Reactive power boosting strategy for hydrogenerators: framework and evaluation using Nordic machine data

Abstract

Modern power systems face operational challenges due to the integration of variable renewable energy, necessitating enhanced reactive power support from synchronous generators. In this study, we propose a comprehensive reactive power boosting strategy (Q-strategy) framework, addressing technical and economic challenges via four pillars: (1) Q-boost defines reactive power limits for continuous/dynamic states by leveraging thermal margins under insulation thresholds (155 °C); (2) Q-model, a capacity twin model (CTM) combining a lumped-parameter thermal network and saturated field current models, enables online hot spot temperature management without finite element analysis; (3) Q-energy optimizes reactive energy delivery by integrating thermal endurance models with aging and start–stop cycles; and (4) Q-value formulates costs for reactive power services, accounting for losses beyond grid codes and lifetime depreciation. The results demonstrate a 57% increase in continuous reactive power capacity, with smaller generators incurring higher costs (up to 1.33 $/Mvarh) than larger units. The proposed framework exploits hydrogenerators’ slow thermal time constants by using authentic historical machine parameters and operational statistics from Nordic installations. To isolate the machine’s direct electro-thermal boundaries and localized voltage-support capabilities without confounding multi-bus network variables, system-level validation is conducted on a single-machine, two-bus power system topology subjected to extreme contingency events and localized load profiles. An online temperature controller ensures safe operation under extreme conditions while aligning with real-world load patterns. The Q-strategy offers a cost-effective alternative to FACTS devices, balancing grid resilience with economic feasibility in high-renewable grids.

1 Introduction

The energy transition is driving power systems toward greater reliance on clean energy and electrification, particularly through the deployment of renewable energy sources such as solar and wind. It is widely expected that by 2050, 75% of electricity generation will come from inverter-based variable renewable energy (VRE), and electricity will account for 52% of total energy consumption (Barani et al., 2026). Although this shift significantly reduces net carbon emissions, it alters the physical architecture of transmission grids, necessitating a comprehensive re-evaluation of structural voltage-support mechanisms.

Modern power systems face acute operational stress from electronic loads, HVDC links, and VRE, which can trigger cascade outages, particularly affecting voltage stability following major disturbances (Adegoke et al., 2024). The fundamental operational problem stems from a stark physical disconnect in reactive current capabilities during grid faults. Traditional power systems relied on heavy iron-and-copper synchronous assets whose large mass naturally withstood large instantaneous overcurrent injections, thereby supporting weak voltage profiles by virtue of their thermal capacity. In contrast, the power-electronic converter interfaces used in VRE and HVDC links have virtually zero thermal overload capacity; to protect fragile semiconductor switches from instantaneous destruction, their internal control loops strictly clip the output current within microseconds during a fault. This structural shift deprives modern networks of the immediate, dynamic, reactive current overshoots that would otherwise mitigate rapid localized voltage degradation and accelerate wide-area voltage collapse (Liao et al., 2026).

Maintaining grid security under these current-constrained conditions requires distinct control mechanisms operating across different timelines. Long-term voltage stability depends on operator control actions and slower-acting equipment, such as generator current limiters. In contrast, short-term voltage stability relies on fast-acting grid infrastructure, including protective devices, FACTS, and synchronous condensers (Dahat and Dhabale, 2024; Nair et al., 2024; Hatziargyriou et al., 2021). Synchronous condensers are typically deployed near HVDC links to improve localized system strength and provide missing short-circuit current capacity (Duan et al., 2022; Tan et al., 2022), an objective that can be advanced by integrating specialized machine-level current control and megavolt-ampere (MVA)-field limiters directly into the excitation loop to enable rapid localized voltage response (Ojo and Shafiul Alam, 2026). The severity of these structural deficits is well documented; for instance, a study using a modified Nordic test system with a 35% share of inverter-based generation shows an increased risk of blackouts due to voltage instability (Heusinger et al., 2021). Moreover, there are currently no clear plans to introduce structural inertia or dynamic reactive reserves as balancing services in the Nordic system (Nordström et al., 2023), reflecting a broader global challenge where the rapid displacement of traditional plants by non-synchronous assets severely degrades baseline grid strength, rotational reserves, and multi-timeline voltage resilience (Dulal et al., 2026).

As VRE penetration increases, conventional synchronous generators are displaced from routine energy production and repurposed as synchronous condensers to maintain relevance and enhance wide-area stability (Halley et al., 2022; Gang et al., 2022; Bichler et al., 2022). Although this repurposing strategy addresses immediate network needs such as baseline inertia and basic reactive power support, existing hydrogenerators offer a unique, highly localized opportunity to provide dynamically enhanced reactive power support while still providing active power (Oyvang and Noland, 2022). However, running these heavy assets under modern, highly variable non-nominal regimes increases thermal cycling and structural stress, necessitating physical refurbishment or rotor pole re-optimization to balance asset lifetime degradation against operational performance benefits (Karekezi et al., 2025). This boosting opportunity stems directly from an unexploited margin in historical manufacturing design: hydrogenerators traditionally operate at conservative Class B insulation limits (130 °C), even though they are frequently manufactured with high-tier Class F insulation materials rated for continuous operation at 155 °C (Pyrhönen et al., 2013). Accounting for a standard 10 K internal hot spot margin, this difference provides an unexploited 25 K physical temperature margin for reactive power boosting, serving as a thermal buffer for the grid during critical voltage-stability contingencies and system optimization via optimal reactive power dispatch (ORPD).

