# Fast Prediction of Characteristics in Wound Rotor Synchronous Condenser Using Subdomain Modeling
## 1 Introduction  
Growing environmental concerns and the impacts of climate change are driving the shift from fossil-fuel-based power generation to renewable energy sources. Integrating renewable energy into power systems worldwide helps reduce greenhouse gas emissions and addresses the significant CO\({}_{2}\) emissions of the energy sector, offering sustainable and eco-friendly electricity to power electronic systems. Achieving net-zero emissions by 2050 is essential to limiting global temperature rise, prompting significant changes in the global energy landscape. By 2050, variable renewable energy sources like wind and solar power are expected to dominate the energy mix, with global electricity production projected to increase by 70%. However, the intermittency of renewable energy requires innovative solutions to maintain grid stability, including advanced energy storage and flexible demand management. Consequently, increasing the installed capacity of renewable energy sources in power systems has become a global priority [1; 2; 3; 4]. Most renewable energy resources (RERs) connect to power systems through power electronic converters, which lack the inertia provided by traditional fossil fuel generators. The high penetration of RERs, coupled with the retirement of aging fossil-fuel-based generators, has reduced power system inertia. Moreover, most RERs are located in rural areas and connect to power systems via long transmission lines. These long lines have highimpedance, which lowers the power system's short circuit ratio [5]. As a result, transmitting active power through the system consumes significant reactive power, reducing the AC grid's dynamic reactive power reserve and making voltage stability a critical issue [6]. A synchronous condenser is a machine used as a reactive power compensation device to enhance AC power system performance. It provides two primary functions: load compensation and voltage support. Load compensation improves power factor, balances loads, and eliminates current harmonics in nonlinear industrial loads. Voltage support aims to reduce voltage fluctuations at transmission line terminals [7; 8], increase the maximum transmittable active power, and enhance AC system stability [9]. Studies in [10; 11] discuss converting existing or retired synchronous generators into synchronous condensers. This conversion involves evaluating and modifying auxiliary equipment, accelerating the generator slightly above synchronous speed, powering the excitation system, and synchronizing the condenser with the transmission system. The system maintains the desired terminal voltage of the condenser using automatic voltage regulation controls. In analyzing electrical machines, particularly synchronous condensers, air-gap flux density is crucial for designers, as it enables determination of parameters such as voltage, torque, and power. Previous studies have mainly relied on finite element method (FEM) simulations, which can be time-consuming, or on magnetic equivalent circuit approaches [12], which often lack precision. Analytical methods offer faster computation with results that align closely with FEM simulations, making them an attractive option for pre-design, analysis, and optimization of wound rotor synchronous condensers (WRSCs). Two main analytical approaches have been explored: subdomain modeling and harmonic modeling. Subdomain modeling involves solving Maxwell's equations and boundary/interface conditions using Fourier series expansion. This approach has been applied to surface permanent magnet (SPM) motors [13]. The second approach, harmonic modeling, addresses permeability issues using a complex Fourier series and a Cauchy product, allowing consideration of nonlinear core characteristics. It was initially introduced in analyzing switched reluctance motors (SRMs) [14]. Based on these analytical approaches, ref. [15] optimized cogging torque by investigating the slot opening and magnet pitch ratio, while [16; 17; 18] optimized output power in magnetically geared machines (MG) and SPMs.