Modal Aggregation Technique to Check the Accuracy of the Model Reduction of Array Cable Systems in Offshore Wind Farms
Abstract
:1. Introduction
2. Modal Aggregation Technique
Application
3. Case Study
4. Aggregation Techniques
4.1. Benchmarked Aggregation Techniques
4.2. Developing a $\pi $Equivalent from the Modal Aggregation Technique
5. Results
6. Discussion
6.1. Equal Current Injection–Monte Carlo Simulation
6.2. DCFR and Higher Frequencies
6.3. Impedance Magnitude Distance (Mode Ratio)
7. Conclusions and Future Works
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations and Subscripts
ADN  Active Distribution Network 
agg  Aggregated 
AS  Aggregated System 
CF  Contribution Factor 
CS  Collector System 
DCFR  Dominant Contribution Factor Ratio 
ISA  Individual String Aggregation 
MCS  Monte Carlo Simulation 
OSS  Offshore Substation 
OWF  Offshore Wind Farm 
PL  Power Loss 
VD  Voltage Drop 
WT  Wind Turbine 
Symbols
${V}^{\prime}$  Representation of V in the modal domain 
$\mathbf{X}$  Matrix or vector formed of elements ${x}_{i}$ 
$\epsilon $  Approximation error 
$\mathsf{\Lambda}$  Eigenvalues in matrix form (diagonally placed) 
r  Right eigenvector 
R  Right eigenvectors in matrix form 
V  Voltage 
I  Current 
Z  Impedance 
C  Capacitance 
M  Number of feeders connected to the offshore substation 
p  Modal turn ratio 
m  Contribution factor or m’th feeder 
$DCFR$  Dominant Capacity Factor Ratio 
Appendix A. TimeDomain Simulation Results for All Case Studies
Appendix B. Errors of TimeDomain Simulation Results for All Case Studies
References
 Chedid, R.; LaWhite, N.; Ilic, M. A Comparative Analysis of Dynamic Models for Performance Calculation of GridConnected Wind Turbine Generators. Wind Eng. 1993, 17, 168–182. [Google Scholar]
 Rahman, M.T.; Hasan, K.N.; Sokolowski, P. Evaluation of wind farm aggregation using probabilistic clustering algorithms for power system stability assessment. Sustain. Energy Grids Netw. 2022, 30, 100678. [Google Scholar] [CrossRef]
 Milanovic, J.V.; Mat Zali, S. Validation of equivalent dynamic model of active distribution network cell. IEEE Trans. Power Syst. 2013, 28, 2101–2110. [Google Scholar] [CrossRef]
 Li, H.; Chen, Z. Overview of different wind generator systems and their comparisons. IET Renew. Power Gener. 2008, 2, 123–138. [Google Scholar] [CrossRef] [Green Version]
 Fernández, L.M.; García, C.A.; Saenz, J.R.; Jurado, F. Equivalent models of wind farms by using aggregated wind turbines and equivalent winds. Energy Convers. Manag. 2009, 50, 691–704. [Google Scholar] [CrossRef]
 Shafiu, A.; AnayaLara, O.; Bathurst, G.; Jenkins, N. Aggregated wind turbine models for power system dynamic studies. Wind Eng. 2006, 30, 171–186. [Google Scholar] [CrossRef]
 Akhmatov, V.; Knudsen, H. An aggregate model of a gridconnected, largescale, offshore wind farm for power stability investigationsimportance of windmill mechanical system. Fuel Energy Abstr. 2003, 44, 161. [Google Scholar] [CrossRef]
 Tapia, G.; Tapia, A.; Ostolaza, J.X. Two alternative modeling approaches for the evaluation of wind farm active and reactive power performances. IEEE Trans. Energy Convers. 2006, 21, 909–920. [Google Scholar] [CrossRef]
 Windenergie Report Deutschland 2018. Available online: https://publica.fraunhofer.de/entities/publication/c3cff22662e84fa2b9c2c2e18fa9328a/details (accessed on 17 October 2022).
 Conroy, J.; Watson, R. Aggregate modelling of wind farms containing fullconverter wind turbine generators with permanent magnet synchronous machines: Transient stability studies. IET Renew. Power Gener. 2009, 3, 39–52. [Google Scholar] [CrossRef]
