# Fractional-order PI based STATCOM and UPFC controller to diminish subsynchronous resonance

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## Abstract

This research article proposes a powerful fractional-order PI controller to mitigate the subsynchronous oscillations in turbine-generator shaft due to subsynchronous resonance (SSR) with flexible AC transmission system devices such as static synchronous compensator (STATCOM) and unified power flow controller (UPFC). The diminution of SSR is achieved by the raising of network damping at those frequencies which are proximate to the torsional mode frequency of the turbine-generator shaft. The increase of network damping is obtained with the injection of subsynchronous frequency component of current and both current and voltage into the line. The subsynchronous component of current and voltage are derived from the measured signal of the system and further the same amount of shunt current is injected with STATCOM and simultaneous injection of current and voltage with UPFC into the transmission line to make the subsynchronous current to zero which is the prime source of turbine shaft oscillations. The insertion and proper tuning of Fractional-order PI controller in the control scheme, the subsynchronous oscillations are reduced to 92 % in case of STATCOM and 98 % in case of UPFC as compared to without controller and 14 % as compared with the results of conventional PI controller. The IEEE first benchmark model has adopted for analyze the effectiveness and speed of the proposed control scheme using MATLAB-Simulink and the corresponding results illustrates the precision and robustness of the proposed controller.

## Keywords

Fractional-order PI controller STATCOM Torque amplification Voltage source converter UPFC## Background

Series capacitor compensation has been broadly employed in Power system to cancel a portion of reactance of the line impedance to increment the power transfer capability of long high voltage (HV) and extra high voltage (EHV) transmission lines further load sharing among parallel lines and boosts the steady state and transient stability limits (Anderson et al. 1989; IEEE SSR Working Group 1985). However, the addition of capacitive compensation in series can cause a new difficulty of turbine-generator shaft oscillations with below the system frequency due to Subsynchronous Resonance (SSR). The subsynchronous oscillations can be excited during the fault or disturbance in the transmission line with series capacitors, when the normal system frequency matches with the complement of any of the mode frequency of the shaft system (Kundur 1994; Padiyar 1998).

Latest advancement of power electronic devices led to the improvement of FACTS devices such as thyristor controlled series capacitor (TCSC), static synchronous compensator (STATCOM), static synchronous series compensator (SSSC) and unified power flow controller (UPFC) (Hingorani and Gyugyi 2000). An astronomically immense number of methods and solutions have been addressed by the different researchers to avoid the problem of SSR with the concern of FACTS devices (Padiyar and Swayam Prakash 2003; Padiyar and Prabhu 2006; Bongiorno et al. 2008b, c; Mohan et al. 2015; Koteswara Raju et al. 2016). The selection of controller is depends on extraction of the subsynchronous frequency components with high speed and accuracy. From the knowledge of subsynchronous frequency component of current and voltage a proper protection system is designed to avoid shaft breakage due to SSR (Bongiorno et al. 2008a).

The mitigation of SSR by the injection of shunt current with STATCOM including proportional integrator controller (PI) is proposed in reference (Umre et al. 2007). The STATCOM with subsynchronous damping controller (SSDC) including type 1 and type 2 controller is proposed based on the tuning of parameters using damping torque method to mitigate the SSR (Padiyar and Swayam Prakash 2003; Padiyar and Prabhu 2006). The parameters of SSDC are tuned to get good performance to provide damping (positive) in the range of torsional mode frequencies. The damping of SSR oscillations with 48 pulses (three-level) VSC based STATCOM using remote signal is proposed in (Salemnia et al. 2008).

The problem of SSR can also be bypassed by a proper combination of hybrid series compensation consists of FACTS controllers (SSSC) along with passive components. The mitigation of SSR with the proper injection of series voltage using PI based SSSC is described in (Bongiorno et al. 2008b, c; Mohan et al. 2015). The injection of subsynchronous frequency component of voltage in series with subsynchronous current suppresser using SSSC based on the knowledge of subsynchronous current to damp out the SSR is proposed in (Panda et al. 2010, 2016;Thirumalaivasan et al. 2013).

