Revenue Requirements Method for Assessing the Cost Impact of Fuel Cladding Corrosion in a Supercritical Water-Cooled Small Modular Reactor: A Methodological Review on Life Cycle Costing Corrosion

An old adage on corrosion is that “an ounce of prevention is worth a pound of cure” [1]. To take economical preventative measures—consistent with pro-active planning [2]—requires an understanding of corrosion phenomena, corrosion control, and their corresponding costs [1,2]. The old adage is expected to apply to advanced nuclear reactor concepts under research and development (R&D) in the Generation IV International Forum (GIF), such as the supercritical water-cooled reactor (SCWR) concept.

Currently, Canadian Nuclear Laboratories (CNL) has an R&D project that aims to develop a model to predict corrosion growth more accurately, and to assess the impact of corrosion on the performance and costs of a small modular Canadian SCWR concept. As part of the first aim, the project's objectives focus on the corrosion of various material candidates in the reactor plant under supercritical conditions, for instance, the general corrosion of chromium-coated zirconium-based and titanium-based alloys [3–5]. A motivation for considering an alternative fuel cladding alloy to a zirconium-based alloy is that “[w]hile zirconium alloys have a low neutron capture cross section and remain the preferred fuel cladding alloy choice from the perspective of neutron economy, it has been long known that these alloys experience unacceptably high corrosion rates” [6–9] in supercritical water (SCW). To support the project, the present investigation aims to identify the economic methods for understanding the impact of corrosion on costs in an SCWR concept.

In addition, the current project supports CNL's collaboration with international partners in the Joint European Canadian Chinese Development of Small Modular Reactor Technology (ECC-SMART) project, which was “created according the joint research activities under the International Atomic Energy Agency” and the GIF umbrella [10]. A part of the ECC-SMART project's main technical objective is to gain an “understanding of the corrosion behavior of the most promising candidate materials at different conditions to support the qualification procedure of the future SCW-SMR [supercritical water-cooled-small modular reactor] constructional materials and assess the relation to the existing standards and guidelines” [11]. Addressing this knowledge gap and challenge is part of the ECC-SMART project's task of developing conceptual design requirements, of which CNL is expected to contribute. Hence, the present study on costing methods for understanding corrosion and its impact on performance in an SCWR concept will also support CNL's on-going participation in the ECC-SMART project.

There are currently no corrosion economics studies for any SCWR concept that can be used to support the CNL and ECC-SMART projects. Moreover, the SCW-SMR concept remains under development, and nuclear economics studies on corrosion are fragmented. To cope with these limitations, the present study reviews SCWR corrosion phenomena and major cost categories on corrosion to gather information useful for identifying a cost model to enable cost and profitability analysis. Furthermore, the present study is primarily concerned with costing methodology (e.g., relation between variables, and steps to calculate), and secondarily with parameter or estimated values. Hence, the present study is a methodological study that prioritizes an economics perspective of corrosion engineering in nuclear power generation. Consequently, discussions on corrosion phenomena may appear elementary to corrosion engineers but essential to the work of corrosion engineering economics.

In the present study, progress on understanding corrosion economics for an SCW-SMR concept begins (Sec. 2) with a review of corrosion phenomena based on past SCWR concept R&D experience, and experience with conventional water-cooled reactors (WCRs) and supercritical (SC) fossil power plants (FPPs). In addition, this review focuses on the expected fuel cladding corrosion phenomenon and associated corrosion control measures (e.g., control parameters).

Section 3 integrates the corrosion related information with corrosion costing categories and a corrosion engineering economics method. These aspects are used as inputs and provide the constraints required to develop a power generation cost model for assessing the cost impact of corrosion and corrosion control of an SCWR concept (Fig. 1). The methodology of the model is guided by an engineering economics method, the revenue requirements approach to life cycle costing (LCC) (Sec. 3). A simple LCC procedure [12] is shown at the bottom of Fig. 1 in connection with the contributions provided by the sections of the present study to identify and develop a power generation cost model. The model is specified in terms of the main corrosion cost categories and a reliability factor used in assessing power generation costs, the costing of chemical species for controlling corrosion, and the present worth of revenue requirements. For each of these elements, potential future changes to the model are discussed to enable future use as the SCW-SMR concept evolves, and as current WCRs continue operating.

In conclusion (Sec. 4), the revenue requirements approach enables incorporating knowledge of corrosion and corrosion control in a power generation cost model that provides a framework for understanding the kind of information available and needed for taking economical preventative corrosion measures for the current generation of WCRs and advanced reactors, such as the SCWR.