Although the domain of reactive power management has been investigated across separate technical and economic subfields, a critical gap persists in linking active component-level boosting capabilities directly with macro-level market cost-formation architectures. Recent state-of-the-art studies have separate pockets of innovation. On the control infrastructure side, Banda et al. (2025) introduced a novel control strategy to achieve reactive power boosting from single-stage BESS-STATCOM systems, showing that isolating the battery pack can maximize the dc-link capacitor’s voltage capability, thereby overcoming low-state-of-charge performance bottlenecks. On the economic and regulatory side, Chiandone et al. (2025) advanced the valuation of dynamic reactive services by proposing a cost allocation framework that accounts for variations in synchronous generator efficiency during under- or overexcitation phases to establish a fair merit order for secondary voltage regulation. However, these streams of research remain decoupled. Hardware-centric power electronic control models are not applicable to heavy iron-and-copper hydro-turbines, and generalized generator efficiency pricing models typically omit the localized insulation thermal degradation, copper loss escalations, or transient hotspot accumulation that physical over-excitation boosting imposes on machine winding profiles.

2 Problem formulation

The untapped potential of hydrogenerators must be adopted to enhance grid optimization and voltage stability margins without incurring extensive capital expenditure on additional FACTS devices, particularly under extreme conditions imposed by a high share of VREs. In this study, we recognize the over-excitation limiter (OEL) as a critical limitation to the hydrogenerator’s untapped potential of Q-boost operation. It proposes a Q-strategy framework to leverage the thermal capabilities of hydrogenerators beyond the standard operation schemes. These thermal capabilities are constrained by the insulation class temperature threshold, aging, and the annual increases in start–stop cycles. The hot spot temperatures require accurate and online predictions to prevent failures and minimize wear and tear during Q-boost operation. Furthermore, the lack of integration between reactive power costs, additional losses incurred by hydrogenerators, and lifetime insulation costs hinders the economic incentives for power producers to provide new reactive power services.

2.1 Voltage stability margin enhancement

Reactive power availability is a critical factor that limits the voltage stability margin, especially when restoring the voltage to its optimal level after power system disturbances. If the system’s reactive power reserve is insufficient, it can lead to a voltage collapse and a system blackout. Hence, generators with Q-boost capabilities already in the grid are a practical and cost-effective solution for providing the system’s reactive power reserves and enhancing voltage stability during disturbances.

The reactive power boundaries of the hydrogenerator are established under nominal conditions for insulation class F (155 °C) to operate at a maximum hot spot temperature of 130 °C, known as the classical Philosophy of Design (POD). Building on the classical hydrogenerator POD with the actual temperature threshold of insulation class F, Equation 1 introduces an approximated Q-boost calculation based on the relation between heat flux and current density (Wang and Ruan, 2022). The variables are primarily substituted to show the field current density’s dependency on temperature. At the same time, the remainder, namely, electrical conductivity, pole coil width, the cross-sectional area of the conductor, and thermal resistance, are considered constant values. To facilitate the analysis, the concept of temperature per unit—defined as the ratio of maximum to nominal operating temperature—is introduced, enabling the evaluation of field current per unit at various temperatures. The results indicate that the field current can be increased by 9.19%, allowing for Q-boost without exceeding the thermal limits of insulation class F.

An example of voltage stability margin enhancement is presented in Figure 1 to demonstrate the impact of Q-boost on system performance, where the field current is obtained using Equation 2 for G1 in Table 1. In this scenario, the active power is set to 0.7767 pu = 80 MW, resulting in a reactive power of 0.6453 pu = 66.47 Mvar. Consequently, the voltage stability margin increases by 21%, attributed to an additional 50% reactive power after losing a transmission line, as depicted in Figure 2. In red, the PV curve for the normal operation is shown. The loss of one line without Q-boost is exhibited in green. Finally, the increased voltage stability margin achieved using Q-boost is depicted in blue. The generator terminal voltage without Q-boost is 1.025 pu, and that with Q-boost is 1.05 pu. The total line reactance is 0.25 pu.

