Loss Modeling of Large Hydrogenerators for Cost Estimation of Reactive Power Services and Identification of Optimal Operation

As a result of the worldwide energy transition, reactive power generation has started to become a more scarce resource in the power grid. Until recently, reactive power has been an auxiliary grid service that classical power generation facilities have provided without necessarily allocating any cost for this valuable service. In this paper, a new approach for predicting the additional costs of reactive power services delivered by large hydrogenerators is proposed. We derive the optimal reactive power (ORP) with minimal losses as a function of the active power level within the generator's capability diagram. This pathway can then be used to calculate additional losses from operational regimes deviating from the ORP. To back up the analysis, a dedicated example study was handpicked consisting of four real-world generators scaled in terms of power rating, i.e., <inline-formula><tex-math notation="LaTeX">$15 \,\mathrm{MVA}$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$47 \,\mathrm{MVA}$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$103 \,\mathrm{MVA}$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$160 \,\mathrm{MVA}$</tex-math></inline-formula>. The objective was to identify how the ORP scale from smaller to larger MVA-sized generators. Moreover, a sensitivity analysis of the machine characteristics is conducted. We find the ratio between the rotor and stator losses as the determining factor. Finally, we show how our framework could justify profit for reactive power services, which are projected to increase in the future.


I. INTRODUCTION
T HE worldwide energy transition will fundamentally transform the future electricity grid to integrate ever-increasing shares of renewable energy sources (RES). At shorter time scales, the intermittency of RES will mainly be balanced by hybridized, fast-response energy storage solutions [1], [2], [3]. This is because the ramping capability of hydropower is usually limited to 10-30 % per minute to reduce wear and tear and avoid too high thermal stresses [4]. Nevertheless, hydropower can provide firm dispatchable power allocated over longer time scales and secure the baseload of the grid. In this way, the variability of RES can play well together with the flexibility of reservoir-based hydropower, where hydropower production is held back at high rates of RES production while stepping in to fill the gaps in other periods of energy drought and get increased revenue. As a result, hydropower is projected to be pushed closer to its capability boundaries and beyond in the near future. It might even contribute to more firm backup power shared between multiple countries [5].
Over diverse set of operating points. This effect is highlighted in [6], [7], [8], where significant changes resulting from the German 'Energiewende' can already be noticed. As a result of the worldwide energy transition, hydropower would need to adapt to completely new operational regimes to enable rapid growth of variable RES [6], [9], where both the active power capability but also the reactive power capacity can be significantly extended [10], [11]. These more diverse operating regimes bring new economic costs into consideration, as the overall efficiency now becomes a significant differentiator [8]. Moreover, to deliver more reactive power, rotor cooling has been found to be a major limiting factor [12], which is known as the overheating problem in the thermal management [13], [14]. Accurate information about the stator's thermal footprint is also essential [15], and combined with the cooling arrangement, can establish a thermal network for the whole machine to predict the overall capability [16]. These issues become even more critical if one tries to allocate short-term reactive power, as envisaged by an extended capability diagram [11].
A price must be paid in terms of reduced efficiency resulting from the dispatch of active and reactive power. To quantify the impact of variable operation, an accumulated average efficiency (AAE) model was recently proposed to improve the accuracy in the determination of the power losses [17]. It was proposed as an alternative to the mainstreamed weighted average efficiency (WAE) model [18], [19], [20]. Nevertheless, the main contribution of this paper is not related to the AAE but rather to how the active losses can be minimized and quantifying the cost of not operating the hydrogenerator optimally. The paper proposes a timely method for the identification of the maximum-efficiency operation of hydrogenerators as a basis for cost estimation of its operating regime based on the reactive power services it provides. The loss modeling presented derives an optimal reactive power profile as a function of active power generation, which can be used to inform the operational characteristic needed to maximize the AAE as the active power generation varies. The identification of the optimal reactive power (Q opt ) pathway in the capability diagram can therefore quantify the impact of the reduced machine efficiency based on the variation of reactive power services provided to the grid. Finally, the most profitable operation of large synchronous generators is deduced, which is the basis of the proposed technique to estimate the cost of an operational regime, including reactive power services.
A sensitivity study is also presented to see how the model parameters and inputs affect the machine efficiency and optimal operational path. First, the impact of the synchronous reactances, the Potier reactance, and the armature voltage are analyzed. Then, direct adjustments in the rated stator and rotor power losses are studied to investigate their impacts on the overall performance. The aim is to explore the link between the machine characteristics and the variable machine losses, optimal reactive power, and optimal efficiency. The sensitivity study offers first-order insight into how the different parameters affect the performance quantities under study. Moreover, our paper goes further beyond studying just one particular generator, and looks at the scalability of our findings by investigating a handpicked collection of machines at different ratings. We identify the underlying causes of the optimal reactive power at different scales, which could also provide valuable input to the design strategy from scratch if one wants in this way to influence the optimal generator operation.
The paper is organized as follows. First, Section II presents the presuppositions for the calculations. Then, Section III presents the concept of optimal reactive loading for loss minimization, while section IV presents a sensitivity analysis of the 103 MVA generator. An example study of four industry generators is provided in Section V. Finally, Section VI presents the clustering of real-world operational data to showcase to what extent calculation simplifications can be made before Sections VII and VIII ends with discussions and conclusions.