In this paper a Fractional-order proportional integrator (FOPI) based STATCOM and UPFC is used for mitigating the subsynchronous oscillations in turbine-generator shaft due to SSR. The mitigation of SSR is done by the injection of shunt current by STATCOM and the simultaneous injection of voltage and current into the line by UPFC with reference to the subsynchronous components which are extracted from the line. The STATCOM and UPFC are constructed with the help of multi-pulse voltage source converters. The fractional-order P I^{λ}D^{µ} controller derived from conventional PID controller with integrator of real order λ and differentiator of real order µ. Taking λ = 1 and µ = 1, we obtain a classical PID controller. If µ = 0 and K_{d} = 0, we obtain a P I^{λ} controller, etc. All these types of controllers are particular cases of the P I^{λ}D^{µ} controller, which is more flexible and gives the adjustment of dynamical properties of the fractional-order control system. In P I^{λ}D^{µ} controller there are five parameters to tune, with respect to the three parameters of the standard PID controller (λ and µ are equal to one) (Igor Podlubny 1999; Shantanu Das 2008). The facility of fine tuning of Fractional-order PI in the control circuit of STATCOM and UPFC controllers the subsynchronous oscillations are reduced to 92 and 98 % as compared to without controller and 14 % as compared to conventional PI controller. Fractional-order PI controllers having larger stability limit with larger phase value as compared to PI controller. Moreover, the Fractional-order PI controllers (FOPI) exhibits a less negative phase than the PI controller and it implies that more robustness (in the sense of stability) to changes in the overall system parameters.

The paper is organized in four sections. “Study system (IEEE FBM) with UPFC controller” section describes UPFC connected IEEE first benchmark model (study system) and the procedure for realization of subsynchronous frequency component of current and voltage. The design of subsynchronous frequency component controller including PI and Fractional-order PI controller and further the study system parameters and specifications of FACTS devices are presented in “Subsynchronous frequency component controller” section. The FFT analysis of LPB-GEN torque signal and simulation results of IEEE first benchmark model without and with FACTS controller under symmetrical (L–L-L) fault using PI and Fractional-order PI controller is given in “Results and discussions” section. The “Conclusion” section represents the conclusion of the complete research work.

## Study system (IEEE FBM) with FACTS controller

### Realization of current and voltage components of subsynchronous frequency

The current and voltage components of below synchronous frequency at the generator terminals are realized by considering the generic case of a transmission line connected with synchronous generator.

_{s}is the magnitude of terminal voltage of generator at rated speed, phase displacement is given by δ, \(\upomega({\text{t}})\) is the speed of rotor in per-unit and \(\upomega_{0}\) is the fundamental angular frequency radians/second. The speed of the generator rotor in terms of fundamental angular frequency \(\upomega_{0}\), and oscillating angular frequency \(\upomega_{\text{m}}\) is given by

When the system subjected to SSR, a small disturbance to generator rotor produces the voltage and current consists of three components: fundamental frequency, bellow synchronous frequency and above synchronous frequency. A little damping of positive is offered by network for super-synchronous frequency; hence the risk offered to generating station by this less (Bongiorno et al. 2008b).

The voltage component of subsynchronous and fundamental frequency is obtained by combining Eqs. (10) and (14). Equations (11) and (15) are used for the estimation of current component of subsynchronous and fundamental frequency.

## Subsynchronous frequency component controller

### Fractional-order PI (FOPI) controller

^{λ}D

^{µ}controller was proposed as a generalization of the PID controller with integrator of real order λ and differentiator of real order µ. The transfer function of such type of controller in Laplace domain has form (Igor Podlubny 1999; Shantanu Das 2008):

_{p}is the proportional constant, K

_{i}is the integration constant and K

_{d}is the differentiation constant. Transfer function (18) corresponds in discrete domain with the discrete transfer function in the following expression (Cao and Cao 2006; Enrico Pisoni et al. 2009):

^{λ}controller substitute µ = 0 and K

_{d}= 0.

_{p}, K

_{i}and λ values, the fine tuning of controller reduces the SSR oscillations with a faster rate. The complete block diagram of subsynchronous frequency component controller is shown in Fig. 3. At First the current and voltage of three-phase are measured from transmission line and further converted into \(\alpha \beta\) –plane, with the help of \(\theta_{f}\)(transformation angle) again converts into to \(dq\)-coordinate system. The outcome of estimation unit is the subsynchronous and fundamental frequency component of current and voltage in \(dq\)-frame. The subsynchronous frequency component in \(dq\)-reference is converted into subsynchronous frequency \({\text{dq}}_{\text{m}}\)-frame with the help of \(\uptheta_{\text{m}}\) (transform angle) which is obtained by integrating \(\upomega_{m}\)(oscillating frequency). The consequential signals are given to the subsynchronous frequency component controller (SSCC).