FIGURE 1

FIGURE 2

2.2 Capacity twin model

A CTM is proposed to achieve online thermal management using a fast H-LPTN, a per-unit loss model, and an electrical model with saturation. The H-LPTN is developed with parameter estimation using actual data for average temperature and finite element analysis (FEA) calculations for hotspots, and subsequently, cross-validated for ultra- and high-loss scenarios; the per-unit loss model calculates the field winding, stator armature, and iron core losses of hydrogenerators (Bortoni et al., 2021), and the saturated field current model (Karekezi et al., 2022) is developed using a polynomial of the form , where is usually 7 or 9. These three models complete the thermo-electrical modeling for the CTM. In this study, 1.0308 and 0.16. It should be noted that the critical component for thermal limits during reactive power boosting is the rotor hotspot for thermal dynamics slower than 10 min.

The H-LPTN implemented in this study is shown in Figure 3, consisting of six ordinary differential equations (ODEs) and the remaining algebraic relations for the heat exchanger dynamics. This low-order H-LPTN is constructed as a double-layer LPTN that connects the hot spot LPTNs to their nearest heat sources in the system LPTN, i.e., the average temperatures of the hydrogenerator.

FIGURE 3

The electrical model for developing, calculating, and validating the capability diagram, Q-boost limits, and the Q-value costs uses the field saturation model developed by Karekezi et al. (2022). On the other hand, the open-circuit characteristic of G1 is directly incorporated into the machine block for the case studies performed in Simulink, as described in Section 4.

2.3 Scaled capacity twin model through thermal scaling

To achieve broader applicability across multiple hydrogenerators using the CTM without relying on computationally expensive FEA, thermal parameter scaling is implemented to create a scaled capacity twin model (s-CTM) that approximates the hot spot LPTNs for air-cooled hydrogenerators, i.e., synchronous generators with similar POD. The parameters to scale are the thermal resistances , the thermal capacitances , and, in consequence, the thermal time constant . The scaling procedure is as follows:

The primary assumption for thermal scaling is the classical POD of a temperature increase of 80 K, as reported by Pyrhönen et al., 2013, which is used to determine the thermal resistance . The thermal capacitance assumption is straightforward: the material is the same; only the mass changes. Therefore, all the thermal parameters, including those of the hot spots and end-winding, can be scaled accordingly using the masses and losses of the actual generator.

2.4 Quadratic thermal model

The temperature-dependent reactive power limits can be approximated using Equation 6—referred to as the quadratic model—which is obtained by combining the rotor hot spot at thermal equilibrium (Equation 4, shown in Figure 3), the rotor loss estimation (Equation 5; Bortoni et al., 2021), and the thermal scaling rule (Equation 3).where , , and are, respectively, the cooling air temperature at thermal equilibrium, rotor hot spot temperature at thermal equilibrium, and an nth-order correction factor for the heat exchanger and thermal dynamics not accounted for in the reduced model, which is expressed as a second-order coefficient in Equation 7. These coefficients are obtained by minimizing the maximum absolute error between the quadratic and the H-LPTN models.

2.5 Effective thermal endurance

The insulation thermal endurance can be approximated using the 10 C rule of thumb (derived from the Arrhenius equation) (Janting et al., 2019), as shown in Equation 8, for a time-varying temperature with a periodic profile per day, i.e., boosting hours per day. Furthermore, hydrogenerators facing an increasing number of start–stop cycles and load changes may reduce the life of the equipment; it is estimated to add an equivalent operating time between 15 h (Nilsson and Sjelvgren, 1997) and 5 h (EPRI, 2017). Therefore, an effective thermal endurance is proposed in Equation 9 that considers the equivalent operating hours of start–stop cycles with annual growth, where, for each year , the added is calculated up to and then subtracted from the purely thermal endurance. Subsequently, all the additional values for each are interpolated using radial basis function (RBF) interpolation and then smoothed using a Gaussian filter to create a continuous function, as shown in Equation 10.

2.6 Reactive boosting energy over the expected lifetime

The reactive boosting energy can be calculated for the additional reactive power injected into the grid, considering the insulation system’s effective thermal endurance or lifetime expectancy. In this regard, the reactive boosting energy over the expected lifetime can be maximized as shown in Equation 11.

2.7 Reactive power boosting cost formation

Optimal reactive power dispatch is crucial for dealing with heavily stressed power systems by reducing grid losses and improving voltage stability. In economic optimal reactive power dispatch, the reactive power cost of synchronous generators is typically modeled as a lost-opportunity cost. We propose the Q-strategy framework, which mitigates the opportunity cost associated with the extended ability to contribute reactive power without reducing active power injection. This framework introduces additional generator losses and lifetime costs for reactive power. Thus, the reactive power income is defined as the difference between the maximum reactive power boost and the grid-code maximum reactive power allowance, minus the cost of the additional generator losses beyond the grid code and the cost of lifetime reduction, as illustrated in Figure 4 and formulated in Equation 12. The raw copper material price baseline is taken as . To translate raw commodity prices into realistic utility project expenditures, an industrial accounting framework is applied, where the specialized labor cost of rewinding constitutes 50% of the total direct refurbishing expenditure, supplemented by an additional 20% multiplier to account for engineering, insulation materials, and manufacturing overheads. This standard practice yields a total structural multiplier of 2.4 relative to the base copper mass, resulting in an effective fully burdened material replacement cost of $15.6/kg. Although this effective rate is slightly higher than the standalone raw material baselines evaluated in the recent hydrogenerator refurbishment literature, such as Karekezi et al. (2025), the explicit integration of labor and overhead metrics ensures a more conservative and comprehensive asset-degradation pricing mechanism that protects the plant operator during extreme dynamic boosting regimes.