II. KEY PRESUPPOSITIONS AND ASSUMPTIONS
This section briefly presents the key presuppositions made in the calculations of this paper. They are as follows.
1) All of the operating points inside the capability diagram are assumed to be steady-state. There are no transients or sub-transients captured in the loading conditions. 2) The generator is connected to a large interconnected grid, implying that the voltage is stiff and equal to 1 pu for all loading points. The generator terminal point of connection is chosen as the reference point. It is, therefore, given the value of 1 pu and zero degrees as a reference. As a result, the armature current is directly proportional to the apparent power.
3) The synchronous reactance used in this manuscript is the reactance between the induced generator voltage and the generator terminal (i.e., step-up transformer reactance is neglected). Thus, the synchronous reactance in the d-axis is the summation of the direct-axis main reactance and the leakage reactance. 4) The downstream profile of the generator terminals was the main focus of all analyses. Consequently, the production profiles of artificially made uniform and synchronous condenser-dominated load distribution do not consider the upstream grid code requirements, where the step-up transformer could limit the reactive power production [8]. However, our modeling could add the reactance of the simple line equivalent of the step-up transformer to the generator's leakage reactance without changing the model. In this way, we have similarly estimated the grid-side reactive power as presented in [8]. 5) The operational regimes presented in this paper assume that there is no large-scale energy storage units other than fast-response storage for balancing RES intermittency at shorter time scales. The hydrogenerator is assumed to have all the dispatchable power reserves needed to dispatch RES at longer time scales. Thus, it highly varies in both active and reactive power levels.

