Integrated simulation of electromechanical and thermal dynamics of voltage source converters

Electrical Power and Energy Systems 155 (2024) 109672 Available online 23 November 2023 0142-0615/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect International Journal of Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Integrated simulation of electromechanical and thermal dynamics of voltage source converters Juan Manuel Mauricio, J. Carlos Olives-Camps, José María Maza-Ortega∗, Antonio Gómez-Expósito Department of Electrical Engineering, Universidad de Sevilla, Camino de los descubrimientos s/n, Seville, 41092, Spain ARTICLE INFO Keywords: Insulated-gate bipolar transistors (IGBTs) Junction temperature consideration Power system simulation Thermal modeling Two-level voltage source converter (VSC) ABSTRACT This paper proposes a simplified yet accurate enough thermal model of Voltage Source Converters (VSC), aimed at circumventing the high computational cost of existing models, which prevents their use in electromechanical simulations. The proposed model reduces to a simple first-order system for thermal dynamics plus two quadratic equations separately modeling the IGBT and diode power losses. In addition, a methodology is provided to derive the proposed VSC thermal model parameters from manufacturer data. The proposed model is tested for two types of devices, both in steady and transient states. The results show that the reduced-order thermal model produces accurate results at a low computational cost, making it especially suitable for the co-simulation of thermal and electrical dynamic phenomena. 1. Introduction The proliferation of Voltage Source Converters (VSCs) in power systems is unstoppable. VSCs are interfacing renewable energy sources (photovoltaic and wind energy systems), storage units (supercapacitors, batteries and flywheels), railway systems and they are also present in High Voltage DC (HVDC) and Flexible AC Transmission Systems (FACTS) applications [1]. The semiconductor switches most commonly used in VSC applications are the insulated-gate bipolar transistors (IGBTs). One of the main limitations of semiconductor switches, determining their safe operating area, refers to the maximum junction temperature, which evolves in accordance to power losses and the design of the thermal dissipator. Commonly, this temperature is limited to values around 125 or 150 ◦C for IGBTs. Exceeding this thermal limit reduces the IGBT useful life and may cause irreversible damages. For this reason, manufacturers offer data about the maximum currents for some operating conditions, usually in steady-state regime, and transient overloads over a short time, supposedly guaranteeing a junction temperature below its maximum value. However, such maximum currents stand just for those standardized conditions. Therefore, it is of utmost importance to develop a simulation tool capable of dealing with the thermal behavior (power losses, heat dissipation and IGBT temperature) of power electronics devices under any circumstances, particularly nowadays, when traditional synchronous machines are being replaced by power converters [2], which are called to provide flexibility in order to keep the network operating in a stable manner. As a matter ∗Corresponding author. E-mail address: [email protected] (J.M. Maza-Ortega). of fact, this is already regulated on the generation side by restrictive grid codes [3,4], where renewable power plants are no longer expected to operate as pure active power injections with unity power factor. These specifications force power converters to offer similar ancillary services to those provided by classical synchronous generators. Reactive power support [5,6], inertia provision [7,8], frequency regulation [9], ramp limitation [10,11] are some of these ancillary services. An adequate computational tool including an accurate thermal model may disclose relevant information about how these ancillary services can be provided, possibly allowing transient overloads above the steady-state limits, but also about their impact on the device, i.e. additional power losses and thermal stress. Such simulation tool must take into account the different dynamics involved in the thermal and electrical parts of the power electronics device. Fig. 1 shows different power system phenomena and their