FIGURE 4

2.7.1 Multi-temperature Pareto front and Chebyshev trade-off analysis

A bi-optimization problem is introduced to analyze the trade-offs between lifetime and income across different operating temperatures. The feasible solution space is defined as in Equation 13:

The minimum active power is set to 0.2 pu as a rule of thumb to prevent cavitation. For each temperature , a bi-objective optimization is solved as in Equation 14:

Since the effective thermal endurance is inversely proportional to temperature and the reactive power income is relatively proportional to temperature, there exists a trade-off between optimizing lifetime and total income over the lifetime. Therefore, a multi-temperature Pareto front analysis is proposed, where a solution is Pareto-optimal if there exists no other feasible solution such that in Equation 15:

The temperature-specific Pareto front is defined as in Equation 16:

To find the local trade-off for each temperature , the first local Utopia points are obtained using Equation 17; then, the Chebyshev distance is applied to find the local trade-offs, as shown in Equation 18, where the trade-off solution for each temperature is defined as the minimum Chebyshev distance, as shown in Equation 19.

The global Pareto front across all temperatures can be represented as in Equation 20. The overall trade-off can be determined using the global Utopia point by minimizing the Chebyshev distance to the global Utopia point, as shown in Equation 21.

Finally, an income analysis is performed, where the active power and temperature are varied for each hour trade-off solution as in Equation 22:

2.8 Hydrogenerator data

We consider four different hydrogenerators for the calculations. The base case for the FEA calculation is G1, while only the scaled modeling calculations are presented for the remaining hydrogenerators. Similarly, the saturation model is based on G1, and the values are scaled according to the nominal operating condition.

3 Method: Q-strategy

The Q-strategy is a comprehensive framework that integrates the hydrogenerator’s available reactive power capacity, hot spot management for online control, and sufficient incentives for the power producer to offer additional reactive power services to enhance voltage support and grid optimization. As visually formalized via the comprehensive operational flowchart in Figure 5, this framework comprises four sequential pillars for execution: Q-boost, Q-model, Q-energy, and Q-value.

FIGURE 5

The Q-boost pillar evaluates the physical capability of the hydrogenerators to provide extended reactive power compensation beyond standard limits, mapping the over-excited active/reactive operational envelopes using a classical machine parameter approach.

These boundaries serve as inputs to the Q-Model pillar, which uses the discrete-time CTM of the hydrogenerator to provide accurate, real-time predictions of hot spot temperature. By iteratively processing internal sub-component losses , the CTM continuously tracks the maximum field current across various dynamic time windows while ensuring that the hotspot never violates the critical continuous thermal threshold of 155 °C.

Next, the Q-energy pillar integrates this thermal profile over an asset-management horizon (up to 100 years). By calculating the cyclic start–stop count and equivalent operating hours , it applies a Gaussian filter smoothing technique to determine the effective insulation lifetime reduction , quantifying the net allowable reactive energy injection .

Finally, the Q-value pillar establishes the service cost formation engine. It evaluates the marginal additions in machine losses and the fully burdened asset degradation cost relative to the baseline grid code requirements (Nøland et al., 2020). By evaluating the boundary condition where the service value function settles to zero , it identifies the minimum acceptable market clearing price .

By seamlessly transitioning from real-time physical constraints to macroeconomic market indices, the overall framework provides transmission system operators (TSOs) with verifiable, thermally secure reactive power-boosting services to enhance grid support.

3.1 Q-boost element

The Q-boost element specifies the reactive power capabilities of hydrogenerators based on thermal limits using a classical POD approach, e.g., the machine designer may provide Q-boost capability. For instance, using Equation 1, the classical Q-boost limits for hydrogenerators at different temperature thresholds are shown in Table 2. It is essential to mention that the ambient and cooling system conditions constrain the maximum temperature, which is assumed to be notwithstanding in this formulation. The IEEE C50.12-2005 states that salient-pole, totally enclosed, water-to-air-cooled hydrogenerators are generally designed for the maximum temperature of the cooling water encountered where the machine is to be installed C50 (2006). Using the proposed H-LTPN for the classical POD approach, the field heating limit for cooling water considerations is between 30 °C and 35 °C, as shown in Figure 6. Moreover, the limits are depicted for the temperature thresholds of 155 °C and 180 °C, where the reactive power is enhanced.