III. OPTIMAL LOADING FOR LOSS MINIMIZATION
In this section, the concept optimal reactive power (Q opt ) for loss minimization will be developed. A loss minima implies maximization of profit for a given active power production (P ). The total losses (P l ) in the generator is a function of both P and where the solution, Q opt (P ), is the solution to reactive power enabling the minimum losses as functions of P . Based on this convention, additional losses (ΔP ) for a given P can be established based on the distance between Q and Q opt , formulated as |ΔQ| = |Q − Q opt |. We will later observe that ΔP l is approximately proportional to a second-order polynomial function of |ΔQ|, as indicated in (2).
The accumulation of ΔP can be used to quantify the total cost of not operating along the optimal reactive power path.
(3) accumulates the additionally wasted energy either using the exact calculation with time intervals or the method using weights representing the probability of occurrence.
To enhance the insights into the loss modeling, this paper separates the losses into rotor, stator, and constant losses as formulated in (4)- (8) with case values in Table I.
The variables in the loss separation model is I a , I f , and U a , where I * a , I * f , and U * a refer to their nominal values. The armature current is calculated in per unit as I a = P 2 + Q 2 /U a , while the field current is found from I f = f (E g , E p ) using the armature reaction parameters of the machine (R a , X d , X q , and X p in Table  II) and the open-circuit saturation characteristics (OCC) [17]. The field current is approximated via the similar simplifications of the Potier method presented in [17]. It is based on the machine parameters and the OCC curve only, which is reduced from more detailed models described in appropriate standards [21].
The loss models are strongly simplified with the assumption that the machine has warm components and is operating at a steady state. Nevertheless, the modeling is well-founded in stationary conditions, even though a warm machine will differ from a cold machine. In the case of a fast dynamic load change, thermal equilibrium needs to settle before the modeled behavior is valid. Moreover, the influence of the operating temperatures on machine losses is neglected, which can strongly affect the field winding losses, in particular. The saturation modeling involved in estimating the field current also makes the field winding losses a sensitive loss component.

A. Initial Case Study of the Hydrogenerator's Loss Minima
By utilizing the machine data of the 103 MVA hydrogenerator provided in Tables I and II, some preview results for the loss minimization and optimal reactive power are provided in Fig.  1. Its power loss contours are validated by seven stationary measurements in Fig. 1-(a). Furthermore, by combining the active power production lines in the capability diagram with the loss curves, an optimal operation path can be proposed, as illustrated in Fig. 1-(b). The active power lines giving the lowest active power losses are those that intersect with the peak of the loss contours, which is about -0.2 pu in reactive power for this particular case. Finally, by subtracting the minimal losses along the optimal P -Q profile, the additional losses are derived in Fig.  1-(c). Fig. 2-(a) further investigates the Q opt (P ) solution of (1), with and without saturation, where the minimum losses are given in Fig. 2-(b). It is shown that one needs to increase the consumption of Q slightly as one increases P . Moreover, Fig. 3 reveals that it is the optimum interaction between the rotor and stator losses that determines Q opt . When extracting out the additional losses from (2), the curves are more or less independent of the active power, as shown in Fig. 3-(a). Moreover, 3-(b) highlights that when plotting the total losses for the full range of active power levels, the shape of the loss profiles is identical with respect to reactive power but has different loss offsets. As a result, the minimum losses for the studied generator is around −0.2 pu reactive power, regardless of the active power level.

B. Basic Cost Modeling of Reactive Power Services
A handpicked average retail electricity price for the US in 2020 is 10.66 cent/kWh [22], which is taken as the basis for income and cost calculations herein. The income for a given P is multiplied by the price and time (Δt), and the same for  (1) and (2). The losses for each active power level (P ) were calculated for the full reactive power (Q) range. Then, the minimum losses were subtracted from the total losses for each active power level. Additional losses occur when deviating from the optimal P -Q load path (indicated in grey).  the cost associated with the P l . The gross profit is proportional to the production income minus the loss costs. As a result, the optimal profit strategy is operating at the optimal operation that minimizes P l . The additional operational costs for a particular load point can be predicted by multiplying ΔP l from (2) with the electricity price (0.1066 $/kWh) and the number of hours ( h) of operation. Table III highlights that Q ≈ -0.2 pu is the most profitable operating point at rated P .
To expand the preliminary cost modeling in Table III, the impact of three different operational regimes has been assessed in terms of cost. A concentrated distribution of measured data (mostly operating at unity power factor) for a whole year of the 103 MVA is compared against two distributions, all depicted in Fig. 4. Table IV reveals that the most profitable operational regime is a concentrated load distribution due to a high income and relatively low operational cost because it mostly operates close to Q opt . In contrast, the synchronous condenser-dominated mode has the lowest profit due to long periods of low P , but it also has more than three times more additional losses and costs due to its reactive power services.