associated time constants. Those transients that are considered fast (from 1 μs to some few seconds) are within the electromagnetic domain while those slower than 1 ms are considered in the electromechanical one. The reduced time steps required for simulating electromagnetic phenomena prevent simulations of more than a dozen generators or so, as CPU times would be unacceptable otherwise. On the contrary, electromechanical simulations may tackle with thousands of them at reasonable computational cost. Note that most of the ancillary services mentioned above fall within this slow dynamic range and, therefore, they are analyzed through electromechanical simulations. https://doi.org/10.1016/j.ijepes.2023.109672 Received 8 November 2022; Received in revised form 24 September 2023; Accepted 16 November 2023 International Journal of Electrical Power and Energy Systems 155 (2024) 109672 2 J.M. Mauricio et al. Nomenclature 𝐼RMS current 𝐼=𝐼𝑒𝑗𝜃 Phasor current 𝑉𝑠=𝑉𝑠𝑒𝑗𝜃𝑠and 𝑉𝑡=𝑉𝑡𝑒𝑗𝜃𝑡Phasor phase-to-neutral grid and VSC voltage 𝑉𝑎𝑐 AC phase-to-phase RMS grid voltage 𝜑Power factor angle 𝑃𝑖and 𝑃𝑑Individual IGBT and diode power losses 𝑃𝑠Single switch (IGBT and diode) power losses 𝑇𝑎Ambient temperature 𝑇𝑖and 𝑇𝑑IGBT and diode juncture temperature 𝑇𝑐and 𝑇𝑠Case and heatsink temperature 𝑇𝑠𝑠 𝑠Heatsink steady-state temperature 𝑁ℎNumber of heatsink switches 𝑅𝑖−𝑠 𝑡ℎ IGBT juncture-to-heatsink thermal resistance 𝑅𝑑−𝑠 𝑡ℎ Diode juncture-to-heatsink thermal resistance 𝑅𝑠−𝑎 𝑡ℎ Heatsink-to-ambient thermal resistance 𝐶𝑠 𝑡ℎ Heatsink thermal capacitance The issue is that thermal analysis of IGBT-based switches is a complex task, because some phenomena belong to the electromagnetic domain, e.g. IGBT juncture and case time constant (𝜏 < 100 ms), while others are in the slower range of the electromechanical domain, e.g. heatsink temperature time constant (𝜏 > 10 s). Several accurate electrothermal models for IGBTs are described and implemented in [12–17]. However the scope of those models is the very fast thermal transient of the juncture temperature. Such models require considering the physics involved in the switching process to determine the switching and conduction losses. This kind of model is typically developed on electromagnetic simulations platforms, such as SPICE, SABER or EMTP, requiring integration time steps smaller than 1 μs [18]. A model simplification is proposed in [19], where the overall thermal model is overly reduced to just one differential equation. However, this equation retains the small time constant of the original model and, therefore, it requires the use of reduced time steps for practical purposes. In addition to modeling the thermal behavior of the VSC, another major issue is that of finding suitable values for the involved model parameters. Semiconductor manufacturers provide software tools for this purpose, but most of them are on-line tools [20–24], hindering the interaction between the electric and thermal simulations. Those tools provide a simulation environment to analyze whether the power electronic switch is able to withstand a given load profile, usually defined as a steady-state load with an overload factor during a long period of time, which is totally unsuitable for the framework of electromechanical studies. Considering the above issues, this paper proposes an original methodology to incorporate the thermal behavior of power electronic devices in electromechanical simulations. For this purpose, a new thermal domain, as depicted in blue in Fig. 1, is defined. The time scale range associated to this new domain is able to consider the most important transients from the point of view of the electromechanical model, while simplifying the fast thermal dynamic complexity. For doing so, the approach considered in [25] is adopted, by which fast dynamics, e.g. fast thermal junction and case thermal phenomena, are modeled as algebraic equations while slow dynamics, e.g. heatsink behavior, are represented by differential ones. Moreover, the model parameters can be computed in a straightforward manner from public data provided by the semiconductor manufacturers. In this way, the parameterized thermal model can be easily integrated into electromechanical simulations, facilitating the analysis of the thermal behavior and limitations of power electronics converters. The rest of the paper is organized