FIGURE 6

Using the proposed H-LPTN, a temperature-dependent capability diagram can be constructed for extended reactive power capabilities according to the rotor hot spot temperature limits, as shown in Figure 7. The difference in reactive power capability between the lowest and highest temperatures, i.e., summer and winter, is presented in Figure 8. The time window consists of 12 h, i.e., thermal equilibrium. Thus, considering seasonal changes in cooling water temperature, the hydrogenerator may operate at different thermal limits to provide reactive power services. For instance, during winter conditions, when load increases due to heating, hydrogenerators may deliver additional reactive power to enhance grid support and optimization.

FIGURE 7

FIGURE 8

Furthermore, a comparison between the quadratic and H-LPTN models is performed, with a reactive power difference of 2.82% for the relationship between the rotor hot spot and cold water temperatures in G1. The correction factor is [−1.704910^-4, 0.9921,−1.461510^-4,−0.8830,−16.3845 °C]. The quadratic model enables ultra-fast calculations of reactive power limits at different seasonal cooling water temperatures, providing a holistic view of reactive power capabilities. For practical purposes, the maximum reactive power, at the maximum active power and maximum temperature of the seasonal cooling water, can be taken as the reactive power boosting limit, as presented in Table 3. The Q-boost approximation limits for different real-world hydrogenerators display that the largest machines possess the lowest continuous boosting per unit capabilities. A lower per-unit value of reactive power may be attributed to earlier saturation of the iron core due to the reduced equivalent core length resulting from radial ventilating ducts (Pyrhönen et al., 2013), which are typically present in large hydrogenerators to dissipate heat (Lipo, 2017; Su et al., 2021; Franc et al., 2019).

3.2 Q-model element

FIGURE 9

These two schemes for reactive power boosting are presented in Figure 9 for insulation Class F at 0.9 pu, where the static Q-boost refers to a service for which the insulation rating class F is utilized without advanced control. In contrast, dynamic Q-boost is the extension of reactive power capability under extreme conditions over different operating time windows, requiring a thermal capacity twin. The simulation uses the following initial temperatures at thermal equilibrium: 20 °C, , , , , , , , , , and .

For DQO, especially during the shortest thermal time window of 15 min, the rotor hot spot remains the limiting component for reactive power boosting; the rate of temperature change remains below the recommended temperature gradient of 2.5 °C/min (Istad et al., 2011). Hence, the DQO can be reliably defined for thermal time windows of 15 min or longer, depending on the requirements of the TSO/DSO or the services planned by plant owners.

3.3 Q-energy element

The Q-energy element incorporates the thermal endurance of the hydrogenerator’s insulation system into calculating reactive energy dispatched into the grid. This calculation enables Q-boost optimization in grid planning and ORPD, ensuring that Q-boost operation is well aligned with the wear-and-tear-associated costs, i.e., the lifetime reduction cost of the insulation system. Q-energy can be expressed as the product of the Q-boost injected into the grid and the corresponding Q-time interval , which can be used to characterize the effective thermal endurance of the insulation system.

A case example is the maximization of the total reactive energy delivered over the lifetime of a hydrogenerator, as expressed in Equation 11. First, the hydrogenerator’s effective thermal endurance is estimated using 300 start–stop cycles (Signore, 2017), with an annual growth 2.8079% (Neupane et al., 2021), considering the added equivalent operating time to be 10 h, and using the on-site measurements of operating points for G1 (Karekezi et al., 2022), the nominal temperature of the rotor hotspots is estimated to be 80 °C at 20 °C.

Figure 10 depicts the insulation’s effective thermal endurance. It shows the impact of Q-boost at different temperatures in reducing the effective thermal endurance concerning the daily boosting hours—referred to as Q-time. This figure shows that the rotor hot spot temperature must be as close as possible to 130 °C to maximize Q-time, which is directly proportional to the reactive energy, because the reactive power variance is 36%, and the Q-time variance is 1,100%. It should be noted that the Q-time governs the reactive power, estimated based on the hotspot temperature allowance, for reactive energy variation. Therefore, the optimal point for reactive boosting energy over the insulation lifetime is at 130 °C, with 12 hours of daily boosting, yielding 5.4 Tvarh over an 18-year lifespan, while active power is reduced to 0.2 pu. In other words, the minimum temperature with the maximum daily boosting hours is needed to maximize lifetime, and the minimum active power is needed to maximize reactive power boosting.

FIGURE 10

3.4 Q-value element

The Q-value element develops a reactive power cost formation mechanism that considers hydrogenerator losses, costs beyond the grid requirements, and insulation lifetime costs to incentivize reactive power services. The minimum cost for reactive power can be obtained for an active power cost of $70 Mwh and a copper cost $6.5kg⁻¹ when , and . The results for G1, G2, G3, and G4 using Equation 12 are $0.6872 Mvarh, $1.0307 Mvarh, $0.9501 Mvarh, and $1.3302 Mvarh, respectively. Here, the smallest hydrogenerator may require a higher incentive for reactive power-boosting services. Therefore, distributed synchronous generators may require higher reactive power values to incentivize the Q-boost services. However, this reactive power cost may require a holistic approach to the distribution grid as a super-node to provide coordinated reactive power services (Melfald and Øyvang, 2023).