The output of Subsynchronous component controller is again transferred into \(\upalpha \upbeta\) and further transferred to abc and are furthermore given to the Pulse width modulation (PWM) generator which issues firing signals to 3-phase 48 pulse (three-level) GTO based voltage source converters (VSC).

### Parameters and specifications of IEEE first benchmark model

IEEE First benchmark network parameters

Resistance of network | \({\text{R}}_{\text{L}}\) | 0.0113 per unit |
---|---|---|

Reactance of transformer | \({\text{X}}_{\text{T}}\) | 0.142 per unit |

Transformation ratio | 22 kV/539 kV | |

Reactance of line | \({\text{X}}_{\text{L}}\) | 0.50 per unit |

Reactance of transmission line | \({\text{X}}_{\text{sys}}\) | 0.080 per unit |

Synchronous machine parameters

Parameter | Value (per unit) | Time constant | Value (sec) |
---|---|---|---|

\({\text{X}}_{\text{a}}\) | 0.130 | \({\text{T}}_{{{\text{d}}0}}^{ '}\) | 4.3 |

\({\text{X}}_{\text{d}}\) | 1.79 | \({\text{T}}_{{{\text{d}}0}}^{''}\) | 0.032 |

\({\text{X}}_{\text{d}}^{ '}\) | 0.169 | \({\text{T}}_{{{\text{q}}0}}^{ '}\) | 0.85 |

\({\text{X}}_{\text{d}}^{''}\) | 0.135 | \({\text{T}}_{{{\text{q}}0}}^{''}\) | 0.05 |

\({\text{X}}_{\text{q}}\) | 1.71 | ||

\({\text{X}}_{\text{q}}^{ '}\) | 0.228 | ||

\({\text{X}}_{\text{q}}^{''}\) | 0.200 |

IEEE first benchmark shaft parameters

Inertia | \({\text{H }}\left[ {{\text{s}}^{ - 1} } \right]\) | Shaft section | Spring constant [per unit T/rad] |
---|---|---|---|

High pressure turbine | 0.092897 | HP-IP | 19.303 |

Intermediate pressure turbine | 0.155589 | IP-LPA | 34.929 |

Low pressure A turbine | 0.858670 | LPA-LPB | 52.038 |

Low pressure B turbine | 0.884215 | LPB-GEN | 70.858 |

Generator | 0.868495 |

### Rating of shunt and series converters of UPFC

A three-phase 48 pulse (three-level) VSC bridge is used for Shunt and Series Converters. The VSC characterized in this research work is a harmonic neutralized 48-pulse GTO based inverter. These arrangements are made to produce harmonic free voltage output by proper connecting 6-pulse VSC’s. 12-pulse configuration is achieved by connecting two 6-pulse VSC’s, two 12-pluse converter are used for a 24-pulse topology and two 24-pulse arrangements are used for obtain a 48-pulse VSC. To produce a 48-pulse waveform with a harmonic content of n = 48 m ± 1, where m = 0, 1, 2, …, the 6-pulse converters requires relative phase displacements accomplished via the gate pulse pattern that determines the angle of the resulting three phase output voltages. Also, PST’s are used and are connected in serially with the phase voltages in the primary side of the MCC transformers to add voltage components in quadrature. These quadrature voltages are obtained from the three phase output voltages of each VSC (Kumar and Ghosh 1999; Hingorani and Gyugyi 2000; Salemnia et al. 2008). The rating of power necessary to diminish the SSR is difficult to conclude and depends on level of compensation (series), duration and location of fault. As the amount of current and voltage components of sub-synchronous frequency is reduced, the Shunt and Series converter rating is also reduced (0.1–5 %). The power rating is 12 MVA and the voltage rating is 8 kV (either capacitor or Direct current source). For an active power of 0.5 pu, the results are obtained and analyzed in the paper.

## Results and discussions

### FFT analysis

The subsynchronous mode with frequency of 24.8 Hz is reduced (mitigated) with the addition of controller and a small value is at 32.23 Hz which is not harm to the system is shown in Fig. 5.

