# Sliding Mode Observer-Based Load Angle Estimation for Salient-Pole Wound Rotor Synchronous Generators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{q}leads the stator terminal voltage U

_{s}, represents a fundamental quantity for power system stability assessment [1,2]. It defines the generator operating point with respect to the stability limits. Load angle is essential in transient stability studies, and its utilization in other real-time applications, such as protection functions, or as an input to excitation control systems and power system stabilizers has been investigated [3,4]. Therefore, the accurate knowledge of load angle value is required.

## 2. Methods

#### 2.1. Synchronous Generator Phasor Diagram

_{s}and the induced voltage E

_{q}in the q-axis of the rotating dq reference frame. The dq components of stator voltages and currents are obtained by applying the dq transformation (Clarke and Park transformations) to the three phase quantities. In the steady-state operation of a synchronous generator, the relationships between the stator voltages and currents may be expressed by a phasor diagram in the complex dq plane, as shown in Figure 1. Generator voltages and currents are expressed as phasors in the complex plane with the d and q axes as coordinates [2].

_{s}represents the stator terminal voltage, I

_{s}represents the stator current, ϕ is the phase angle between the stator voltage and current (power factor angle), E

_{q}is the induced voltage in the q-axis, δ is the load angle, X

_{q}is the synchronous quadrature (q) axis reactance, and R

_{s}is the stator resistance.

#### 2.2. SMO-Based Load Angle Estimator

_{q}is the synchronous q-axis inductance. Equivalent EMF components are defined as:

_{d}is the synchronous d-axis inductance, L

_{ad}is the mutual inductance between stator and rotor windings in the d-axis, ω is the rotor electrical angular velocity, i

_{d}is the d-axis stator current component, i

_{fd}is the field current, and θ is the rotor angular position.

_{c}is the filter cut-off frequency.

#### 2.3. Simulation Model

## 3. Results and Discussion

- Case 1: P = 0.89, Q = 0.41 ind.
- Case 2: P = 0.89, Q = 0
- Case 3: P = 0.89, Q = 0.11 cap.

- Mean squared error (MSE):$$MSE=\frac{1}{N}{\displaystyle \sum _{n=1}^{N}{\left(\widehat{\delta}(n)-\delta (n)\right)}^{2}}$$
- Mean absolute error (MAE):$$MAE=\frac{1}{N}{\displaystyle \sum _{n=1}^{N}\left|\widehat{\delta}(n)-\delta (n)\right|}$$
- Maximum absolute error (MAXE):$$MAXE=\underset{n=\left\{1,\dots ,N\right\}}{\mathrm{max}}\left\{\left|\widehat{\delta}(n)-\delta (n)\right|\right\}$$

#### 3.1. Reactive Power Disturbances

#### 3.1.1. Case 1

#### 3.1.2. Case 2

#### 3.1.3. Case 3

#### 3.2. Active Power Disturbances

#### 3.2.1. Case 1

#### 3.2.2. Case 2

#### 3.2.3. Case 3

#### 3.3. Parameters Sensitivity Analysis

_{s}and quadrature axis synchronous inductance L

_{q}. The actual values of the machine parameters may differ from the provided nominal values or from the values obtained by some of the parameters estimation methods. Moreover, these parameters depend on operating conditions and their values may change during the generator operation, due to temperature (R

_{s}) or magnetic saturation (L

_{q}).

_{s}and L

_{q}refer to the values used in the estimator’s structure, while R

_{sn}and L

_{qn}refer to the nominal values which were used in the simulation model of a synchronous generator.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Parameter | Symbol | Value | Unit |
---|---|---|---|

Apparent power | S_{n} | 155 | MVA |

Voltage | U_{sn} | 15.75 | kV |

Current | I_{sn} | 5.682 | kA |

Power factor | cosφ_{n} | 0.89 | |

Speed | n_{n} | 600 | rpm |

Frequency | f_{n} | 50 | Hz |

Moment of inertia | J | 260 000 | kgm^{2} |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Stator resistance | R_{s} | 0.00189 | Ω |