In Figure 11, the reactive power income [$/h] is presented for , in which each subplot refers to a different active power (P). Each subplot depicts different bounds for reactive power boosting because the rotor hot spot temperature limits them. Furthermore, effective endurance is shown for different hours of boosting per day. The least reactive power boosting income occurs at high temperatures, long boosting hours per day, and high active power (losses). In contrast, the most significant income comes from the lowest boosting hours, low temperature, and reduced active power (losses).

FIGURE 11

The bi-objective optimization analysis of total reactive energy and the lifetime at various temperatures are shown in Figure 12, where the optimal solution is 1700 k$ at 130 °C and 0.2 pu (the lowest active power threshold); the overall Chebyshev trade-off point is 1160 k$ at 130 °C and 0.2 pu, with a lifetime expectancy of 55 years; and the Chebyshev trade-off point for 155 °C is 299 k$ at 0.2 pu, with a lifetime expectancy of 29 years. The multi-temperature Pareto front is performed to illustrate the total income for optimal, utopia, and Chebyshev trade-off points, as shown in Figure 12a; the relationship of the Chebyshev trade-off solutions for temperature and hour considering lifetime is depicted in Figure 12b; and the total income for the Chebyshev trade-off points by variation of active power is presented in Figure 12c. Note that the total income includes the lifetime cost expenses. For instance, the lowest trade-off is presented at 155 °C and a nominal active power of 0.9 pu, where the lifetime expectancy is 28.7 years and the total income is 95 k$. Moreover, for nominal active power operation at 130 °C, the lifetime expectancy increases to 54.8 years, with a total income of 191 k$. These lowest trade-off points have a 67% income variance for a range of 28–55 years of lifetime expectancy. On the other hand, for the trade-off points at the lowest active power, the income variance is 118% over the same lifetime period.

FIGURE 12

As shown in Figures 11, 12, the optimized Q-value will tend to reduce the operating temperature to achieve the highest daily boosting hours. At the same time, the maximum Q-value is obtained at the lowest possible active power operation because it provides a broader temperature allowance for reactive power boosting. However, the extent of reactive power boosting is ultimately determined by the grid and TSO/DSO requirements.

It is essential to mention that two security measures to protect the hydrogenerators may be present in this approach: 1) the water inlet temperature of the heat exchanger is set to 20°C and 2) the initial condition or the pre-boost point of operation corresponds to the rated values of the hydrogenerators and, therefore, represents the highest temperature under normal operating conditions.

4 Model validation

The model validation assesses the suitability of different models using real-world data. In this regard, three paths are followed: (1) the H-LPTN cross-validation demonstrates the accuracy and computational efficiency of hot spot temperature predictions. (2) The thermal scaling rule demonstrates the validity of parameter scaling for temperature predictions of four hydrogenerators of different sizes. (3) Two case studies for online thermal control evaluate the adequacy of reactive power boosting for enhanced voltage regulation with temperature control.

Crucially, it must be emphasized that the high-fidelity thermal network validated in this section directly and exclusively governs the Q-model pillar—which constitutes one-fourth of the integrated Q-strategy framework architecture. Although the remaining operational, lifetime, and economic loops are processed by the Q-boost, Q-energy, and Q-value pillars, respectively, the thermal tracking engine within the Q-model serves as the indispensable physical foundation of the entire strategy. It acts as the machine’s guardian of structural integrity, ensuring that aggressive grid-side voltage optimization loops never drive the stator or rotor windings into unmonitored localized thermal runaway or catastrophic insulation degradation.

4.1 Simulation environment and machine configuration

To rigorously evaluate the transient capabilities of the proposed Q-strategy, system-level validations are implemented within a single-machine two-bus (SMIB) power system configuration. This simplified electrical topology is intentionally selected to isolate the localized dynamic thermal responses, insulation safety margins, and direct voltage-stability sensitivity of the hydrogenerator, preventing wide-area network impedances or multi-machine control actions from masking the core physical phenomena. Crucially, although the network model is simplified, the context is inherently grounded in real-world Nordic systems; all structural machine parameters, slow open-circuit thermal time constants, and historical start–stop cycle profiles are directly extracted from operational Nordic hydropower statistics. This ensures that the evaluation reflects the exact physical capabilities of the regional asset base under extreme localized stress.

