{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,7]],"date-time":"2026-07-07T20:56:43Z","timestamp":1783457803558,"version":"3.55.0"},"reference-count":55,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,29]],"date-time":"2025-06-29T00:00:00Z","timestamp":1751155200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Robust dynamic equivalents of large power networks are essential for fast and reliable stability analysis of bulk power systems. This is because the dimensionality of modern power systems raises convergence issues in modern stability-analysis programs. However, even with modern computational power, it is challenging to find reduced-order models for power systems due to the following factors: the tedious mathematical analysis involved in the classical reduction techniques requires large amounts of computational power; inadequate information sharing between geographical areas prohibits the execution of model-dependent reduction techniques; and frequent fluctuations in the operating conditions (OPs) of power systems necessitate updates to reduced models. This paper focuses on a measurement-based approach that uses a deep artificial neural network (DNN) to estimate the dynamics of an external system (ES) of a power system, enabling stability analysis of a study system (SS). This DNN technique requires boundary measurements only between the SS and the ES. However, machine learning-based techniques like this DNN are known for their extensive training requirements. In particular, for power systems that undergo continuous fluctuations in operating conditions due to the use of renewable energy sources, the applications of this DNN technique are limited. To address this issue, a Deep Transfer Learning (DTL)-based technique is proposed in this paper. This approach accounts for variations in the OPs such as time-to-time variations in loads and intermittent power generation from wind and solar energy sources. The proposed technique adjusts the parameters of a pretrained DNN model to a new OP, leveraging symmetry in the balanced adaptation of model layers to maintain consistent dynamics across operating conditions. The experimental results were obtained by representing the Queensland (QLD) system in the simplified Australian 14 generator (AU14G) model as the SS and the rest of AU14G as the ES in five scenarios that represent changes to the OP caused by variations in loads and power generation.<\/jats:p>","DOI":"10.3390\/sym17071023","type":"journal-article","created":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T03:54:28Z","timestamp":1751255668000},"page":"1023","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Transfer-Learning-Based Approach to Symmetry-Preserving Dynamic Equivalent Modeling of Large Power Systems with Small Variations in Operating Conditions"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7407-8826","authenticated-orcid":false,"given":"Lahiru","family":"Aththanayake","sequence":"first","affiliation":[{"name":"Australian Energy Market Operator (AEMO), Melbourne, VIC 3000, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Devinder","family":"Kaur","sequence":"additional","affiliation":[{"name":"Australian Energy Market Operator (AEMO), Melbourne, VIC 3000, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2354-7960","authenticated-orcid":false,"given":"Shama Naz","family":"Islam","sequence":"additional","affiliation":[{"name":"School of Engineering, Deakin University, Geelong, VIC 3220, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4635-4993","authenticated-orcid":false,"given":"Ameen","family":"Gargoom","sequence":"additional","affiliation":[{"name":"School of Engineering, Deakin University, Geelong, VIC 3220, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nasser","family":"Hosseinzadeh","sequence":"additional","affiliation":[{"name":"Energy Queensland, Townsville, QLD 4810, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"109917","DOI":"10.1016\/j.epsr.2023.109917","article-title":"Power system reduction techniques for planning and stability studies: A review","volume":"227","author":"Aththanayake","year":"2024","journal-title":"Electr. Power Syst. Res."},{"key":"ref_2","unstructured":"Kron, G. (1939). Tensor Analysis of Networks, J. Wiley & Sons."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1109\/EE.1949.6444973","article-title":"Equivalent circuits for power-flow studies","volume":"68","author":"Ward","year":"1949","journal-title":"Electr. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1109\/59.54565","article-title":"Application of the REI equivalent for operations planning analysis of interchange schedules","volume":"5","author":"Oatts","year":"1990","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/0142-0615(90)90007-X","article-title":"Singular perturbation analysis of large-scale power systems","volume":"12","author":"Chow","year":"1990","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/S1474-6670(17)64799-2","article-title":"Order reduction of linear state-space models via optimal approximation of the nondominant modes","volume":"13","author":"Litz","year":"1980","journal-title":"IFAC Proc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2049","DOI":"10.1109\/TPWRS.2014.2301032","article-title":"Dynamic-feature extraction, attribution, and reconstruction (DEAR) method for power system model reduction","volume":"29","author":"Wang","year":"2014","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1109\/TPWRS.2005.846109","article-title":"Model reduction in power systems using Krylov subspace methods","volume":"20","author":"Chaniotis","year":"2005","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Scarciotti, G. (2015, January 26\u201330). Model reduction of power systems with preservation of slow and poorly damped modes. Proceedings of the 2015 IEEE Power & Energy Society General Meeting, Denver, CO, USA.","DOI":"10.1109\/PESGM.2015.7285719"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1068","DOI":"10.1109\/TPWRS.2004.825825","article-title":"Coherency and aggregation techniques incorporating rotor and voltage dynamics","volume":"9","author":"Joo","year":"2004","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5401","DOI":"10.1109\/TPWRS.2018.2809548","article-title":"Power system coherency identification under high depth of penetration of wind power","volume":"33","author":"Khalil","year":"2018","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3487","DOI":"10.1109\/TPWRS.2024.3518593","article-title":"Model Order Reduction of Large-Scale Wind Farms: A Data-Driven Approach","volume":"40","author":"Gong","year":"2025","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"11997","DOI":"10.1109\/ACCESS.2019.2892060","article-title":"A black-box external equivalent method using tie-line power mutation","volume":"7","author":"Yan","year":"2019","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1109\/TPWRS.2003.821459","article-title":"Transient power system analysis with measurement-based gray box and hybrid dynamic equivalents","volume":"19","author":"Stankovic","year":"2004","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_15","unstructured":"Shahzad, U. (2022). Artificial Neural Network For Transient Stability Assessment: A Review. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1822","DOI":"10.1016\/j.apenergy.2018.07.084","article-title":"A survey of artificial neural network in wind energy systems","volume":"228","author":"Perez","year":"2018","journal-title":"Appl. Energy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"136848","DOI":"10.1016\/j.scitotenv.2020.136848","article-title":"Solar irradiance measurement instrumentation and power solar generation forecasting based on Artificial Neural Networks (ANN): A review of five years research trend","volume":"715","author":"Pazikadin","year":"2020","journal-title":"Sci. Total Environ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"106025","DOI":"10.1016\/j.epsr.2019.106025","article-title":"A novel fuzzy-based ensemble model for load forecasting using hybrid deep neural networks","volume":"178","author":"Sideratos","year":"2020","journal-title":"Electr. Power Syst. Res."},{"key":"ref_19","first-page":"2419","article-title":"Hybrid renewable energy based smart grid system for reactive power management and voltage profile enhancement using artificial neural network","volume":"43","author":"Chandrasekaran","year":"2021","journal-title":"Energy Sources Part A Recover. Util. Environ. Eff."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1478","DOI":"10.1109\/TPWRS.2003.818704","article-title":"Identification of nonparametric dynamic power system equivalents with artificial neural networks","volume":"18","author":"Stankovic","year":"2003","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.ijepes.2009.03.016","article-title":"Identification of a continuous time nonlinear state space model for the external power system dynamic equivalent by neural networks","volume":"31","author":"Shakouri","year":"2009","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_22","unstructured":"Cai, C., Wu, M., Dou, L., Dou, Z., and Zheng, J. (2014, January 20\u201322). Micro-grid dynamic modeling based on RBF Artificial Neural Network. Proceedings of the 2014 International Conference on Power System Technology, Chengdu, China."