Note that magnetite is a naturally occurring iron oxide mineral [19,20].

Controlling or minimizing corrosion is a design engineer's objective [1,13,15,16,21–23]. Moreover, “[f]rom an SCWR designer's perspective, a key output required of any program to study corrosion is a means to predict the total metal loss (or metal penetration) at the end of in-service life of the component and the uncertainties associated with that prediction” [7]. For design engineers to develop and assess corrosion control methods and strategies, “[p]erformance criteria for in-core materials can be developed from the requirement that the material should not fail during the in-service life” [7,24]. Although materials for both in-core and out-of-core components must be specified, SCWR R&D focuses on in-core components due to the extreme in-core environment in an SCWR concept [7,24,25].

A prime example of in-core component studied for the SCWR is the fuel cladding, since

Given the aforementioned emphasis on fuel cladding in SCWR R&D, fuel cladding is selected as an example for understanding the potential economic impact of corrosion in an SCWR concept.

To understand the relevance of fuel cladding corrosion phenomena for economics, the key control parameters are identified that enable performance and corrosion control. In addition, the topics discussed are the in-core environment, fuel cladding, corrosion product transport, and modeling fuel failure. While this discussion is concise given the wide array of issues and depth to consider on corrosion phenomena in an SCWR concept, more extensive discussions and references are in Refs. [7], [24], and [27–29].

Thermal power plants require a medium to transfer heat from a source, and pipework to contain the heat transfer medium [7]. The heat transfer medium (or fluid) in any SCWR concept is supercritical water. An SCWR concept typically consists of a direct steam cycle configuration for the balance of plant, such as shown in Fig. 2 [7]. For most SCWR concepts, the coolant system is comprised of the reactor core, the turbine set, and the feedtrain [24]. A “portion of the in-core piping, plus the main steam line and high-pressure turbines, are at $T>$ the critical temperature $Tc$⁠, and only the former are irradiated. The rest of the system operates under conditions for which significant operating experience and a well-developed knowledge base exist” [24] (Fig. 3)—similar concerns exist for an indirect cycle [7]. Furthermore, since for any SCWR concept supercritical water is the coolant, the properties of water change when crossing the in-core environment [24] (Fig. 4). Hence, “the properties of supercritical water directly affect general and localized corrosion (such as environmentally assisted cracking) of system materials and the transport of corrosion products to and from the core” [7]. Material and water chemistry studies on corrosion in an SCWR concept, therefore, tend to focus on SCWR in-core materials due to the combination of changing water properties, the intense radiation field, and high temperature and pressure that result in an “extreme” in-core environment in an SCWR [7,24].

In assessing fuel cladding materials, performance criteria are developed and compared to experimental measurements. Modern experimental data are generated in accordance to the type of materials study required in a four-tier testing strategy (Fig. 5) that was developed by the GIF Materials and Chemistry Project Management Board (M&C PMB) for materials R&D in support of SCWR concepts [7]. Furthermore,

Fuel cladding failure may occur due to any of several factors, such as: through-wall penetration by general corrosion; oxide buildup that can impede heat transfer; stress corrosion cracking (SCC); stress exceeding the yield limits of the material; and pellet-cladding interaction [28]. Since current knowledge of general corrosion is one of the most advanced knowledge areas of fuel cladding [7], and current experimental investigations for a small modular SCWR concept [3,4] focus on general corrosion, the discussion below is concerned with general corrosion phenomena.

While “[t]he ultimate goal of engineering studies of general corrosion is to define the total metal loss (metal penetration) at the end of in-service life, enabling designers to add a corrosion allowance to the component thickness required on the basis of other (typically mechanical) material properties.” [7] The corrosion allowance is the difference between a minimum and maximum thickness of fuel cladding.

at the end of in-service life of 3.5 years (30,600 h) [26].