4.2 Hot spot lumped parameter thermal network validation

To ensure the practical reliability of the Q-Model safety loops, the underlying H-LPTN was subjected to a rigorous parameter calibration and validation process. The network is calibrated using an optimization pipeline combining a heuristic particle swarm optimization initialization with a formal Bayesian inference framework via Markov chain Monte Carlo (MCMC) sampling. The calibration data comprise empirical-transient and steady-state thermal measurements extracted from official industrial factory shop-floor heat-run tests (HRTs) conducted in compliance with IEC 60034-1, supplemented by localized spatial loss distributions calculated using high-fidelity 2D and 3D electromagnetic FEA solvers.

Although the complete multi-variable parameter estimation algorithm, detailed probability density distributions, and the raw experimental datasets are comprehensive and compiled within the foundational companion study by Tome-Robles et al. (2025), these validated error boundaries directly justify the conservative temperature safety buffers applied to the online temperature limiter (OTL) schemes implemented herein.

4.3 Thermal scaling modeling

The model validation for thermal parameter scaling is performed for three different hydrogenerators. The estimated parameters are presented in Table 4, where G1 is the base case for the scaled thermal parameters of G2, G3, and G4. For instance, the highest percentage difference for the rotor winding hot pot is observed for G4, as expected due to its small size, where the thermal resistance and time constant percentage differences of 394.86% and 6.00%, respectively. Note that the thermal time constant slightly changes due to the intrinsic slow thermal dynamics. In contrast, the thermal resistance exhibits the greatest variation because it determines the temperature variation with respect to losses and the steady-state operating temperature.

The validation for G2 and G3 is performed using heat-run test data obtained during commissioning. The temperature results of the thermal scaling for G2 and G3 are shown in Table 5 and 6, where the limiting component—the rotor temperature—shows differences of up to 4.4% and 12.2% for G2 and G3, respectively. Finally, for G4, the validation is achieved with 24 h real-world temperature measurements. The maximum percentage error over the 24 h comparison is 6.80% at 18.87 h, the valley after the decrease in active power, as presented in Figure 13. The assumptions for G4 are 3.3389 K and 3.2 K, the air and water temperature difference, respectively, for the heat exchanger dynamics.

FIGURE 13

4.4 Case studies for online thermal control

This section introduces an OTL scheme, as shown in Figure 14, where a temperature PI controller serves as an outer loop for the AVR to limit the field voltage when the rotor hot spot temperature reaches 150 °C. A secondary voltage control (SVC) uses the load voltage as a pilot voltage to set the reference voltage of the AVR. The two-bus case example shown in Figure 1 is used for the contingency and extreme scenarios. For these examples, the proposed H-LPTN is used for temperature estimation.

FIGURE 14

4.4.1 Contingency

Figures 15a,b show the electrical and thermal dynamics over an interval of 480 min, during which different contingencies occurred, with the maximum field voltage limited by the rotor hot spot temperature threshold. Notably, during short-circuit and line disconnections, the temperature dynamics are minimally affected because their time constants are much larger than those of the electrical dynamics. The rotor temperatures are kept constant, whereas the stator and iron core temperatures vary with current and voltage to meet the active power requirements, thereby affecting the stator and iron core losses. Furthermore, during the worst-case contingency period, i.e., the 100–155-min interval, when the line is disconnected, 3-SC occurs. The load is increased by 25%, and the load voltage reaches 0.96 pu compared to 0.99 pu at the hydrogenerator’s terminal, representing a 3.08% difference. Therefore, the grid is supported with additional reactive power by increasing the terminal voltage of the hydrogenerator. A critical remark is that the field voltage is limited to 2.5 pu, which prevents it from reaching the SVC-specified load voltage of 1.025 pu. Nonetheless, the OLT clamps the voltage reference to the AVR to avoid trespassing the maximum allowed hot spot temperature.

FIGURE 15

FIGURE 16

FIGURE 17

4.4.2 Extreme scenario

The 24-h extreme scenario shows how reactive power boosting supports the grid under different contingencies for a typical load profile, as shown in Equations 16, 17. The voltage control keeps the rotor hot spot temperature below the threshold and provides sufficient field voltage to prevent voltage collapse during the 785–960-min interval, when a line is disconnected, the peak load is reached, and a 3-SC is present. Nonetheless, during peaks, the thermal limit is reached, and the voltage at the load no longer follows the set value of 1.025 pu.

In both scenarios, the hydrogenerator provided Q-boost following the SVC pilot voltage while maintaining synchronism during various contingencies and extreme grid conditions.

5 Discussion

The main challenges faced by reactive power boosting are thermal cycling, power-electronics thermal limits, and harmonization with modern grid codes. The Q-strategy implementation may require several technical considerations. A grid-independent excitation system for extended dynamics may be necessary, alongside careful evaluation of adaptive cooling systems. In the case of power electronics, they have a poor thermal constant during extended operation (Rodriguez et al., 2021; Kuprat et al., 2023). Excitation system modifications that incorporate capability curves and a dedicated current-limiter, as proposed by Ojo and Shafiul Alam (2026) for safe synchronous condenser operation of hydrogenerators, illustrate the kind of control upgrades that a comprehensive Q-Strategy demands. Therefore, resizing the hydrogenerator’s reactive power-boosting capabilities will require a new design philosophy aligned with these new services. The approach demands a comprehensive digital twin model that encompasses the generator, step-up transformers, excitation systems, and other components, supported by additional sensors and advanced control systems.