Stator leakage reactance | X_{l} | 0.208 | Ω |

d-axis synchronous reactance | ${X}_{d}$ | 1.825 | Ω |

d-axis transient reactance | ${X}_{d}^{\prime}$ | 0.560 | Ω |

d-axis subtransient reactance | ${X}_{d}^{\u2033}$ | 0.288 | Ω |

q-axis synchronous reactance | ${X}_{q}$ | 1.088 | Ω |

q-axis subtransient reactance | ${X}_{q}^{\u2033}$ | 0.304 | Ω |

d-axis transient open-circuit time constant | ${T}_{d0}^{\prime}$ | 9.8 | s |

d-axis subtransient open-circuit time constant | ${T}_{d0}^{\u2033}$ | 0.073 | s |

q-axis subtransient open-circuit time constant | ${T}_{q0}^{\u2033}$ | 0.270 | s |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Apparent power | S_{tn} | 155 | MVA |

Frequency | f_{tn} | 50 | Hz |

Primary voltage | U_{t1n} | 400 | kV |

Secondary voltage | U_{t2n} | 15.75 | kV |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Resistance | R_{L} | 0.0370 | Ω/km |

Inductance | L_{L} | 0.0012 | H/km |

Length | l_{L} | 20 | km |

## References

- Anderson, P.M.; Fouad, A.A. Power System Control and Stability, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2003; ISBN 0-471-23862-7. [Google Scholar]
- Kundur, P. Power System Stability and Control, 3rd ed.; McGraw-Hill Education: New York, NY, USA, 1994; ISBN 0-07-035958-X. [Google Scholar]
- Sumina, D.; Bulić, N.; Erceg, I. Three-dimensional power system stabilizer. Electr. Power Syst. Res.
**2010**, 80, 886–892. [Google Scholar] [CrossRef] - Sumina, D.; Bulić, N.; Vražić, M. Load Angle Control of a Synchronous Generator. Prz Elektrotechniczn
**2012**, 88, 225–231. [Google Scholar] - Kaňuch, J.; Girovský, P. The device to measuring of the load angle for salient-pole synchronous machine in education laboratory. Measurement
**2018**, 116, 49–55. [Google Scholar] [CrossRef] - Despalatović, M.; Jadrić, M.; Terzić, B. Real-time power angle determination of salient-pole synchronous machine based on air gap measurements. Electr. Power Syst. Res.
**2008**, 78, 1873–1880. [Google Scholar] [CrossRef] - Sumina, D.; Šala, A.; Malarić, R. Determination of Load Angle for Salient-pole Synchronous Machine. Meas. Sci. Rev.
**2010**, 10, 89–96. [Google Scholar] [CrossRef] - Idžotić, T.; Erceg, G.; Sumina, D. Synchronous Generator Load Angle Measurement and Estimation. Automatika
**2004**, 45, 179–186. [Google Scholar] - Venkatasubramanian, V.; Kavasseri, R.G. Direct computation of generator internal dynamic states from terminal measurements. In Proceedings of the 37th Annual Hawaii International Conference on System Sciences, Big Island, HI, USA, 5–8 January 2004. [Google Scholar]
- Del Angel, A.; Glavic, M.; Wehenkel, L. Using artificial neural networks to estimate rotor angles and speeds from phasor measurements. In Proceedings of the Intelligent System Applications in Power ISAP2003, Lemnos, Greece, 2003; Available online: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.151.7413&rep=rep1&type=pdf (accessed on 27 April 2019).
- Del Angel, A.; Geurts, P.; Ernst, D.; Glavic, M.; Wehenkel, L. Estimation of rotor angles of synchronous machines using artificial neural networks and local PMU-based quantities. Neurocomputing
**2007**, 70, 2668–2678. [Google Scholar] [CrossRef] - Mišković, M.; Erceg, G.; Mirošević, M. Load angle estimation of a synchronous generator using dynamical neural networks. Energija
**2009**, 2, 174–191. [Google Scholar] - Wang, Y.; Wang, X.; Xie, W.; Dou, M. Full-Speed Range Encoderless Control for Salient-Pole PMSM with a Novel Full-Order SMO. Energies
**2018**, 11, 2423. [Google Scholar] [CrossRef] - Wang, M.-S.; Tsai, T.