### Stress analysis

_{a}). Due to this torque, varying stress is produces in the rotor shaft. When this stresses exceeds the endurance limit i.e. 45 × 10

^{7}N/m

^{2}, the shaft will be damaged. The torsional fatigue of turbine generator shaft is primarily as a function of amplitude of the stress and secondarily on its limit (Kundur 1994). The calculation of mechanical stress needs the calculation of mechanical angle (twist angle between shafts) δ and is as fallows.

_{m}] consists of mechanical torque acting on various masses, [T

_{e}] consists of electrical torque produced by various masses.

^{2}, δ

_{i}, δ

_{i+1}: twist angle of ith mass to the (i + 1)th mass in radians, G: modulus of rigidity in N/m

^{2}; L: length of shaft in meters; R: radius of shaft in meters.

### Simulation results

To realize the performance of the suggested control scheme to diminish the SSR due to Torque Amplification, the IEEE FBM with STATCOM and UPFC has been simulated using Matlab-Simulink software. A three-phase fault is applied to the grid at 1 s for time duration of 0.05 s with 55 % series compensation. The simulation results are shown for three cases, without controller, with PI and with FOPI based STATCOM and UPFC controller. After the clearance of fault at 1.05 s the system has to regain its previous state that is the turbine-generator shaft oscillations are at normal level.

^{7}N/m

^{2}which will damage the entire shaft system. To avoid the damage due to torque amplification effect of SSR the PI controller based STATCOM or UPFC is connected to the line. The PI based STATCOM injects a shunt current or simultaneous injection of shunt current and series voltage with UPFC into the line in order to reduce the capacitive reactance from the capacitor bank and further shift the electrical resonance of the system, thus avoiding the risk of SSR. The simultaneous injection of shunt current and series voltage with UPFC, the turbine oscillations and stress are reduced to such a low value about 94 % as compared to without controller. With PI based STATCOM controller the torque between LPB and Generator is reduced from 2.2 to 0.3 pu and further reduced from 0.46 to 0.09 pu in case of UPFC. Similarly the Mechanical stress between LPB and Generator is reduced from 2.2 × 10

^{7}to 0.4 × 10

^{7}N/m

^{2}with STATCOM and 0.45 to 0.06 × 10

^{7}N/m

^{2}with UPFC for a time interval of 10 s shown in Figs. 6b and 7b. With the facility of fractional tuning of FOPI controller the torque between LPB and Generator is further reduced from 2.1 to 0.28 pu in case of STATCOM and from 0.42 to 0.08 pu with UPFC. Similarly the Mechanical stress is furthermore reduced from 2.1 × 10

^{7}to 0.34 × 10

^{7}N/m

^{2}in case of STATCOM and from 0.42 × 10

^{7}to 0.05 × 10

^{7}N/m

^{2}in case of UPFC for a time interval of 10 s shown in Figs. 7a and 8.

By considering the Figs. 9b, 10a, b and 11 the conclusion is that both the controllers reduce the effect of SSR on Electromagnetic torque and Rotor speed and further the FOPI based controllers are more effective than conventional PI based controllers.

Comparison of different performance indexes of IEEE first benchmark model with PI and with FOPI based STATCOM and UPFC controllers

1. LPB-GEN Torque (per unit) With PI based controller the torque is reduced from 2.2 to 0.3 pu in case of STATCOM and from 0.46 to 0.09 pu with UPFC for a time interval of 1.05 s to 10 s. The torque is further more reduced with FOPI controller from 2.1 to 0.28 pu in case of STATCOM and from 0.42 to 0.08 in case of UPFC | |

2. LPB-GEN Stress (10 The Mechanical stress between LPB and Generator is reduced from 2.2 × 10 | |

3. Change in Electro-Magnetic Torque (per unit) The Electromagnetic torque is reduced and reaches to a safe value from 0.9 to 0.05 and 0.27 to 0.01 pu with PI based STATCOM and UPFC controllers for a time interval of 1.05 to 10 s. With the use of FOPI in place of PI controller the change in Electromagnetic torque is further reduced from 0.86 to 0.04 pu and 0.25 to 0.01pu in case of STATCOM and UPFC | |

4. Change in Rotor Speed (per unit) Without SSR controller the rotor speed is changes from 1.05 to 1.28 times. The change in Rotor speed is reduced from 1.008 to 1.002 and 1.0015 to 1.001 pu with PI based STATCOM and UPFC controllers for a time interval of 1.05 to 10 s. The use of FOPI in place of PI controller the change in Rotor speed is further reduced from1.0077 to1.0018 pu and 1.0012 to 1.001 with STATCOM and UPFC |

By observing the Table 5 the FOPI based UPFC controller is more effective and faster in mitigation of Torque amplification due to SSR as compared to PI based STATCOM, UPFC and FOPI based STATCOM controllers.