A first step in validating models for understanding the corrosion of fuel cladding material, and control strategies, is to obtain experimental results on metal penetration. Research groups report experimental results (e.g., [6,7,26,28,35–38]) as the change in weight of the samples of surface area exposed to the test solution (measured by weight after$–$weight before (⁠$mg·dm−2$⁠)—for alternative measurements see Longton [35] and Svishchev et al. [39]). The change in weight presented in the literature is presented as either a weight gain (⁠$WG$⁠, or change in weight without descaling) or loss after descaling (i.e., descaled weight loss, $WL$⁠) [7,26]. In experiments, when the weight gain is used for assessing the performance of the coupon, which has some inherent uncertainties, the weight gain is used as a conservative approximation [7,26,36]. Eventually, the weight loss after descaling the surface oxide should be used to evaluate general corrosion rate of a material for SCWR fuel cladding [7], since it is the descaled weight loss that measures the true corrosion of the material, i.e., “the only unambiguous method to measure the corrosion rate” [36].

where $Ea$ is the activation energy of the rate controlling phenomenon (assuming that it is thermally activated), $kB$ is the Boltzmann constant, $T$ is the temperature, $t$ is the exposure time, $z$ is a constant, and x and y are parameters such as flowrate or dissolved oxygen (DO) concentration experimentally determined to be important [28]—for alternatives see [7,40]. Once the parameters are determined in a corrosion rate law function, such as in Eq. (3), the parameters are validated by comparing the predicted results to experimental corrosion data, and the importance of the parameters can be assessed.

Based on experimental data, the expected major physical and chemical environmental parameters affecting general corrosion of a given alloy under SCWR conditions in order of decreasing importance are [31]

temperature $≈$ surface finish $≈$ grain size $>$ water chemistry $>$ SCW density

“In general, over the entire temperature range [of SCW], the corrosion rate increases with increasing temperature” [28] as exposure time increases. For understanding the effects of the remaining parameters and more discussion on temperature, the reader is referred to Refs. [7] and [28] and the references therein.

Section 2.2 partially considered corrosion of the fuel cladding material in an extreme in-core environment, which can lead to fuel failure and associated costs. Another aspect is that the oxidation of fuel cladding material contributes to the formation of corrosion products that become active and transported from the core (Fig. 6). The “[f]ormation of thick deposits on SCWR fuel cladding could result in (1) overheating of the cladding surface or under deposit corrosion, (2) changes in-core reactivity (crud-induced power shifts), and (3) increased radiation fields on out-of-core piping” [24]. In addition, corrosion product deposition on fuel cladding will have a negative impact on the neutron economy “because of parasitic absorption of neutrons within the deposited material” [41]. The selection of water chemistry, therefore, depends on both how the environment is affected by and affects the fuel cladding material.

The selection of water chemistry is expected to provide guidance for materials testing in support of the SCWR concept, especially the testing of candidate fuel cladding alloys (see Fig. 7, adapted from Ref. [32]) by identifying key factors, such as the oxide solubility. In turn, the test results will support the high-level goals to predict and mitigate corrosion product transport (including activity transport) and to predict the in-core chemistry conditions, in particular the effects of water radiolysis.

To control corrosion and corrosion products, reactor operators require knowing the concentration level measurements by monitoring and sampling the types of corrosion products associated with oxide deposition in a reactor [43–45]. The principal oxide expected to deposit in SCW is magnetite [28]. A major activation product associated with magnetite is cobalt, primarily cobalt-60 (⁶⁰Co), which has a high adsorption affinity within magnetite [46]. Minimizing iron concentration is, therefore, an important parameter to control for selecting appropriate chemical additives and water purification systems [28,45–47]. To compliment the water chemistry, SCWR studies recommend “feedtrain materials with corrosion rates of ∼ $7.1×10−4$ mm/y would minimize in-core deposition” [28]. The selection of the feedtrain material has several implications. For instance, the original work on ammonia addition to boiling water reactors (BWRs) was guided by a desire to reduce capital costs by substituting carbon steel for stainless steel [21,48,49], a strategy successfully deployed in the Gentilly-1 reactor [50]. However, the use of ammonia dosing would increase turbine activity [50]. An alternative to selecting between different alloys is to consider selecting between different production methods of the same alloy, e.g., different steel making techniques [51,52]. In close connection to the feedtrain material selection is the design configuration for the purification system [53,54]. Minimizing corrosion and corrosion products also supports avoiding costly repairs and minimizes downtime (or enhances plant availability) [21,55,56], in part because ⁶⁰Co affects maintenance work during outages [46,57,58].

Even though “water radiolysis remains the key outstanding issue preventing definition of the SCWR in-core environment” [7], to support ongoing experimentation current SCWR R&D on corrosion product transport indicates that

While there are different SCW oxygenated treatment technologies [59] and costs [60,61] for SCFPPs readily available, any study of adjustments to the technology for an SCWR concept are not yet available. Moreover, Guzonas and Cook [62], speaking of a direct-cycle SCWR, noted that while iron transport might be minimized by an oxygenated feedtrain chemistry, it would likewise be necessary to suppress net radiolysis in the reactor by adding hydrogen. Since this hydrogen would be ejected at the turbines, this implies a continuous consumption of hydrogen, adding to the operating costs of a direct-cycle SCWR.