-M. Sliding Mode and Neural Network Control of Sensorless PMSM Controlled System for Power Consumption and Performance Improvement. Energies
**2017**, 10, 1780. [Google Scholar] [CrossRef] - Luo, X.; Niu, S. Maximum Power Point Tracking Sensorless Control of an Axial-Flux Permanent Magnet Vernier Wind Power Generator. Energies
**2016**, 9, 581. [Google Scholar] [CrossRef] - Sellami, T.; Berriri, H.; Jelassi, S.; Darcherif, A.M.; Mimouni, M.F. Short-Circuit Fault Tolerant Control of a Wind Turbine Driven Induction Generator Based on Sliding Mode Observers. Energies
**2017**, 10, 1611. [Google Scholar] [CrossRef] - Fang, H.; Derong, L.; Chengwei, L.; Zhuo, L.; Gongping, W. Cascaded Robust Fault-Tolerant Predictive Control for PMSM Drives. Energies
**2018**, 11, 3087. [Google Scholar] - Chavira, F.; Ortega-Cisneros, S.; Rivera, J. A Novel Sliding Mode Control Scheme for a PMSG-Based Variable Speed Wind Energy Conversion System. Energies
**2017**, 10, 1476. [Google Scholar] [CrossRef] - Merabet, A. Adaptive Sliding Mode Speed Control for Wind Energy Experimental System. Energies
**2018**, 11, 2238. [Google Scholar] [CrossRef] - Zribi, M.; Alrifai, M.; Rayan, M. Sliding Mode Control of a Variable- Speed Wind Energy Conversion System Using a Squirrel Cage Induction Generator. Energies
**2017**, 10, 604. [Google Scholar] [CrossRef] - Song, Q.; Li, Y.; Jia, C. A Novel Direct Torque Control Method Based on Asymmetric Boundary Layer Sliding Mode Control for PMSM. Energies
**2018**, 11, 657. [Google Scholar] [CrossRef] - Ji, K.; Huang, S. Direct Flux Control for Stand-Alone Operation Brushless Doubly Fed Induction Generators Using a Resonant-Based Sliding-Mode Control Approach. Energies
**2018**, 11, 814. [Google Scholar] - Zoghlami, M.; Kadri, A.; Bacha, F. Analysis and Application of the Sliding Mode Control Approach in the Variable-Wind Speed Conversion System for the Utility of Grid Connection. Energies
**2018**, 11, 720. [Google Scholar] [CrossRef] - Utkin, V.; Guldner, J.; Shi, J. Sliding Mode Control in Electro-Mechanical Systems, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2009; ISBN 978-1-4200-6560-2. [Google Scholar]
- Vas, P. Sensorless Vector and Direct Torque Control; Monographs in Electrical and Electronic Engineering; Oxford University Press: Oxford, UK, 1998; ISBN 978-0-19-856465-2. [Google Scholar]
- Liu, J.; Nondahl, T.A.; Schmidt, P.B.; Royak, S.; Harbaugh, M. Rotor Position Estimation for Synchronous Machines Based on Equivalent EMF. IEEE Trans. Ind. Appl.
**2011**, 47, 1310–1318. [Google Scholar] - Qiao, Z.; Shi, T.; Wang, Y.; Yan, Y.; Xia, C.; He, X. New Sliding-Mode Observer for Position Sensorless Control of Permanent-Magnet Synchronous Motor. IEEE Trans. Ind. Electron.
**2013**, 60, 710–719. [Google Scholar] [CrossRef] - Fan, Y.; Zhang, L.; Cheng, M.; Chau, K.T. Sensorless SVPWM-FADTC of a New Flux-Modulated Permanent-Magnet Wheel Motor Based on a Wide-Speed Sliding Mode Observer. IEEE Trans. Ind. Electron.
**2015**, 62, 3143–3151. [Google Scholar] [CrossRef] - Zhao, Y.; Zhang, Z.; Qiao, W.; Wu, L. An Extended Flux Model-Based Rotor Position Estimator for Sensorless Control of Salient-Pole Permanent-Magnet Synchronous Machines. IEEE Trans. Power Electron.
**2015**, 30, 4412–4422. [Google Scholar] [CrossRef] - Lin, S.; Zhang, W. An adaptive sliding-mode observer with a tangent function-based PLL structure for position sensorless PMSM drives. Int. J. Elec. Power.
**2017**, 88, 63–74. [Google Scholar] [CrossRef] - Liang, D.; Li, J.; Qu, R.; Kong, W. Adaptive Second-Order Sliding-Mode Observer for PMSM Sensorless Control Considering VSI Nonlinearity. IEEE Trans. Power Electron.
**2018**, 33, 8994–9004. [Google Scholar] [CrossRef] - Sheng, L.; Li, W.; Wang, Y.; Fan, M.; Yang, X. Sensorless Control of a Shearer Short-Range Cutting Interior Permanent Magnet Synchronous Motor Based on a New Sliding Mode Observer. IEEE Access.
**2017**, 5, 18439–18450. [Google Scholar] [CrossRef] - Ouassaid, M.; Maaroufi, M.; Cherkaoui, M. Observer-based nonlinear control of power system using sliding mode control strategy. Electr. Power Syst. Res.
**2012**, 84, 135–143. [Google Scholar] [CrossRef] - Jiang, L.; Wu, Q.H.; Zhang, C.; Zhou, X.X. Observer-based nonlinear control of synchronous generators with perturbation estimation. Int. J. Electr. Power Energy Syst.
**2001**, 23, 359–367. [Google Scholar] [CrossRef] - Loukianov, A.G.; Canedo, J.M.; Utkin, V.I.; Cabrera-Vazquez, J. Discontinuous Controller for Power Systems: Sliding-Mode Block Control Approach. IEEE Trans. Ind. Electron.
**2004**, 51, 340–353. [Google Scholar] [CrossRef]