IV. PARAMETER SENSITIVITY TO THE LOSS MODELING
To gain a deeper understanding of the underlying modeling mechanisms determining the optimal reactive power, a parametric sensitivity study is done of the 103 MVA hydrogenerator. The analysis is conducted in normalized quantities, which means that the same principles apply to other generators, yielding a level of generalisability. The parameters under study, namely X d , X q , X p , and U a , are altered to affect the machine efficiency and consequently change the optimal reactive power (Q opt ), using the loss model presented in (4)-(7). In Table V, three different values for each parameter under study are investigated, highlighting changes in losses, efficiency, and optimal reactive power. When one of the parameters changes, the other ones are kept constant at the original value according to Table II. Furthermore, another simplification is made assuming that the standard parameters X d , X q , and X p are not affecting the no-load curve of the machine in the sensitivity study, where   [17]. Nevertheless, the parameters at the design stage are very much correlated with each other. The sensitivity analysis results are also plotted in Fig. 5.
There is a slight negative trend between the reactance values (X d , X q , X p ) and the machine efficiency. Higher synchronous reactances (X d and X q ) increase both the induced voltage (E g ) and the Potier voltage magnitude (E p ) and angle. Thus, the field current is increased and indirectly affected. This again leads to higher rotor losses according to (5). However, the reactance values do not influence the stator losses nor the other constant losses in this simplified modeling approach. The physical limitation is that, in reality, the stray load losses would be affected due to the armature reaction's magnetomotive force depending on the air gap, which is influenced by X d and vice versa.
A change in the armature voltage has the overall largest impact on the efficiency and Q opt . E.g., increasing the armature voltage causes a decrease in the armature current, consequently reducing stator losses. However, both rotor and constant losses are, in general, increasing from higher values of U a . As seen from Fig.  5, the efficiency is highest around nominal condition (U a = 1.0).
The AAE for the three operational regimes described in Fig.  4 stays mostly constant in relation to each other. Machine reactances do not affect the AAE in any significant way. The armature voltage has the largest impact on the AAE, while the load distribution is a more substantial factor.

A. Sensitivity to Changes in Rotor and Stator Losses
A second part of the sensitivity study is presented in Table  VI and Fig. 6. The rotor and stator losses at the nominal point (described in Table I) are incrementally increased from their default value using an adjustment factor. This is done to see how the optimal reactive power and efficiency are affected when either the rotor or stator losses become more dominant. The rated losses in the stator and the rotor are scaled directly in the calculation and adjusted from their original values. This can be seen quantitatively in Table VI and in Fig. 6. By increasing the stator losses, the optimal reactive power moves asymptotically toward zero (i.e., the unity power factor is optimal). However, when stator losses approach zero, the optimal reactive power will exceed -1 pu and will then be constrained by the outer limits of the capability diagram. In contrast, zero rotor loss implies that the optimal reactive power of the generator is 0 pu. These findings provide valuable insight into what determines the optimal reactive power in the trade-off between the stator and rotor losses. To expand on these preliminary insights, an example  VI  THE IMPACT OF ALTERING THE RATED STATOR AND ROTOR LOSSES ON THE OPTIMAL REACTIVE POWER AND THE AAE  study of several generators of different sizes and characteristics will be the focus of the next section.

V. EXAMPLE STUDY OF FOUR INDUSTRY GENERATORS
This section presents an example study consisting of four power industry generators, i.e., G1, G2, G3, and G4, where generator G2 represents the already investigated preliminary study case. To generalise the findings, a comparative study of the optimal reactive loading of generators is provided.

A. Measured Performance Data
The key rated quantities of G1-G4 can be seen in Table VII, and the rated losses of each generator are provided in Table VIII.  When not explicitly known, the Potier reactance, which is useful to estimate the field current in loaded conditions, was estimated using the approximation where X p ≈ 0.7X d [23], which applies for generators G2 and G4. The reported measured losses were evaluated according to the guidelines of the IEC 60034-1 [24] and IEC 60034-2-3 [25], respectively.