 Kunjumuhammed, L.P.; Pal, B.C.; Oates, C.; Dyke, K.J. The Adequacy of the Present Practice in Dynamic Aggregated Modeling of Wind Farm Systems. IEEE Trans. Sustain. Energy 2017, 8, 23–32. [Google Scholar] [CrossRef] [Green Version]
 Marinopoulos, A.; Pan, J.; Zarghami, M.; Reza, M.; Yunus, K.; Yue, C.; Srivastava, K. Investigating the impact of wake effect on wind farm aggregation. In Proceedings of the 2011 IEEE Pes Trondheim Powertech: The Power of Technology for a Sustainable Society, Powertech, Trondheim, Norway, 19–23 June 2011; p. 6019323. [Google Scholar] [CrossRef]
 Ruan, J.Y.; Lu, Z.X.; Qiao, Y.; Min, Y. Analysis on Applicability Problems of the AggregationBased Representation of Wind Farms Considering DFIGs’ LVRT Behaviors. IEEE Trans. Power Syst. 2016, 31, 4953–4965. [Google Scholar] [CrossRef]
 Fernandez, L.M.; Garcia, C.A.; Jurado, F.; Saenz, J.R. Aggregation of doubly fed induction generators wind turbines under different incoming wind speeds. In Proceedings of the 2005 IEEE Russia Power Tech, Powertech, St. Petersburg, Russia, 27–30 June 2005; p. 4524685. [Google Scholar] [CrossRef]
 Liu, H.; Chen, Z. Aggregated modelling for wind farms for power system transient stability studies. In Proceedings of the 2012 Asiapacific Power and Energy Engineering Conference, Shanghai, China, 27–29 March 2012; p. 6307118. [Google Scholar] [CrossRef]
 Fernández, L.M.; Jurado, F.; Saenz, J.R. Aggregated dynamic model for wind farms with doubly fed induction generator wind turbines. Renew. Energy 2008, 33, 129–140. [Google Scholar] [CrossRef]
 MercadoVargas, M.J.; GómezLorente, D.; Rabaza, O.; AlamedaHernandez, E. Aggregated models of permanent magnet synchronous generators wind farms. Renew. Energy 2015, 83, 1287–1298. [Google Scholar] [CrossRef]
 Wang, H.; Buchhagen, C.; Sun, J. Methods to aggregate turbine and network impedance for wind farm resonance analysis. IET Renew. Power Gener. 2020, 14, 1304–1311. [Google Scholar] [CrossRef]
 Cheng, X.; Lee, W.J.; Sahni, M.; Cheng, Y.; Lee, L.K. Dynamic equivalent model development to improve the operation efficiency of wind farm. In Proceedings of the 2015 IEEE/IAS 51st Industrial and Commercial Power Systems Technical Conference, I and Cps, Calgary, AB, Canada, 5–8 May 2015; p. 7266410. [Google Scholar] [CrossRef]
 Mat Zali, S.; Milanovic, J.V. Generic model of active distribution network for large power system stability studies. IEEE Trans. Power Syst. 2013, 28, 3126–3133. [Google Scholar] [CrossRef]
 Brogan, P. The stability of multiple, high power, active front end voltage sourced converters when connected to wind farm collector systems. EPE Wind Energy Chapter Semin. 2010, 1–6. [Google Scholar] [CrossRef]
 MartínezTurégano, J.; AñóVillalba, S.; BernalPerez, S.; BlascoGimenez, R. Aggregation of type4 large wind farms based on admittance model order reduction. Energies 2019, 12, 1730. [Google Scholar] [CrossRef]
 Zou, J.; Peng, C.; Xu, H.; Yan, Y. A Fuzzy Clustering AlgorithmBased Dynamic Equivalent Modeling Method for Wind Farm with DFIG. IEEE Trans. Energy Convers. 2015, 30, 1329–1337. [Google Scholar] [CrossRef]
 Muljadi, E.; Butterfield, C.P.; Ellis, A.; Mechenbier, J.; Hochheimer, J.; Young, R.; Miller, N.; Delmerico, R.; Zavadil, R.; Smith, J.C. Equivalencing the collector system of a large wind power plant. In Proceedings of the 2006 IEEE Power Engineering Society General Meeting, Montreal, QC, Canada, 18–22 June 2006; p. 1708945. [Google Scholar] [CrossRef]
 Muljadi, E.; Pasupulati, S.; Ellis, A.; Kosterov, D. Method of equivalencing for a large wind power plant with multiple turbine representation. In Proceedings of the IEEE Power and Energy Society 2008 General Meeting: Conversion and Delivery of Electrical Energy in the 21st Century, Pittsburgh, PA, USA, 20–24 July 2008; p. 4596055. [Google Scholar] [CrossRef]