**Figure 5.**Generator power change during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.41 ind.: (

**a**) Active power; (

**b**) Reactive power.

**Figure 6.**Load angle variation during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.41 ind.: (

**a**) Comparison of the actual and estimated load angle values; (

**b**) Load angle estimation error.

**Figure 8.**Generator power change during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0: (

**a**) Active power; (

**b**) Reactive power.

**Figure 9.**Load angle variation during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0: (

**a**) Comparison of the actual and estimated load angle values; (

**b**) Load angle estimation error.

**Figure 11.**Generator power change during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.11 cap.: (

**a**) Active power; (

**b**) Reactive power.

**Figure 12.**Load angle variation during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.11 cap.: (

**a**) Comparison of the actual and estimated load angle values; (

**b**) Load angle estimation error.

**Figure 14.**Generator power change during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.41 ind.: (

**a**) Active power; (

**b**) Reactive power.

**Figure 15.**Load angle variation during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.41 ind.: (

**a**) Comparison of the actual and estimated load angle values; (

**b**) Load angle estimation error.

**Figure 16.**Generator power change during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0: (

**a**) Active power; (

**b**) Reactive power.

**Figure 17.**Load angle variation during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0: (

**a**) Comparison of the actual and estimated load angle values; (

**b**) Load angle estimation error.

**Figure 18.**Generator power change during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.11 cap.: (

**a**) Active power; (

**b**) Reactive power.

**Figure 19.**Load angle variation during dynamic disturbances of active power for operating point P = 0.89, Q = 0.11 cap.: (

**a**) Comparison of the actual and estimated load angle values; (

**b**) Load angle estimation error.

**Figure 20.**Mean absolute error (MAE) and Maximum absolute error (MAXE) of the SMO based load angle estimator due to the variation of the estimator’s parameter: (

**a**) Stator resistance R

_{s}; (

**b**) Quadrature axis inductance L

_{q}.

**Table 1.**Performance indices of the SMO (sliding mode observer)-based estimator and the phasor diagram based estimator during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.41 ind. MSE: Mean squared error; MAE: Mean absolute error; MAXE: Maximum absolute error.

Estimator Type | MSE | MAE | MAXE |
---|---|---|---|

SMO-based estimator | 0.1165 | 0.0977 | 2.1682 |

Phasor diagram based estimator | 0.1279 | 0.1312 | 2.3217 |

**Table 2.**Relative performance improvement of the SMO-based estimator over the phasor diagram based estimator during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.41 ind.