B. Extraction of Open-Circuit Characteristics
The open-circuit characteristics (OCC) of generators G1-G4 are modeled using coefficients presented in Table IX, which are used to be matched against the measured data [17], [26]. The air gap line and the OCC are found using [27] and the OCC data for G1-G4 are given in the standard datasheets of these real-world generators. Already, G2 and G3 are provided in [17] and [28], respectively. The estimated OCC profiles for G1-G4 are compared against the recordings in Fig. 7. It can be noted Fig. 5. Efficiency, optimal reactive power, and losses as the standard parameters X d , X q , X p , and U a are changed for the 103 MVA hydrogenerator.  that there are some deviations between the model fit and the measurements because the unsaturated and saturated part of the OCC is modeled with a unified approximation that covers all regions, i.e.,

C. Detailed Example Study of Optimal Reactive Loading
The validation of the iso-efficiency maps of G1-G4 is presented in Fig. 8, which are used to verify the calculation by experimentally cross-checking the calculated curves against the measured efficiencies given at specific load points. Moreover, the optimal reactive power operations (Q opt ) of the four industry generators are visualized in Fig. 9. They highlight how the exact optimal reactive path differs for the four hydrogenerators with different characteristics. Fig. 9 also presents the separated stator, rotor, and constant losses, in addition to the total losses, which provide further insights into how the total losses are distributed. Fig. 7. Modeled, [17], versus the measured OCC of G1-G4 in the example study. The base value for the plots are the airgapline rated field current (I fu ) and the rated terminal voltage (U a ) (see Table VII). The modeled OCC curves are based on coefficients provided in Table IX.   (1) What is noticeable is that even though the losses vary, the overall loss curve for all the different generators still has a minimum value of very close to -0.2 pu reactive power, as already found in the preliminary study. These values of Q opt are summarized in Table X for nominal active power, where also the corresponding grid-side values have been approximated.

VI. SIMPLIFIED LOSS MODELING OF OPERATIONAL DATA
When the optimal reactive power has been identified for a particular generator, the additional losses and costs from reactive services can be established. The estimation of those can be made easier based on justified simplifications of real-world operational data. The power industry is currently using a method of zone clustering [19], with probability-of-occurrence maps of a specified resolution. However, the error from this simplification has not been explored in terms of loss and cost modeling or through efficiency evaluation. The power producer manually determines the number of clustered zones, with more zones leading to more accurate representations of the operational area. Usually, the  Table XI assesses the precision of clustering the data in the concentrated dataset that are identical to Fig. 4-(a). The accuracy is shown to be strongly dependent on the number of weighted zones, which approximates the production dataset. Four sample cases of zone resolutions are depicted in Fig. 10. From Fig.  11 and Table XI, we can conclude that the practice of zone clustering is a viable solution. However, suppose the zones do not Fig. 9. Mapping of the total losses (P l ), in addition to the varying stator, total rotor, and constant losses separately (P l,s , P l,r , and P l,c ). The exact course of the optimal reactive path (Q opt ) for all four industry generators, G1-G4, are presented from the top to bottom. E.g., the top row shows the total loss mapping, exact Q opt , separated losses and total losses at different active power levels for generator G1. correctly approximate the clustering of the load points. In that case, it comes at the expense of the efficiency calculation, which would then lead to inaccuracies in the loss modeling, and, consequently, also impact the estimation of the costs of the reactive power service that the generator provides. A more computationally expensive practice is to use all the actual operating points in the loss calculation. In Fig. 11, the AAE (η a ) starts to converge as the number of zones is increased, where it begins to approximate the load density of the concentrated dataset correctly. Even though a lower number of zones overestimates the AAE, it underestimates the mean loss (P l ) of the data, which also underestimates to cost of the reactive power service the generator provides throughout the year. Still, a realistic estimation of machine losses can be achieved through the clustering of operational data. However, the accuracy of such approximate models is quite dependent upon the number of weighted zones. Fig. 11. Impact of the AAE efficiency for a yearly dataset ofÅbjøra hydropower plant (i.e, 103 MVA hydro generator) for an increasing number of weighted zones, using the AAE method developed in [17].
It is, therefore, important to treat a specific dataset correctly according to its distribution.