 Abbes, M.; Allagui, M.; Hasnaoui, O.B. An aggregate model of PMSGbased, grid connected wind farm: Investigation of LVRT capabilities. In Proceedings of the 2015 6th International Renewable Energy Congress, Irec 2015, Sousse, Tunisia, 24–26 March 2015; p. 7110976. [Google Scholar] [CrossRef]
 Ali, M.; Ilie, I.S.; Milanovic, J.V.; Chicco, G. Wind farm model aggregation using probabilistic clustering. IEEE Trans. Power Syst. 2013, 28, 309–316. [Google Scholar] [CrossRef]
 Kocewiak, L.H.; Hjerrild, J.; Bak, C.L. Wind farm structures’ impact on harmonic emission and grid interaction. In Proceedings of the European Wind Energy Conference and Exhibition 2010, Ewec 2010, Warszawa, Poland, 20–23 April 2010; Volume 4, pp. 2829–2836. [Google Scholar]
 Pöller, M.; Achilles, S. Aggregated Wind Park Models for Analyzing Power System Dynamics. In Proceedings of the 4th International Workshop on LargeScale Integration of Wind Power and Transmission Networks for Offshore Wind Farms, Billund, Denmark, 2014. [Google Scholar]
 QuinonezVarela, G.; Ault, G.W.; AnayaLara, O.; McDonald, J.R. Electrical collector system options for large offshore wind farms. IET Renew. Power Gener. 2007, 1, 107–114. [Google Scholar] [CrossRef]
 Lin, W.; Wen, J.; Liang, J.; Cheng, S.; Yao, M.; Li, N. A ThreeTerminal HVDC System to Bundle Wind Farms with Conventional Power Plants. IEEE Trans. Power Syst. 2013, 28, 2292–2300. [Google Scholar] [CrossRef]
 Dutta, S.; Overbye, T.J. A clustering based wind farm collector system cable layout design. In Proceedings of the 2011 IEEE Power and Energy Conference at Illinois, Peci 2011, Urbana, IL, USA, 25–26 February 2011; p. 5740480. [Google Scholar] [CrossRef]
 Kocewiak, Ł.H.; Kramer, B.L.Ø.; Holmstrøm, O.; Jensen, K.H.; Shuai, L. Resonance damping in array cable systems by wind turbine active filtering in large offshore wind power plants. IET Renew. Power Gener. 2017, 11, 1069–1077. [Google Scholar] [CrossRef]
 Chow, J.H. Power System Coherency and Model Reduction; Springer: New York, NY, USA, 2013. [Google Scholar]
 Sundararajan, D. Introductory Circuit Theory; Springer: Cham, Switzerland, 2020. [Google Scholar]
 Chechurin, V.L.; Korovkin, N.V.; Hayakawa, M.; Hayakawa, M. Inverse Problems in Electric Circuits and Electromagnetics; Springer: New York, NY, USA, 2007. [Google Scholar]
 Kundur, P.S. Power System Stability and Control; McGrawHill: New York, NY, USA, 1994. [Google Scholar]
 Ørsted: Anholt Offshore Wind Farm. Available online: https://orstedcdn.azureedge.net//media/www/docs/corp/com/ourbusiness/windpower/windfarmprojectsummary/anholt_uk_2018.ashx?la=en&rev=3b2f175b341e4ca2b906bf2a2ff68c8f&hash=51B4609E53A631D29E0EF49936C33A84 (accessed on 26 September 2022).
 6–36 kV Medium Voltage Underground Power Cables. Available online: https://www.powerandcables.com/wpcontent/uploads/2016/12/Nexans633kVMediumHighVoltageUndergroundPowerCables.pdf (accessed on 18 October 2022).
 IEC 61400271 Ed. 2; Wind TurbinesPart 271: Electrical Simulation Models for Wind Power Generation—Wind Turbines. International Electrotechnical Commission: Geneva, Switzerland, 2020.
 PSCAD Type 4 Wind Turbine Generators. Available online: https://www.pscad.com/knowledgebase/article/227 (accessed on 25 August 2022).
 Kocewiak, L. Harmonics in Large Offshore Wind Farms. Ph.D. Dissertation, Department of Energy Technology at Aalborg University, Aalborg, Denmark, 2012. [Google Scholar]
 600601:2011; HighVoltage Test Requirements, Part 1: General Definitions and Test Requirements. International Electrotechnical Commission (IEC): Geneva, Switzerland, 2011.
Cable Type  R [m${\Omega}$/km]  L [mH/km]  C [$\mathsf{\mu}$F/km] 