Performance Indices Improvement | MSE | MAE | MAXE |
---|---|---|---|

Relative improvement (%) | 8.91 | 25.53 | 6.61 |

**Table 3.**Performance indices of the SMO-based estimator and the phasor diagram based estimator during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.

Estimator Type | MSE | MAE | MAXE |
---|---|---|---|

SMO-based estimator | 0.4372 | 0.2184 | 3.7164 |

Phasor diagram based estimator | 0.4688 | 0.2516 | 3.9163 |

**Table 4.**Relative performance improvement of the SMO-based estimator over the phasor diagram based estimator during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.

Performance Indices Improvement | MSE | MAE | MAXE |
---|---|---|---|

Relative improvement (%) | 6.74 | 13.20 | 5.10 |

**Table 5.**Performance indices of the SMO-based estimator and the phasor diagram based estimator during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.11 cap.

Estimator Type | MSE | MAE | MAXE |
---|---|---|---|

SMO-based estimator | 0.6505 | 0.3013 | 4.1189 |

Phasor diagram based estimator | 0.6960 | 0.3334 | 4.3391 |

**Table 6.**Relative performance improvement of the SMO-based estimator over the phasor diagram based estimator during dynamic disturbances of reactive power for generator operating point P = 0.89, Q = 0.11 cap.

Performance Indices Improvement | MSE | MAE | MAXE |
---|---|---|---|

Relative improvement (%) | 6.54 | 9.63 | 5.07 |

**Table 7.**Performance indices of the SMO-based estimator and the phasor diagram based estimator during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.41 ind.

Estimator Type | MSE | MAE | MAXE |
---|---|---|---|

SMO-based estimator | 0.0195 | 0.0415 | 0.9185 |

Phasor diagram based estimator | 0.0227 | 0.0710 | 1.0073 |

**Table 8.**Relative performance improvement of the SMO-based estimator over the phasor diagram based estimator during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.41 ind.

Performance Indices Improvement | MSE | MAE | MAXE |
---|---|---|---|

Relative improvement (%) | 14.10 | 41.55 | 8.81 |

**Table 9.**Performance indices of the SMO-based estimator and the phasor diagram based estimator during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.

Estimator Type | MSE | MAE | MAXE |
---|---|---|---|

SMO-based estimator | 0.0393 | 0.0626 | 1.2412 |

Phasor diagram based estimator | 0.0433 | 0.0907 | 1.3347 |

**Table 10.**Relative performance improvement of the SMO-based estimator over the phasor diagram based estimator during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.

Performance Indices Improvement | MSE | MAE | MAXE |
---|---|---|---|

Relative improvement (%) | 9.24 | 30.98 | 7.01 |

**Table 11.**Performance indices of the SMO-based estimator and the phasor diagram based estimator during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.11 cap.

Estimator Type | MSE | MAE | MAXE |
---|---|---|---|

SMO-based estimator | 0.0481 | 0.0736 | 1.3355 |

Phasor diagram based estimator | 0.0526 | 0.1009 | 1.4263 |

**Table 12.**Relative performance improvement of the SMO-based estimator over the phasor diagram based estimator during dynamic disturbances of active power for generator operating point P = 0.89, Q = 0.11 cap.

Performance Indices Improvement | MSE | MAE | MAXE |
---|---|---|---|

Relative improvement (%) | 8.55 | 27.06 | 6.37 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lopac, N.; Bulic, N.; Vrkic, N.
Sliding Mode Observer-Based Load Angle Estimation for Salient-Pole Wound Rotor Synchronous Generators. *Energies* **2019**, *12*, 1609.
https://doi.org/10.3390/en12091609

**AMA Style**

Lopac N, Bulic N, Vrkic N.
Sliding Mode Observer-Based Load Angle Estimation for Salient-Pole Wound Rotor Synchronous Generators. *Energies*. 2019; 12(9):1609.
https://doi.org/10.3390/en12091609

**Chicago/Turabian Style**

Lopac, Nikola, Neven Bulic, and Niksa Vrkic.
2019. "Sliding Mode Observer-Based Load Angle Estimation for Salient-Pole Wound Rotor Synchronous Generators" *Energies* 12, no. 9: 1609.
https://doi.org/10.3390/en12091609