VII. DISCUSSION
This paper's proposed approach is able to track the optimal reactive power for a given active power to maximize the efficiency within the capability diagram of the hydrogenerator. However, in the context of the power system, one will not be able to select the optimal reactive power but rather serve the power grid. Nevertheless, the cost of operating non-optimally within the capability diagram can be quantified by identifying the optimal reactive power.
It has also been identified that the coupling between both the rotor and stator performances becomes the primary deciding factor for optimal operation. In particular, the stator and rotor losses directly originate from the stator and rotor designs and their associated parameters. A sensitivity study focused on the uncertainty of how the optimal reactive path is affected when machine parameters and loss distributions are altered. However, it is evident from the design stage that there is a strong correlation between the machine parameters. Nevertheless, the relative altering of individual parameters is valuable in providing insights into parameter sensitivities. Moreover, it gives an in-depth understanding of how isolated changes will influence overall performance.
For general applicability, the optimal reactive paths are investigated at different scales of machine sizes and ratings. The generators under study were generally optimized for overexcited loading conditions (i.e., cos(ϕ) ≤ 0.9, inductive). Results show that the optimal reactive path is a consequence of the design, as the exact optimal reactive path differs slightly from case to case. It is evident that when machine size changes, so will the machine parameters, copper and iron weight, and field and armature current loading change accordingly. However, such scaling effects have less impact on the optimal reactive power, which seems to remain more or less unchanged. Different operational regimes have been investigated to provide enhanced near-term revenue possibilities of reactive power services for power producers. The results presented indicate that deviation from the optimal operation could result in incentives that substantiate an increased revenue for power producers. It can support decision-making for power plant operators and allow them to compare additional losses against revenue for the provision of reactive power. However, it depends strongly on country-specific ancillary service policies. Moreover, the concentrated dataset with high active power production targets a reactive power close to the optimal path and is to be desired to reduce the cost of operation. Compensation for any sub-optimal operation might be an important topic when considering future operational regimes that tend to be less concentrated within the capability diagram. We also show that the cost of reactive power services can be established either by evaluating specific load points or complete operating regimes, where simplifications, such as zone clustering, apply.

VIII. CONCLUSION
This paper presents a methodology to identify and reveal an operation profile within the capability diagram of hydrogenerators to achieve optimal reactive power dispatch. Our work is based on an example study of four generators of different sizes and ratings, where a comparison is made based on loss modeling and verification against measured data. Finally, we provide evidence for proof-of-concept in terms of cost modeling of reactive power services. The main highlights from our paper are as follows.
1) The model is reasonably accurate for steady-state operation but will differ for a cold machine or in the case of a recent dynamic response. I.e., temperature dependency on losses is neglected. 2) The optimal reactive path referred to the generator terminals is around −0.2 pu reactive power for the generators studied in this example study. Nevertheless, the optimal reactive path is always a consequence of the machine design. 3) In general, one can say that the optimal reactive path will always be ≤ 0 pu in reactive power because the minimum stator losses are at zero reactive power. Moreover, the optimum reactive power will also be ≥ −1 pu because the minimum rotor losses occur when there is a complete consumption of reactive power at the left border of the capability diagram. Future work will focus on combining the optimal reactive path of the machine with grid and turbine models in an overall loss minimization framework. Moreover, the modeling framework could further be combined with an advanced control system for optimal operation. Such a control system will be relevant for enhanced system operational security and could help avoid voltage collapses of interconnected power systems.