Winding refurbishment to a higher thermal threshold should be studied. For instance, using the thermal endurance rule-of-thumb equation, the total lifetime of the insulation class F for 12 h of boosting at 155 °C per day is 4.4511 years. However, if an insulation system upgrade is considered, this can be increased for an insulation class H up to 25.6885 years. In this regard, 21.1474 years are added to the hydrogenerator, with the possibility of delivering reactive power-boosting services to the grid, that is, an increase of 130.45%. Complementary to insulation upgrades, rotor pole redesign and field winding refurbishment can further reduce ohmic losses and shift the optimal reactive power characteristics, with net economic benefits shown to be four to five times the incurred material costs (Karekezi et al., 2025). This new design philosophy can be accompanied by an update of IEEE (2006) to incorporate peak capabilities for insulation classes F and H, similar to the IEEE (2014), where class H insulation can be the next step for rotor winding.

Regarding the grid code harmonization, the modern grid code establishes a static envelope (Q/) (Nøland et al., 2020). However, if the reactive power output for G1 may be up to Q = 0.79 pu for continuous operation, the maximum active power at the connection point should be 1.6 pu to comply with the grid code. Therefore, making the codes flexible through dynamic operating envelopes is indispensable, especially under restorative and emergency operating conditions. Future work should address fast reactive power-boosting response, reactive power-boosting coordination, and unit commitment. Q-energy can be further developed considering the TSO/DSO grid codes, contingencies, and optimization for multi-machine reactive power compensation, coordination, and synchronization through the ORPD. In multi-machine coordination scenarios (Melfald and Øyvang, 2023; Dragosavac et al., 2012; Dragosavac et al., 2013), the Q-strategy can maximize reactive power allocation benefits by incorporating these dynamic envelopes for grid codes and managing OEL thresholds through online hot spot temperature predictions using a CTM, while considering insulation system lifetime costs.

Furthermore, thermal control is crucial for minimizing wear and tear through temperature change rate management and Q-boost time limitations to comply with the standard for thermal cycling. The daily boosting duration, defined as Q-time , may be used to minimize the tear and wear of the hydrogenerator while considering the power plant owner’s requirements.

Therefore, hot spot temperature management demands rethinking the machine philosophy of design, control schemes, cooling systems, and copper insulation class temperature margins. In this study, a CTM for air-cooled hydrogenerators is proposed as a significant tool to mitigate the risks and costs associated with thermal dynamics, demonstrating how advanced control systems can accurately predict the operational efficiency, safety, and longevity of hydrogenerators.

This study not only presents a pioneering approach to addressing the challenges posed by the integration of VREs but also lays the foundation for future research to refine and expand the adaptive multi-component thermal hot spot management methodology to enhance the operational efficiency and reliability of hydropower plants.

5.1 Framework boundaries and limitations

6 Conclusion

This novel framework enhances hot spot management in the hydrogenerator to optimize their reactive power capacities, keeps temperatures within the insulation class thermal thresholds under contingencies due to its relatively large thermal time constants, and enables reactive power cost formation by considering additional losses and lifetime costs. The H-LPTN can be used as a generic CTM for air-cooled hydrogenerators of different sizes and shows good agreement with temperature measurements and hot spot FEM calculations. Furthermore, the quadratic thermal model is shown to agree well with the H-LPTN for ultra-fast calculations based on rotor losses and field current.

The disruptive step toward implementing the Q-strategy framework comes from the economic incentives of reactive power income, considering additional losses and lifetime cost, where it is found that a trade-off solution to maximize lifetime and income can be in a range of 299–1,700 k$ for 155 °C to 130 °C, respectively, when the active power is reduced to 0.2 pu. That means an income percentage difference of 140%. On the other hand, when the active power is at a nominal value of 0.9 pu, the income range is 95–191 k$ for 155 °C to 130 °C, respectively. The income percentage difference is 67%. Note that this income already includes the lifetime cost for winding refurbishment.

This study finds that smaller generators require higher reactive power costs to incentivize the Q-boost services. Hence, distributed generation might require a separate assessment of reactive power costs. Moreover, the most efficient generator will produce cheaper reactive power when the generators are similar. For a 12-MVA hydrogenerator, the reactive power cost is 1.3302 Mvarh. In contrast, the most efficient hydrogenerators have the lowest costs, i.e., for a 103-MVA hydrogenerator with a reactive power cost of 0.6872 Mvarh. The percentage difference between the highest and lowest reactive power costs is 63.75%.

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