150 mm^{2}  124  0.39  0.19 
240 mm^{2}  75.4  0.36  0.23 
500 mm^{2}  36.6  0.32  0.32 
Case  Number of Strings  Branches  Symmetry  ${\mathit{N}}_{\mathit{WT}}$  ${\mathit{S}}_{\mathit{base}}$ [MW] 

1  1  ✓  9  32.4  
2  2  ✓  9  32.4  
3  1  ✓  ✓  18  64.8 
4  2  ✓  ✓  18  64.8 
2*  2  13  46.8  
4*  2  ✓  13  46.8 
Voltage Drop (VD)  Power Loss #1 (PL1)  Power Loss #2 (PL2)  

${Z}_{m}$: Equivalent impedance of the individual feeder, m 
$${Z}_{\mathit{m}}=\frac{1}{{N}_{m}}\sum _{n}^{{N}_{m}}{w}_{n}{Z}_{\mathit{n},\mathit{m}}$$

$${Z}_{m}=\frac{1}{{N}_{m}^{2}}\sum _{n}^{{N}_{m}}{w}_{n}^{2}{Z}_{n,m}$$

$${Z}_{m}=\frac{1}{{N}_{m}^{2}}\sum _{n}^{{N}_{m}}{w}_{n}^{2}{Z}_{n,m}$$

${Z}_{\mathit{eq}}$: Equivalent impedance of the whole wind farm 
$${Z}_{\mathit{eq}}=\frac{1}{{\sum}_{i=1}^{M}\frac{1}{{Z}_{m}}}$$

$${Z}_{\mathit{eq}}=\frac{1}{{\sum}_{i=1}^{M}\frac{1}{{Z}_{m}}}$$

$${Z}_{eq}=\frac{{\sum}_{m}^{M}{N}_{m}^{2}{Z}_{m}}{{\left[{\sum}_{m}^{M}{N}_{m}\right]}^{2}}$$