Some costing aspects of chemical treatments for WCRs are known, for instance, the coolant treatment and purification equipment cost for a pressurized water reactor (PWR) [63,64], the unit cost and consumption of boron [65], and cost modeling decontamination [66] are available. The boron treatment is relevant for a pressure vessel design of the SCWR concept [43,67] but not for a pressure-tube design of the SCWR concept [47]. Hydrogen consumption costs for BWRs are not readily available, but these costs could be scaled to SCWR conditions. Furthermore, cost estimates for the hydrogen injection system are available [68]. Boiling water reactors operating with hydrogen water chemistry (HWC) plus noble metal chemical addition (NMCA) consume a small amount of hydrogen in order to mitigate intergranular stress corrosion cracking (IGSCC) [69]; a direct-cycle SCWR is anticipated to consume up to 100 times more hydrogen to suppress net radiolysis [70,71]. The addition of hydrogen is used to control electrochemical corrosion potential to control in mitigating IGSCC, which is a part of the environmentally assisted cracking [7,28,72–75]. An additional cost consideration associated with IGSCC are repair costs [56,76].

Differences in cycle design play a major role in both plant construction and O&M costs. For instance, an indirect-cycle SCWR would have the added capital cost of a steam generator [7]. On the other hand, a direct-cycle SCWR would require heavy shielding of the main steam lines and high pressure turbines [7].

Corrosion studies on models of the fuel failure of different fuel cladding materials due to corrosion for SCWR concepts do not seem to be available. Instead, corrosion studies for supporting SCWR concepts, for instance, Ru and Staehle [77–79] use past experience of WCRs [80–83] to understand potential failure rate of equipment due to corrosion in terms of a probability distribution. Typically, these studies use a Weibull distribution. A similar strategy is suggested in the present study to develop probabilistic models for assessing the fuel cladding failure rate of an SCWR concept. A variety of studies on fuel failure due to corrosion in WCRs exist, for example, [46,84–97]. Amongst these studies, Kritsky et al.'s studies [46,85] incorporate general corrosion and water chemistry (including corrosion product transport).

where $n1$ is the actual number of failures for light water reactors (LWRs), $n0$ is a nominal number of failures for LWRs, $G$ is the number of parameters considered (corrosion included), $be$ is a coefficient defining the effect of corrosion on fuel element failure in the data series observed, and $Ve$ is the variation of the corrosion parameter during a year of observation.

In Eq. (4), the main factor that relates to Secs. 2.1–2.3 is the parameter $Ve$⁠. This parameter is derived in Kritsky et al. [46,85] from a corrosion rate equation, which accounts for multiple factors including water chemistry (to suppress the effects of water radiolysis), temperature, vapor quality, neutron flux, oxide layer thickness and thermal conductivity, and impurities. According to Kritsky et al. [46,85], the approach to Eq. (4) is supported by the data in Table 1, which is based on operating reactors. Unlike the WCRs considered by Kritsky et al. [46,85], SCWR concepts must overcome the knowledge gap afforded to years of operating experience. A similar approach to that used in Kritsky et al. [46,85] may, therefore, be used for supporting an SCWR concept where the failure rate depends on the fuel cladding corrosion rate, which in turn is determined by the known factors affecting the corrosion process. In this case, the corrosion model will be expected to evolve as new and better knowledge is gained.

In assessing corrosion control options, utilities and analysts use the revenue requirements and LCC approaches as part of understanding reliability and maintenance of a power plant [76,98–101]. This section, therefore, develops a revenue requirements approach for the LCC of an SCWR that includes a model for corrosion that incorporates the information from Sec. 2. Alternative engineering economics methods for corrosion are discussed elsewhere [102–105]. Furthermore, the SCWR concept's similarity to WCRs would suggest considering past studies with engineering economics evaluations of corrosion in WCRs. Nonetheless, this path has methodological challenges. One challenge is that some past studies may mention an engineering economics method and provide the values used but not elaborate on the methodology used to obtain them. Questions then arise about how to modify these values to account for the underlying engineering design choices and their effects on fuel failure [106]. Others studies can be found that relate more directly to fuel failure [107], but only cover a portion of the power generation costs, and do not consider costs across time. Both kinds of studies lack details about the O&M costs.