as follows. Section 2provides the methodology proposed in this paper for the integrated analysis of electromechanical and thermal simulation of VSCs. Section 3presents a mathematical model of the power losses generated in the operation of power switches and the thermal model describing the heat transfer. In the same section, a method to obtain the constants that appear in these models is detailed. The implementation of the proposed thermal model in current VSC electromechanical models is explained in Section 4. Section 5validates the proposed thermal model by comparing its results with the manufacturer solutions. Finally, the document closes with the conclusions. 2. Proposed methodology As the thermal and electromechanical simulation tools are implemented in separate software platforms, power engineers dealing with these issues necessarily follow the workflow depicted on the top diagram of Fig. 2. The first step is to perform dynamic network studies with electromechanical simulation programs, e.g. PSS/E or DigSILENT, where the device under study is faced to the worst scenario conditions. After that, the required input information for the thermal analysis tool must be extracted from the simulated data and exported to the manufacturer application. Usually this corresponds to a reduced time series of VSC RMS current, power factor and modulation index, whose size mainly depends on the manufacturer software characteristics. Then, the thermal behavior is simulated, yielding typically information about temperatures and power losses. If the temperature exceeds the imposed limits, the designer has to run again the electromechanical simulation with new limits or a different control strategy. Therefore, the current methodology is a tedious iterative process involving data transfers back and forth between separate simulation tools. The proposed methodology can be explained with the help of the bottom diagram of Fig. 2. Basically, it relies on obtaining a thermal model from the manufacturer simulation tool and incorporating it within the electromechanical simulation. This greatly simplifies the process for three reasons. First, for a given power electronic stack, it is required to obtain the thermal model only once by using the manufacturer simulation tool. Second, power losses and temperatures are obtained in a straightforward manner from the power system simulation tool, just by incorporating the tuned thermal model. Third, as the temperature variable is explicitly handled within the electromechanical simulation, it should be possible to incorporate additional constraints, such as temperature protections or even controllers that consider the temperature evolution, which is very awkward to do using existing models. 3. VSC thermal model This section presents the thermal model required to obtain the different operating temperatures of a VSC. The section is subdivided into three different subsections dealing respectively with the VSC thermal dynamic model, which relates temperatures with the heat sources, the power losses model and a method for computing all the model parameters using available data from manufacturers. It is worth noting that the proposed VSC thermal model relies on simulation tools provided by manufacturers. It is assumed that these thermal simulation tools have been previously validated by the manufacturers through experimental testing, given the fact that they are widely used to rate and design the power electronic stacks. For this reason, the experimental validation of the proposed thermal model is out of the scope of this paper. International Journal of Electrical Power and Energy Systems 155 (2024) 109672 3 J.M. Mauricio et al. Fig. 1. Time scale of dynamic phenomena in power systems and their relationship with simulation domains. Fig. 2. Methodologies for assessing the thermal behavior of VSC in electromechanical simulations. Top diagram: current iterative method. Bottom diagram: proposed integrated or joint approach. International Journal of Electrical Power and Energy Systems 155 (2024) 109672 4 J.M. Mauricio et al. Fig. 3. VSC circuit and physical VSC schemes with different heatsink arrangements. 