${C}_{\mathit{eq}}$: Equivalent capacitance of the whole wind farm 
$${C}_{eq}=\sum _{m}^{M}\sum _{n}^{{N}_{m}}{C}_{n,m}$$

$${C}_{eq}=\sum _{m}^{M}\sum _{n}^{{N}_{m}}{C}_{n,m}$$

$${C}_{eq}=\sum _{m}^{M}\sum _{n}^{{N}_{m}}{C}_{n,m}$$

Case  Voltage Drop  Power Loss #1  Power Loss #2 

1  408.29  0.80  0.80 
2  1170.77  1.90  2.08 
3  227.79  1.19  1.19 
4  3238.18  24.37  1.51 
2*  510.52  1816.51  2.41 
4*  1258.84  2041.78  1.22 
Average  1135.73  647.76  1.53 
Case  Measure  VD  PL1  PL2  PL2^{ISA}  Modal  Best 

1  P  2.35  2.91  2.91  2.91  2.86  VD 
Q  4.80  2.92  2.92  2.92  2.90  Modal  
I  2.06  1.30  1.30  1.30  1.30  PL1, PL2, PL2${}^{ISA}$, Modal  
V+  0.25  0.15  0.15  0.15  0.15  PL1, PL2, PL2${}^{ISA}$, Modal  
2  P  4.22  4.56  4.56  4.56  4.62  VD 
Q  3.35  2.69  2.69  2.69  2.73  PL1, PL2, PL2${}^{ISA}$  
I  1.13  1.11  1.11  1.10  1.12  PL2${}^{ISA}$  
V+  0.38  0.32  0.32  0.32  0.33  PL1, PL2, PL2${}^{ISA}$  
3  P  0.88  0.47  0.47  0.47  1.22  PL1, PL2, PL2${}^{ISA}$ 
Q  2.22  0.43  0.43  0.43  0.65  PL1, PL2, PL2${}^{ISA}$  
I  0.43  0.52  0.52  0.52  0.87  VD  
V+  0.12  0.03  0.03  0.03  0.04  PL1, PL2, PL2${}^{ISA}$  
4  P  4.05  4.83  4.88  2.89  4.88  PL2${}^{ISA}$ 
Q  3.18  2.68  2.70  1.81  2.70  PL2${}^{ISA}$  
I  1.00  0.90  0.90  0.68  0.90  PL2${}^{ISA}$  
V+  0.36  0.31  0.31  0.22  0.31  PL2${}^{ISA}$  
2*  P  2.84  4.51  2.90  2.18  2.83  PL2${}^{ISA}$ 
Q  1.54  2.66  1.59  1.18  1.56  PL2${}^{ISA}$  
I  1.56  1.92  1.57  1.25  1.55  PL2${}^{ISA}$  
V+  0.13  0.19  0.13  0.10  0.13  PL2${}^{ISA}$  
4*  P  1.37  1.57  0.76  1.84  0.76  PL2, Modal 
Q  1.53  1.75  1.56  1.70  1.56  VD  
I  4.04  4.06  4.01  4.06  4.01  PL2, Modal  
V+  0.14  0.15  0.14  0.15  0.14  PL2, Modal  
Average  1.83  1.79  1.62  1.48  1.67  PL2 
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. 
© 2022 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 (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Bakhshizadeh, M.K.; Vilmann, B.; Kocewiak, Ł. Modal Aggregation Technique to Check the Accuracy of the Model Reduction of Array Cable Systems in Offshore Wind Farms. Energies 2022, 15, 7996. https://doi.org/10.3390/en15217996
Bakhshizadeh MK, Vilmann B, Kocewiak Ł. Modal Aggregation Technique to Check the Accuracy of the Model Reduction of Array Cable Systems in Offshore Wind Farms. Energies. 2022; 15(21):7996. https://doi.org/10.3390/en15217996
Chicago/Turabian StyleBakhshizadeh, Mohammad Kazem, Benjamin Vilmann, and Łukasz Kocewiak. 2022. "Modal Aggregation Technique to Check the Accuracy of the Model Reduction of Array Cable Systems in Offshore Wind Farms" Energies 15, no. 21: 7996. https://doi.org/10.3390/en15217996