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1109\/72.298229","article-title":"Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems","volume":"5","author":"Vt","year":"1994","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Tong, N., Jiang, Z., You, S., Zhu, L., Deng, X., Xue, Y., and Liu, Y. (2020, January 1\u20133). Dynamic equivalence of large-scale power systems based on boundary measurements. Proceedings of the 2020 American Control Conference (ACC), Denver, CO, USA.","DOI":"10.23919\/ACC45564.2020.9147425"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1016\/S0893-6080(99)00067-2","article-title":"HyFIS: Adaptive neuro-fuzzy inference systems and their application to nonlinear dynamical systems","volume":"12","author":"Kim","year":"1999","journal-title":"Neural Netw."},{"key":"ref_26","unstructured":"Han, C., Changhong, D., and Dalu, L. (June, January 30). Recurrent neural network-based dynamic equivalencing in power system. Proceedings of the 2007 IEEE International Conference on Control and Automation, Guangzhou, China."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1049\/ip-gtd:20041070","article-title":"Artificial neural network-based dynamic equivalents for distribution systems containing active sources","volume":"151","author":"Azmy","year":"2004","journal-title":"IEEE Proc.-Gener. Transm. Distrib."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2611","DOI":"10.1109\/TNNLS.2018.2885219","article-title":"A novel equivalent model of active distribution networks based on LSTM","volume":"30","author":"Zheng","year":"2019","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1016\/j.egyr.2020.11.041","article-title":"Dynamic equivalent modeling for microgrid based on GRU","volume":"6","author":"Li","year":"2020","journal-title":"Energy Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"23120","DOI":"10.1109\/ACCESS.2020.2966238","article-title":"Microgrid equivalent modeling based on long short-term memory neural network","volume":"8","author":"Cai","year":"2020","journal-title":"IEEE Access"},{"key":"ref_31","first-page":"639","article-title":"Dynamic equivalent modeling for black-box microgrid under multi-operating-point by using LSTM","volume":"10","author":"Li","year":"2022","journal-title":"Csee J. Power Energy Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"113067","DOI":"10.1016\/j.asoc.2025.113067","article-title":"Strategic integration of adaptive sampling and ensemble techniques in federated learning for aircraft engine remaining useful life prediction","volume":"175","author":"Xu","year":"2025","journal-title":"Appl. Soft Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4639","DOI":"10.1109\/TII.2025.3545079","article-title":"Robust Transfer Learning for Battery Lifetime Prediction Using Early Cycle Data","volume":"21","author":"Kang","year":"2025","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"111948","DOI":"10.1016\/j.ymssp.2024.111948","article-title":"An embedded physical information network for blade crack detection considering dynamic multi-level credibility","volume":"224","author":"Shen","year":"2025","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1186\/s40537-016-0043-6","article-title":"A survey of transfer learning","volume":"3","author":"Weiss","year":"2016","journal-title":"J. Big Data"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Meghdadi, S., Tack, G., Liebman, A., Langren\u00e9, N., and Bergmeir, C. (2021, January 26\u201329). Versatile and robust transient stability assessment via instance transfer learning. Proceedings of the 2021 IEEE Power & Energy Society General Meeting (PESGM), Washington, DC, USA.","DOI":"10.1109\/PESGM46819.2021.9638195"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4856","DOI":"10.1109\/TPWRS.2021.3089042","article-title":"An integrated transfer learning method for power system dynamic security assessment of unlearned faults with missing data","volume":"36","author":"Ren","year":"2021","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"118729","DOI":"10.1016\/j.apenergy.2022.118729","article-title":"Transfer learning based multi-layer extreme learning machine for probabilistic wind power forecasting","volume":"312","author":"Liu","year":"2022","journal-title":"Appl. Energy"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Sarmas, E., Dimitropoulos, N., Marinakis, V., Mylona, Z., and Doukas, H. (2022). Transfer learning strategies for solar power forecasting under data scarcity. Sci. Rep., 12.","DOI":"10.1038\/s41598-022-18516-x"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1016\/j.enbuild.2016.01.030","article-title":"Unsupervised energy prediction in a Smart Grid context using reinforcement cross-building transfer learning","volume":"116","author":"Mocanu","year":"2016","journal-title":"Energy Build."