3.1. VSC thermal dynamics This subsection describes the model representing the VSC thermal behavior, assuming that the only heat source corresponds to the power losses of the semiconductor devices, to be analyzed in the next subsection. It is considered that any switch is composed of an IGBT and its corresponding anti-parallel diode, as shown on the top of Fig. 3. It will be also assumed that 𝑁ℎswitches are installed on a single heatsink. Note that 𝑁ℎdoes not represent the total number of switches of the power electronic stack, but just those sharing the same heatsink. According to this, 𝑁ℎ= 6 and 𝑁ℎ= 1 for the configurations shown in the middle and lower schemes of Fig. 3, respectively. In case of power stacks with several heatsinks, no thermal interaction between them will be assumed. The objective of the thermal model is to determine how the power losses are dissipated from the semiconductor switches to the environment. Considering the involved elements illustrated in Fig. 3, the resulting heat transfer can be split into three different parts: (i) from the IGBT/diode junction to the corresponding case; (ii) from the IGBT/diode case to the heatsink; (iii) from the heatsink to the environment. Usually, the thermal model is represented by using an analogy with an electric equivalent circuit, as proposed by Cauer [26,27]. In this analogy, the heat generators are current sources while the environment International Journal of Electrical Power and Energy Systems 155 (2024) 109672 5 J.M. Mauricio et al. Fig. 4. Cauer thermal model of the heatsink with semiconductor switches and simplified model. with a constant temperature is represented as a voltage source [21]. Between these two extremes, from the heat source to the environment, the thermal dynamic behavior of the different elements involved in the heat flow is represented using multiple pairs of resistors and capacitors. The capacitor voltages represents the different element temperatures, which are state variables of the dynamic problem. Fig. 4 shows the derived electric circuit of the thermal model, comprising 𝑁ℎpower electronic switches with a common heatsink where only one switch, composed of an IGBT and diode, is depicted. A detailed analysis of the internal thermal dynamics corresponding to the IGBT/diode junction-case and case-heatsink reveals that it is really fast, with time constants of just a few milliseconds [20]. Therefore, in the thermal domain illustrated in Fig. 4, this fast process can be reduced to an algebraic equation, converting temperatures from state variables to algebraic quantities. It is important to point out that this simplification does not eliminate the faster temperature dynamics. The transformation into an algebraic relationship simply yields a discrete temperature evolution up to the steady-state condition. Moreover, this simplification offers conservative results, because any sudden change in the boundary conditions, e.g. power losses or environment temperature, may cause a smooth evolution of the actual temperature whereas the simulated one varies stepwise. Note that this simplification implies eliminating the capacitors of the equivalent circuit, as shown in the bottom diagram of Fig. 4. Therefore, and assuming the same power losses for the 𝑁ℎswitches, the thermal model accounting for the variation of the IGBT and diode temperature as a function of the heatsink temperature can be easily derived: 𝑇𝑖=𝑇𝑠+𝑃𝑖𝑅𝑖−𝑠 𝑡ℎ ,(1) 𝑇𝑑=𝑇𝑠+𝑃𝑑𝑅𝑑−𝑠 𝑡ℎ .(2) It is worth noting that this part of the thermal model neglects the thermal cross-coupling between the IGBT and diode [28], an information which, in any case, is not currently included in the manufacturers’ data sheets. However, the proposed methodology could be easily updated to consider this effect just by adding the corresponding cross-coupling thermal resistors to the thermal model shown in Fig. 4. Conversely, the dynamics of the heatsink temperature are within the time frame analyzed in the electromechanical simulations and, therefore, according to Fig. 4, can be modeled as: 𝑑𝑇𝑠 𝑑𝑡 =1 𝐶𝑠 𝑡ℎ (𝑁ℎ𝑃𝑠−𝑇𝑠−𝑇𝑎 𝑅𝑠−𝑎 𝑡ℎ )(3) where 𝑃𝑠=𝑃𝑖+𝑃𝑑represents the sum of IGBT and diode power losses. The heatsink steady-state temperature, 𝑇𝑠 𝑠𝑠, can be obtained by forcing the left side of (3) to zero: 𝑇𝑠𝑠 𝑠=𝑇𝑎+𝑅𝑠−𝑎 𝑡ℎ 𝑁ℎ𝑃𝑠.(4) Then, it is possible to compute the normalized heatsink temperature, defined as 𝛩=𝑇𝑠∕𝑇𝑠𝑠 𝑠, which will be used to estimate precisely the IGBT and diode power losses, as explained in the next subsection. According to its definition, this variable is equal to one in the steadystate case, and can be considered almost constant for most studies. However, it must be taken into account in case a precise analysis of the thermal dynamics is required. 