## Conclusion

In this work, a robust Fractional-order PI based STATCOM and UPFC controllers are developed to diminish the oscillations in turbine-generator shaft due to torque amplification effect of SSR. Based on the control system the mitigation of subsynchronous resonance is achieved by raising the damping of network with proper estimation and injection of subsynchronous component quantities into the line using UPFC and the results are compared with STATCOM controller. For the system studied, the superiority of proposed FOPI based controller is demonstrated by comparing the results of conventional PI controller for all cases. From the proposed study, it was observed that, fine and fast SSR mitigation is achieved by Fractional-order PI controller based UPFC as compared to conventional PI based STATCOM, UPFC and Fractional order PI based STATCOM controllers. Moreover, the performance of proposed controller is evaluated under three-phase fault with different performance indices namely; torque and mechanical stress between LPB-Generator, change in electromagnetic torque and change in rotor speed.

## Notes

### Authors’ contributions

DKR carried out the design of PI and FOPI based STATCOM and UPFC controllers to mitigate the SSR. BSU and ASJ performed the mathematical analysis, mode shapes and dominant mode of turbine generator shaft under SSR condition. MPT, RM and CS performed the analysis of mechanical stress between LPB and Generator. All authors read and approved the final manuscript.

### Acknowledgements

The authors are very thankful for the facilities provided by the Department of Electrical Engineering, VNIT, Nagpur to carry out their research work.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ information

**D. Koteswara Raju** recieved the B.Tech. degree in Electrical and electronics engineering from JNTU Hyderabad, India and M.Tech. degree in Power system engineering from Acharya Nagarjuna University (ANU), Vijayawada, Andhrapradesh, India in 2006 and 2009 respectively. He had been with RVR&JC College of engineering from June 2009 to May 2011 and Gudlavalleru engineering college from June 2011 to Till date as an Assistant professor in Electrical and Electronics Engineering Department. At present he is pursing Ph.D. from Visvesvaraya National Institute of Technology, Nagpur, India. His research interests include power system protection, operation and control; **Bhimrao S. Umre** is working as Associate professor in the Department of Electrical Engineering, Visvesvaraya National Institute of Technology Nagpur. He received the B.E., M.Tech. and Ph.D. from Govt. College of Engineering, Visvesvarya Regional College of Engineering and RSTM of Nagpur University, India in 1983, 1986 and 2009 respectively. His research interests include electrical machines, power system operation, control and torsional oscillations; **Anjali S. Junghare** is working as Associate professor in the Department of Electrical engineering, Visvesvaraya National Institute of Technology, Nagpur. She has recieved the B.E., M.Tech. and Ph.D. from Visvesvaraya National Institute of Technology, Nagpur, India in 1981, 1985 and 2008, respectively. Her research interests include control systems, power system operation and control; **Mohan P. Thakre** received the B.Tech. and M.Tech. degree in electrical engineering from Dr. Babasaheb Ambedkar Technological University, Raigad, India, in 2009 and 2011, respectively. He is awaiting Ph.D. degree in power system protection from Visvesvaraya National Institute of Technology, Nagpur, India. His area of research interest includes power system protection, operation and control ; **Rambabu Motamarri** received the B.E and M.Tech in electrical and electronics engineering and power electronics and drives from Chaitanya Engineering College, Andhra Pradesh and Visvesvaraya National Institute of Technology, Nagpur, India in 2010 and 2013 respectively. His research interests include application of power electronics in power system operation and control; **Chaitanya Somu** received the M.Tech. in power electronics and drives from Visvesvaraya National Institute of Technology, Nagpur, India from Visvesvaraya National Institute of Technology, Nagpur, India in 2014. At present he is working as Assistant professor in EEE Department in Aditya Engineering college, Kakinada, Andhrapradesh, India. His area of research interest includes Application of power electronics in power system operation and control.

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