},{"key":"ref_41","unstructured":"Xu, B., Guo, F., Wen, C., Deng, R., and Zhang, W. (2021). Detecting False Data Injection Attacks in Smart Grids with Modeling Errors: A Deep Transfer Learning Based Approach. arXiv."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4483","DOI":"10.1109\/TPWRS.2015.2509481","article-title":"Power system dynamic model reduction based on extended Krylov subspace method","volume":"31","author":"Zhu","year":"2016","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_43","unstructured":"Wang, X., Song, Y., and Irving, M. (2010). Modern Power Systems Analysis, Springer Science & Business Media."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1109\/TEC.2003.822296","article-title":"Identification of physical parameters of a synchronous generator from online measurements","volume":"19","author":"Karrari","year":"2004","journal-title":"IEEE Trans. Energy Convers."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.solener.2020.05.053","article-title":"Stability and control of power systems with high penetrations of inverter-based resources: An accessible review of current knowledge and open questions","volume":"210","author":"Kenyon","year":"2020","journal-title":"Sol. Energy"},{"key":"ref_46","unstructured":"Kundur, P.S., and Malik, O.P. (2022). Power System Stability and Control, McGraw-Hill Education."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1109\/JPROC.2020.3004555","article-title":"A comprehensive survey on transfer learning","volume":"109","author":"Zhuang","year":"2020","journal-title":"Proc. IEEE"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Iman, M., Arabnia, H.R., and Rasheed, K. (2023). A review of deep transfer learning and recent advancements. Technologies, 11.","DOI":"10.3390\/technologies11020040"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Aththanayake, L., Mahmud, A., Hosseinzadeh, N., and Gargoom, A. (2021, January 25\u201328). Performance Analysis of Regression and Artificial Neural Network Schemes for Dynamic Model Reduction of Power Systems. Proceedings of the 2021 3rd International Conference on Smart Power & Internet Energy Systems (SPIES), Shanghai, China.","DOI":"10.1109\/SPIES52282.2021.9633912"},{"key":"ref_50","unstructured":"Aththanayake, L., Hosseinzadeh, N., Mahmud, A., Gargoom, A., and Farahani, E.M. (December, January 29). Comparison of different techniques for voltage stability analysis of power systems. Proceedings of the 2020 Australasian Universities Power Engineering Conference (AUPEC), Hobart, Australia."},{"key":"ref_51","unstructured":"(2019). IEEE Draft Guide for Synchronous Generator Modeling Practices and Parameter Verification with Applications in Power System Stability Analyses (Standard No. IEEE P1110\/D06)."},{"key":"ref_52","unstructured":"Gibbard, M., and Vowles, D. (2014). Simplified 14-Generator Model of the South East Australian Power System, The University of Adelaide."},{"key":"ref_53","unstructured":"SIEMENS (2017). PSSE Program Application Guide Volume 1, SIEMENS."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Aksan, F., Li, Y., Suresh, V., and Janik, P. (2023). CNN-LSTM vs. LSTM-CNN to Predict Power Flow Direction: A Case Study of the High-Voltage Subnet of Northeast Germany. Sensors, 23.","DOI":"10.3390\/s23020901"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"8110","DOI":"10.1109\/ACCESS.2025.3527047","article-title":"Uniform Physics Informed Neural Network Framework for Microgrid and Its Application in Voltage Stability Analysis","volume":"13","author":"Feng","year":"2025","journal-title":"IEEE Access"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1023\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:01:17Z","timestamp":1760032877000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1023"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,29]]},"references-count":55,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["sym17071023"],"URL":"https:\/\/doi.org\/10.3390\/sym17071023","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,29]]}}}