3.2. Power losses model The power losses depend on a wide variety of factors such as the AC currents, AC and DC side voltages, switching frequency, ambient temperature, gate resistors, etc. Some of these variables can vary over a wide range depending on operating conditions of the grid-connected VSC, particularly: 1. AC RMS injected currents. 2. Power factor. 3. AC RMS grid voltages, which may also vary widely if the VSC has to operate under short-circuit conditions. On the contrary, it is reasonable to consider that the following variables are almost constant: 1. Switching frequency is almost constant due to the use of PWM strategies. 2. DC voltage is constant because of the use of a proper DC regulator. 3. Ambient temperature remains constant. This is a feasible hypothesis considering the usual simulation intervals or, moreover, considering that the large power electronic stacks are within cabinets with controlled ambient temperature by means of Heat, Ventilation and Air Conditioning (HVAC) systems. Those assumptions simplify to a greater extent the complexity of the power losses model and are fully feasible considering conventional engineering implementation of power electronic devices. International Journal of Electrical Power and Energy Systems 155 (2024) 109672 6 J.M. Mauricio et al. With the above considerations, it is possible to formulate the IGBT and diode power losses as follows: 𝑃𝑖(𝐼, 𝛼) = 𝛩(𝑎𝑖+(𝑏𝑖+𝑐𝑖𝛼)𝐼+(𝑑𝑖+𝑒𝑖𝛼)𝐼2)(5) 𝑃𝑑(𝐼, 𝛼) = 𝛩(𝑎𝑑+(𝑏𝑑+𝑐𝑑𝛼)𝐼+(𝑑𝑑+𝑒𝑑𝛼)𝐼2)(6) where 𝐼is the VSC AC injected current, 𝛩is the normalized heatsink temperature and 𝛼is the product of the VSC modulation index, 𝑚, and power factor, cos 𝜑. Basically, the model is composed of two factors: the steady-state power losses and a correction term which takes into account the dependence of the power losses with the dynamic evolution of the temperature. The first factor computes the IGBT/diode power losses depending on electrical variables at steady-state conditions. Therefore, the influence of the steady-state temperature is inherently considered in the IGBT/diode power losses. This factor depends on quadratic functions of some electrical variables (RMS current, modulation index and the power factor) whose coefficients (𝑎𝑥,𝑏𝑥,𝑐𝑥,𝑑𝑥and 𝑒𝑥where 𝑥=𝑖, 𝑑) must be conveniently adjusted to accurately represent the off-state blocking, conduction and switching losses of electronic power devices. The second factor corrects the IGBT/diode power losses considering the dynamic evolution of the temperature. Note that the higher the temperature the higher the losses. Therefore, it is proposed to model this dependence in a linear fashion by introducing the normalized heatsink temperature parameter 𝛩. This way, given that the heatsink temperature is a state variable in the thermal model, it is possible to accurately represent the transient variation of IGBT/diode power losses. 3.3. Estimation of thermal model parameters This subsection is devoted to define the procedure for obtaining all the parameters of the thermal and power losses models. Regarding the thermal model, it is required to determine the involved thermal resistors and capacitors of (1),(2) and (3), as shown in Fig. 4. Particularly, 𝑅𝑖−𝑠 𝑡ℎ and 𝑅𝑑−𝑠 𝑡ℎ are directly provided by manufacturers or derived from related data, such as the junction, 𝑇𝑠𝑠 𝑖or 𝑇𝑠𝑠 𝑑, and heatsink temperatures, 𝑇𝑠𝑠 𝑠, in steady-state conditions for a given scenario with power losses, 𝑃𝑖or 𝑃𝑑: 𝑅𝑖−𝑠 𝑡ℎ =𝑇𝑠𝑠 𝑖−𝑇𝑠𝑠 𝑠 𝑃𝑖 , 𝑅𝑑−𝑠 𝑡ℎ =𝑇𝑠𝑠 𝑑−𝑇𝑠𝑠 𝑠 𝑃𝑑 . Similarly, 𝑅𝑠−𝑎 𝑡ℎ can be determined as: 𝑅𝑠−𝑎 𝑡ℎ =𝑇𝑠𝑠 𝑠−𝑇𝑎 𝑃𝑖+𝑃𝑑 . The thermal capacitor is obtained from the manufacturers’ software through the simulation of a step change on the AC RMS current, which will cause a temperature evolution as shown in Fig. 5. This exponential curve allows to compute the time constant of the thermal capacitance as follows: 𝜏𝑠=𝑅𝑠−𝑎 𝑡ℎ 𝐶𝑠 𝑡ℎ ⇒𝐶𝑠 𝑡ℎ =𝜏𝑠 𝑅𝑠−𝑎 𝑡ℎ ,(7) Regarding the parameters of the power losses model, according to (5) and (6), it is required to obtain ten parameters: 𝑎𝑥,𝑏𝑥,𝑐𝑥,𝑑𝑥and 𝑒𝑥 where 𝑥= {𝑖, 𝑑}. Given the fact that the power losses depend linearly on these parameters, a straightforward procedure consists of evaluating (5) and (6) for 𝑛operating points characterized by different 𝐼𝑗and 𝛼𝑗 with 𝑗= 1,…, 𝑛 and 𝑛≥5. This can be formulated in matrix form as follows: 𝐀𝐲𝐱=𝐛𝐱𝑥= {𝑖, 𝑑}(8) Fig. 5. Dynamic evolution of the heatsink temperature when an RMS current step reference is applied to the VSC. Fig. 6. Converter connected to the grid through a filter. 