ms and 181 ms, for lower and higher penetration of DFIG wind farm, which illustrates a 17 ms and 20 ms improvement in CCL, respectively, for the proposed controller (STRATEGY B) in the fixed wind speed operation of the DFIG wind farm.

### 5.2 Performance of the control strategies of the UPFC in the DFIG wind farm penetrated two area four machine system (Fig. 5) with the DFIG wind farm subjected to variable wind speed

*AFITSMPC*for the AM of the UPFC is verified for the stability enhancement of the MPN (Fig. 5) under both the lower as well as higher level of penetrations of the DFIG wind farm, for a three phase fault initiated on bus 7 at the timing instants ‘A’, ‘B’ ‘C’ and ‘D’, respectively, with a sporadic wind profile input to the DFIG wind farm and the results are illustrated in Figs. 9, 10, 11 and 12, respectively. Analogous to the previous section, in order to evaluate the

*T*

_{ CCL }(see Appendix for details), the performance of the proposed as well as conventional controls of the AM of the UPFC has been constantly observed by gradually increasing the duration of fault’

*t*

_{ f }’ for all the cases discussed in this case. Similar observations as in case of previous case has been observed in this case. For the particular cases illustrated in these figures, it is observed that for the fault duration resembling the

*T*

_{ CCL }for the conventional PI control (where the generators loose synchronism in the post fault region), the performance of the proposed

*AFITSMPC*for the AM of the UPFC is very much stable there is drastic improvement in the

*T*

_{ CCL }, which is evident in the subfigures (a) to (d) in the Figs. 9, 10, 11 and 12, respectively.

*(*AFITSMPC

*)*for the AM of the UPFC over the conventional STRATEGY A, for demonstrating its (proposed controller) robustness in enhancing the transient stability of the DFIG wind farm penetrated two area four machine power system have been illustrated in Tables 1 and 2 respectively. Form these tables, it illustrates the

*T*

_{ CCL }for both the conventional and proposed controls as well as the improvement in the

*T*

_{ CCL }for the proposed control for all the cases discussed in this case, with the lower and higher penetration levels of the DFIG, respectively.

Critical Clearing time for the conventional and proposed controllers subjected to Lower penetration of DFIG Power

Operating Points | | | Improvement in T |
---|---|---|---|

A | 498 | 514 | 16 |

| 514 | 523 | 09 |

| 485 | 504 | 19 |

| 507 | 521 | 14 |

Critical Clearing time for the conventional and proposed controllers subjected to Higher penetration of DFIG Power

Operating Points | | | Improvement in T |
---|---|---|---|

A | 391 | 414 | 23 |

| 411 | 426 | 15 |

| 387 | 392 | 05 |

| 355 | 371 | 16 |

The improvement in *T*_{ CCL } ranges between 9 ms − 19 ms in the lower DFIG wind power penetration case, where as it lies between 5 ms − 23 ms for the higher DFIG wind power penetration case. Thus as the proposed *AFITSMPC* for the AM of the UPFC illustrates an improved result in terms of damping out the oscillations between the generators of the multimachine power system for a number of intrinsic operating conditions, hence it guarantees the robustness of the method and also the applicability of the method in real time applications (which is considered as a future work in this paper).

## 6 Discussion

It is observed that from previous two subsections and Figs. 7, 8, 9, 10, 11 and 12, that after the initiation of disturbance, the simulation of the overall model (Fig. 5), illustrates some low frequency oscillations (as in case of the interarea oscillation between the generators 1 and 4) which is uncontrollable. On the contrary, the above simulation with the proposed *AFITSMPC* for the AM of the UPFC, i.e., STRATEGY B, also exhibits some low frequency oscillation (as in case of the interarea oscillation between the generators 1 and 4), which deviates approximately between − 2 and 2 Rad/S. But in spite of it, the active power or the reactive power based *AFITSMPC* control of an AM of the UPFC considerably damps out the respective inter-area low-frequency oscillations exhibited by the network. In addition, the DFIG wind farm is also equipped with the PI control mechanism on active and reactive powers for both the rotor side as well as grid side converters [21], which also boosts up in damping the above low frequency interarea oscillations [22]. It is also observed that, [22] illustrates a method by which the low-frequency oscillation modes of the presented power system can be calculated through a low- frequency oscillation modal analysis combined with the dynamic small signal mathematical models of DFIG wind turbines and synchronous generators, including their eigenvalues, oscillation frequencies, and damping ratios. This is an important and interesting topic which will be given full consideration as a future work.

In addition, the adaptive nature of the controller gains of the proposed *AFITSMPC* for the AM of the UPFC, is very much significant in quickly stabilizing the admittance model of the UPFC where the DFIG wind farm in the multimachine power system has been subjected to fixed as well as sporadic wind profile. In all the cases, there has been a significant improvement in the interarea oscillations exhibited by the MPN by the strategy B, which is justified in subfigure (d) of the Figs. 6, 7, 8, 9, 10 and 11, respectively. The *T*_{ CCL } for both the strategy A and B have been tabulated which justifies its improvement for proposed STRATEGY B for almost all the cases illustrated in this section. Further it is observed that, with the increase in penetration level of the DFIG wind farm, there is a significant increase in the maximum overshoots of the *P*_{ dg }. In spite of this, the proposed *AFITSMPC* for the AM of the UPFC, in comparison to the conventional PI controller, is very much significant and robust to enhance the stability of the MPN, subjected to both the higher and lower level DFIG wind farm penetration with fixed as well as sporadic wind profiles subjected to different intrinsic operating conditions.

## 7 Conclusion

An adaptive fractional integral terminal sliding mode power control strategy of the admittance model (AM) of the UPFC, in order to damp out the oscillation between the generators in a DFIG wind farm penetrated multimachine power system is proposed in this paper. Taking the critical clearance time (*T*_{ CCL }) as the basis, the performance of the conventional PI is compared with the proposed *AFITSMPC* for the AM of the UPFC. It is observed that the adaptive nature of the proposed strategy B is very much significant in maintaining the synchronism between the generators in the multimachine system for a larger period of time (*T*_{ CCL }), which is evident by improvement in TCCL for the proposed strategy for almost all the case studies illustrated in the paper. It is also observed that the proposed *AFITSMPC* for the AM of the UPFC significantly stabilizes the active power output of the DFIG wind farm. This improves the electric power of the nearby generators, which subsequently improves (diminish) the irrespective rotor angle deviations and hence leads to the stability enhancement of the multimachine power system. The proposed controllers for the UPFC has been tested for the stability enhancement of the MPN for higher and lower penetrations of the DFIG power, at various operating points, where the DFIG wind farm has been subjected to a fixed as well as a sporadic wind profile, respectively. It is observed that, with an increase in penetration level of the DFIG, the oscillations shown by its active power in the post fault region increases. In spite of this, the proposed controller for the AM of the UPFC has outperformed the conventional one, by inheriting larger *T*_{ CCL }, for the DFIG penetrated multimachine system, under a number of intrinsic operating conditions. This has been analyzed in the simulation and result section, where the outputs shown are satisfactory and vindicates the superiority of the proposed controller.

## Notes

### Authors’ contributions

Author RKP designed the Adaptive fractional sliding mode algorithm for the DFIG wind farm with UPFC along with some simulations. Author P.K.Dash conceived the original problem for detailed study along with results verification and coordition of the various sections of the manuscript.. Author S.P.Mishra did some simulations and provided data and took part in revising the paper.. All authors read and approved the final manuscript.

## Competing interests

The authors declare that they have no competing interests.” Also no fund is received from any financial or non-financial organization.

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