Equation (5) is first calculated on an annual basis, which is useful for budgeting purposes [108]. Second, Eq. (5) is multiplied by a present worth factor to consider the costs over the useful life of the product [108,111]. To make the LCC method useful for studying the cost impact of corrosion in an SCWR concept requires specifying the LCC in terms of a power generation cost model that can incorporate the main corrosion cost categories, a reliability factor to address corrosion, and the cost of chemical additives for controlling corrosion. For the present study, the present worth of revenue requirements is applied to the power generation cost model to correspond to the final step in the LCC method.

The discussion below focuses on an engineering economics method rather than on a macro-economic method for assessing the economic impact of corrosion for an SCWR concept. Applying a macro-economic method is left for future development, though a discussion on this type of method is available in Biezma and San Cristóbal [112].

where $fcr$ is the annual fixed charge rate, $Ctpc$ is the total plant cost (sense of the concept will vary according to the cost information available and needed), $COM$ is the annual O&M cost, $CF$ is the annual fuel cost, $Srcap$ is the grid system replacement capacity cost, $ΔP$ is the replaced capacity (MW_el), $Srene$ is the grid system replacement energy cost, $ΔE$ is the replaced energy (MWh), and $E$ is the expected annual electric power to be produced (MWh). The first term in Eq. (6) deals with initial plant investment and operating costs, while the second term deals with the cost of replacement energy of an alternate plant (e.g., an LWR or FPP) needed to run a grid reliably while the plant of interest (e.g., an SCWR) is under maintenance. Implicitly, Eq. (6) includes the depreciation cost category through the $fcr$ term since it consists of the rate of return, depreciation, insurance, taxes, and administrative and general expenses as a fraction of the total plant cost [117,118,124].

A missing cost category is decommissioning, which can be affected by water chemistry and decontamination choices for controlling activity transport [125,126]. The inclusion of decommissioning will be similar to the total plant cost, except that its financial assumption will reflect payments to recover for a deferred expense after the end of the plant's life. In addition, replacement costs of major equipment associated with corrosion can be added in a similar manner as the total plant cost, with financial assumptions reflecting economic conditions that can differ from those of the original investment (see Federal Energy Regulatory Commission (FERC) rules [127] for cost classification). Other omitted cost categories are R&D, testing and qualification (see FERC rules [127] for cost classification). Their inclusion can be incorporated in a similar manner as the total plant cost (see FERC rules [127] for cost classification), though the extent of its inclusion in the $COE$ will also depend on the extent to which governments share the R&D, testing and qualification costs.

By modeling $R$ in terms of a failure rate that corresponds to corrosion, such as in Krisky [46] (see Sec. 2.4), Eqs. (6) and (9) may be used to address the impact of fuel cladding corrosion on reliability and costs of an SCWR concept. As an SCWR concept evolves, the reliability factor is also expected to account for the reliability of the whole SCWR plant concept, in which the reliability of the fuel will be a part.

The capacity factor may also be expressed in terms of availability in alternative ways (either Ref. [122] or [132]) to explicitly address derating due to, for example, corrosion or deposition on fuel cladding. In addition, derating is implicitly addressed in lost production costs in Eq. (6)—see [107] for additional discussion on derating and costs of lost production associated with fuel failure.

Elements implicit in Eq. (6) that are affected by reliability that may be made explicit in the future are: the competing power generation technology in a power grid that determines replacement energy and capacity cost (i.e., lost production costs); and, overall plant thermal efficiency [121,123]—for a case relating to corrosion see, e.g., [133]. These factors may be affected by energy policies that affect planning for reliability in the power systems, and the mix and expansion of the system. Other related factors to energy policies are the state of an economy and economic policies affecting the economy.

Incorporating reliability in the $COE$ is expected to provide insight on the cost impact of increasing the service life of fuel, structural materials, and components by controlling corrosion. For instance, past experience in evaluating a PWR project over its service life by Électricité de France (EdF) indicates that there are two opposing economic forces captured in an annual cost bathtub diagram (Fig. 8—for illustrative purposes) [134]. In Figure 8, the annual costs weigh the construction cost and the maintenance cost in unit production costs during different years of operation [134]. Lengthening the service life reduces annual cost of construction but may increase the annual maintenance cost due to, for example, replacement costs [134]. The optimal service life is, therefore, a result of weighing each of these opposing factors. Similarly for the corrosion of system components in a nuclear power plant, “[c]hemistry performance requirements are set by the sometimes conflicting desires to minimize corrosion (general and localized), fouling and activity transport, optimize thermal performance and maximize heat transport system (HTS) lifetime” [23] (emphasis added). Thus, the optimum specification of the “coolant chemistry is always a compromise” [23].