𝐀= ⎡⎢⎢⎢⎢⎢⎢⎢⎣ 1𝐼1𝐼1𝛼1𝐼2 1𝐼2 1𝛼1 1𝐼2𝐼2𝛼2𝐼2 2𝐼2 2𝛼2 1𝐼3𝐼3𝛼3𝐼2 3𝐼2 3𝛼3 ⋮ ⋮ ⋮ ⋮ ⋮ 1𝐼𝑛𝐼𝑛𝛼𝑛𝐼2 𝑛𝐼2 𝑛𝛼𝑛 ⎤⎥⎥⎥⎥⎥⎥⎥⎦ (9) 𝐲𝑥=[𝑎𝑥𝑏𝑥𝑐𝑥𝑑𝑥𝑒𝑥]𝑇(10) 𝐛𝑥=[𝑃𝑥(𝐼1, 𝛼1)𝑃𝑥(𝐼2, 𝛼2)⋯𝑃𝑥(𝐼𝑛, 𝛼𝑛)]𝑇(11) The parameters are obtained by solving a linear least-squares problem which can be formulated as follows: min ‖ ‖𝐀𝐲𝐱−𝐛𝐱‖ ‖2𝑥= {𝑖, 𝑑}(12) 4. Thermal model initialization A key aspect of any electromechanical model is to define a feasible initial state before doing a small signal analysis or time domain simulations. This process, known as backward initialization [25], has to be extended to the thermal model introduced in this paper. The objective of the initialization is to compute the VSC power losses and temperatures of IGBTs, diodes and heatsink. The initialization of gridconnected VSC is usually based on the solution of the power flow, that yields the voltage phasor 𝑉𝑠at the point of interconnection (POI) given the injected VSC complex power 𝑆=𝑃+𝑗𝑄, as shown in Fig. 6. Then, it is possible to compute the current through the AC coupling filter from: 𝐼=(𝑆𝑠 3𝑉𝑠)∗ ,(13) and the VSC voltage considering that 𝑍is the impedance of the AC coupling filter: 𝑉𝑡=𝑉𝑠+𝐼 𝑍, (14) For the computation of 𝛼, both the modulation index and the power factor are required. The former parameter is derived by relating the peak value of the AC voltage with the DC bus voltage: 𝑚=2√2𝑉𝑡 𝑉𝑑𝑐 ,(15) International Journal of Electrical Power and Energy Systems 155 (2024) 109672 7 J.M. Mauricio et al. Table 1 Characteristics of the VSCs. Parameter S-VSC M-VSC L-VSC Manufacturer Semikron Infineon ABB Model SKiiP25AC12F4V1 FF800R17KE 5SNA1500E33030 𝑉𝑎𝑐 (V) 400 400 1100 𝑉𝑑𝑐 (V) 750 800 2200 𝐼(A) 45 650 1500 𝑆(kVA) 30 500 3000 𝑓𝑠𝑤 (kHz) 20 10 1 𝑇𝑎(◦C) 40 40 25 On the other hand, the power factor is computed by considering the phase angle difference between the VSC voltage, 𝑉𝑡=𝑉 𝑒𝑗𝜃𝑡, and the VSC injected current, 𝐼=𝐼𝑒𝑗𝜃 , as follows: cos 𝜑= cos (𝜃𝑣−𝜃).(16) The IGBT and diode power losses are computed form (5) and (6), considering steady-state conditions (𝛩= 1). The initial heatsink temperature, 𝑇𝑠𝑠 𝑠is computed by means of (4) and, after that, the IGBT and diode initial temperatures can be calculated using (1) and (2), respectively. Finally, it is possible to compute the power on the VSC DC side by just applying a power balance: 𝑃𝑑𝑐 =ℜ(3𝑉𝑡𝐼∗) ± 𝑁ℎ𝑃𝑠,(17) where the power losses sign depends on the power flow direction. Note that this backward initialization can be performed after a power flow and does not require an iterative process. 5. Model validation This section is devoted to validating the thermal model by comparing the estimated temperatures and power losses with the corresponding values provided by the manufacturers’ software tools. The section is divided into three subsections, respectively detailing the targeted VSCs where the model has been derived, the steady-state results and dynamic simulations. 5.1. Targeted VSCs Any general-purpose model, such as the one proposed in this paper, should fit to the actual system irrespective of its operating point, rated values and manufacturer. For this reason, in order to emphasize this versatility, the thermal model has been applied to a small (S-VSC), medium (M-VSC) and large (L-VSC) VSC of different manufacturers, according to the details shown in Table 1. Note that the switching frequencies have been selected according to the VSC power rating. In addition, different environment temperatures have been selected to evidence that the model perfectly fits to this operating condition. The thermal model parameters of the targeted VSCs are summarized in Table 2. It is important to highlight that different information sources have been used for obtaining those parameters, because each manufacturer provides its own tool. Semikron provides the on-line software Semisel [20] while Infineon and ABB offer their corresponding spreadsheets [21,22]. The sets of parameters have been computed using just five operating points: 15%, 60% and 100% of the rated current with cos 𝜑= 1 and 60%, 100% with cos 𝜑= 0. 5.2. Steady-state results Figs. 7 and 8show the power losses and temperatures corresponding to the L-VSC case (large ABB VSC), as a function of the injected AC RMS for cos 𝜑= 1 and cos 𝜑= 0 respectively. The continuous curves correspond to the values computed by the proposed thermal model, while the discrete points refer to the data provided by the manufacturer Table 2 Thermal model parameters for the targeted VSCs. Parameter S-VSC M-VSC L-VSC 𝑅𝑖−𝑠 𝑡ℎ 0.517 0.056 0.019 𝑅𝑑−𝑠 𝑡ℎ 0.839 0.078 0.038 𝑅𝑠−𝑎 𝑡ℎ 0.663 0.072 0.007 𝐶𝑠 𝑡ℎ 15.08 279.66 2855.21 𝑎𝑖∕𝑎𝑑1.958∕0.963 0.0000∕43.582 434.0125∕145.245 𝑏𝑖∕𝑏𝑑0.324∕0.027 0.508∕0.280 0.559∕0.594 𝑐𝑖∕𝑐𝑑−0.227∕0.051 −0.187∕0.172 −0.015∕0.193 𝑑𝑖∕𝑑𝑑0.016∕0.013 0.0004∕0.0002 0.0009∕0.0003 𝑒𝑖∕𝑒𝑑−0.011∕0.014 −0.0003∕0.0002 −0.0005∕0.0002 simulation tool. The operating points used for adjusting the model parameters have been highlighted in the figures, using bigger discrete markers. It can be observed that the power losses and temperatures of IGBT, diode and heatsink fit almost perfectly in all the operational range for both power factors. This good matching between the proposed thermal model results and manufacturer data is achieved for different power factors, not only those used to tune the model parameters, cos 𝜑= 1 and cos 𝜑= 0 in this application. This fact can be observed in Fig. 9, where power losses and temperatures are provided for the S-VSC and M-VSC operating at cos 𝜑= 0.8inductive. Again the analysis of this figure clearly shows that the fit is almost perfect, which highlights the robustness of the thermal model against different operating conditions. 5.3. Dynamic results This section analyzes the dynamic behavior of the proposed thermal model with varying operating conditions, involving changes of AC RMS injected current, modulation index and power factor. For this purpose, the large ABB VSC is chosen, and the results are shown in Fig. 10. As in the case of the steady-state analysis of the previous section, the continuous curves corresponds to the thermal model results while discrete markers are manufacturer data. These data have been obtained from the available manufacturer tool, which allows the simulation of 10 different operating points with a total simulation time of 180 s. Therefore, several simulations have been sequentially run to obtain the results depicted in Fig. 10. The final heatsink temperature of a simulation period is used as initial temperature for the next one. The simulation starts with a no-load condition and a steady-state heatsink temperature of 27◦C, which is slightly above the environment temperature (25◦C) because of the IGBT and diode losses. Note that the semiconductor switches incur in power losses just by being energized from the DC bus, even in absence of injected current, which is considered in the power losses model through the parameters 𝑎𝑖and 𝑎𝑑. At the time instant 𝑡= 10 s, a step change is applied to the current, up to 1 p.u. with cos 𝜑= 0.8. Note that, considering the existing POI voltage, a modulation index 𝑚= 0.8is required. It is worth noting also the different dynamic evolution of the IGBT and diode temperatures, with respect to the heatsink one. The current step affects directly the IGBT and diode temperatures, due to the reduced time constant of the heat transfer across the junction-case-heatsink circuit. Conversely, the heatsink temperature follows a slow first-order dynamics with similar time constant to those analyzed in electromechanical simulations. The power losses are not only affected by the AC RMS current but also by the modulation index and power factor. For this reason, the dynamic simulation also includes step changes on these parameters in order to assess this issue. With this in mind, a step increase is simulated on the POI voltage at the time instant 𝑡= 40 s, leading to a step increase of the modulating index to maintain the VSC injected current. Note that this increase in the modulation index yields a slight variation of the IGBT and diode power losses, which directly affects their temperatures but not so significantly the heatsink temperature. The influence of the International Journal of Electrical Power and Energy Systems 155 (2024) 109672 8 J.M. Mauricio et al. Fig. 7. L-VSC case: Power losses and temperatures with the proposed thermal model and manufacturer data with cos 𝜑= 1. Fig. 8. L-VSC case: Power losses and temperatures with the proposed thermal model and manufacturer data with cos 𝜑= 0. International Journal of Electrical Power and Energy Systems 155 (2024) 109672 9 J.M. Mauricio et al. Fig. 9. M-VSC and S-VSC cases: Power losses and temperatures with the proposed thermal model and manufacturer data with cos 𝜑= 0.8. power factor is analyzed at the time instant 𝑡= 100 s, where a step change from cos 𝜑= 0.85 to cos 𝜑= 0 is performed. At the same time instant, the POI voltage is decreased to its original value and, therefore, the corresponding modulation index. Note that this change implies a redistribution in the IGBT and diode currents (higher current for the diode), which immediately translates to their internal temperatures. The next step change at 𝑡= 120 s corresponds to a decrease of the AC current, considerably reducing the VSC power losses and therefore its temperatures. Finally, at the time instant 𝑡= 160 s the power factor is changed to cos 𝜑= 1, which modifies the current sharing between the IGBT and diode just in the opposite fashion as in the previous case, therefore increasing the IGBT power losses. The dynamic simulation reveals that the developed thermal model perfectly reproduces the VSC thermal behavior, which fits the data provided by manufacturers in their corresponding dynamic simulation software. This matching is achieved even with operating points quite different from those used in the estimation of the thermal model parameters. It is worth noting that the model exactly reproduces not only the temperature evolution but also the IGBT and diode power losses. In this regard, it is interesting to highlight the dynamic evolution of the IGBT and diode power losses, which do not exactly follow the current steps because, according to the thermal model of (5)-(6), the IGBT and diode power losses also depend on the temperature dynamics. 6. Conclusions This paper has presented a methodology for analyzing the thermal behavior of VSCs in electromechanical simulations. A thermal domain is introduced where two different dynamics have been identified. The fast dynamics corresponds to the evolution of the temperatures of IGBT and diode junctions and the corresponding cases, while the slower one refers to the heatsink temperature. A proposal is made to simplify the fast dynamics, which reduces to a set of algebraic equations. Conversely, the slower dynamics has been fully represented by a first-order thermal model. The model has been validated with VSC from different sizes and manufacturers with accurate results both in steady-state and dynamic conditions. The discrepancies between the temperatures and power losses estimated by the proposed thermal model and the ones provided by the manufacturer are negligible, even for operating points far away from the ones used for the estimation of the model parameters. To the best of the authors’ knowledge, there is no other method in the specialized literature dealing with the integration of thermal models in electromechanical simulations. This simplification allows the evolution of all VSC temperatures to be analyzed with the larger simulation steps customarily used in electromechanical simulation programs. Therefore, the assessment of the VSC thermal behavior is extraordinarily simplified since it is possible to address the electromechanic and thermal problems simultaneously without the need of applying a tedious iterative process using different simulation platforms. Moreover, the paper has also provided a methodology for computing all the thermal model parameters from the available manufacturer data in a straightforward way. The proposed thermal model opens new research lines dealing with the incorporation of the thermal limits in the VSC control algorithms rather than using the conservative steady-state rated AC RMS currents. This may allow, for instance, to transiently overload the VSC, which may be useful in the new power system paradigm with a massive penetration of renewable energies, where ancillary services such as inertia and primary frequency response are required to contribute to the system stability. The incorporation of the thermal model to the control algorithm may broaden the transient operational area of the VSC-based generators, providing an extended flexibility to the power system.