The main tradeoff to consider, based on Sec. 2, for the SCWR concept is the temperature of SCW. Throughout R&D programs on SCWR concepts, the underlying advantage sought in superheating steam is to increase the overall plant thermal efficiency higher than WCRs [67,136]. This advantage is expected to translate throughout the plant [136]. However, achieving a higher efficiency requires reaching a SC temperature. As indicated in Sec. 2, the corrosion increases with temperature. Higher temperatures are, thus, expected to limit the service life of materials in an SCWR. A part of the reliability effect of corrosion corresponds to wear-out failures [137], in which case costs increase with service life (the latter part of the bathtub curve in Fig. 8). Hence, the advantages of the SCWR may be offset to an extent by the effect of corrosion on the service life of materials—for additional tradeoffs related to fuel cladding based on LWRs experience see, e.g., [138]. To weigh the balance of the tradeoff, results from either the LCC or the revenue requirements approach to LCC may be incorporated into performance-to-cost [139] and weighted risk [140,141] methods for tradeoff analysis.

Various studies that assume Eq. (6) for estimating the $COE$ assume that O&M costs can be broken down into fixed and variable components. For addressing corrosion of an SCWR concept, that assumption is an option as indicated by the three common methods of estimating O&M costs: the fraction method [142–144], the plant size scaling method [143,145–150], and the labor scaling method [122,144]—each method is classified by the primary adjustment factor used to estimate O&M costs.

Once an O&M costing method is selected, certain aspects of corrosion control relating to fuel cladding can be estimated, such as water chemistry and ultrasonic fuel cleaning [7,57,151,152]. Other corrosion control aspects, such as the material selection for the fuel cladding and coating protection fall under fuel fabrication costs in the fuel cost category, while the purification system, and the material selection of structures and components fall under total plant costs. The purification system would also have an operating cost component.

Equation (11) sums up all the chemical species unit costs (⁠$ucwc, m$⁠) and quantities (⁠$qwc,m$⁠) required for each mth species used for corrosion control, and $φ$ is not needed. Similar strategies may be used for fuel cladding and coating as part of fuel fabrication costs. The selection of the costing strategy will likely depend on the information available (including, e.g., water chemistry and design parameters, quantity required, and unit costs).

Consists of a physical quantity (⁠$qpur$⁠, which represents either the resin or the installed equipment), and a unit cost (⁠$ucpur$⁠) [65,159,160,165–167]—see these and previous references on water treatment/purification/ion exchange/decontamination for more details. For a new nuclear build, the equipment will be part of the total nuclear plant cost. If a purification system is being replaced, then the equipment will likely be capitalized (depends on the accounting rules—see, e.g., Rothwell [122,168] on accounting for capital additions for the American case).

To meet an ECC-SMART project requirement of comparing an SCW-SMR concept to an alternative SMR based on PWR technology, the present worth of the $COE$ can be used to compare costs over the lifetime of each technology. More specifically, the present worth of revenue requirements can be used as follows.

Revenue requirements are “the revenues which must be obtained in order to cover all expenses incurred, associated with and including the company's minimum acceptable return (MAR) on investors' committed capital, no more and no less” [169]. The MAR (i.e., the cost of capital—a percent return) is the distinguishing characteristic of the revenue requirements approach to determine the present worth of the $COE$ [117,169].

where $n$ is the service life [169]. Second, the rate of return in the $fcr$ is also the MAR. In both parts the MAR may represent either the rate of return for the pool of investor's committed capital or the rate of return for equity only depending on the purpose of the economic study [169].

Section 2 and Secs. 3.1–3.3 identified the corrosion of fuel cladding in an SCWR concept, the main corrosion cost categories for power generation, the incorporation of the corrosion failure rate, and the annual O&M cost estimating methods. These discussions need to be combined with two objectives that are relevant to nuclear power plant owners:

1. minimize costs, and

2. minimize corrosion

   to provide good availability while maintaining safety [170,171]. Examining how these elements are to be combined will require understanding the role of managerial, engineering, and regulatory decisions in achieving the aforementioned objectives [2,73,172,173]. These decisions may be studied under the perspective of institutional economics [174–176], since this perspective includes engineering economics decision-making under managerial and bargaining transactions. An institutional perspective of corrosion engineering economics may, thus, be used to frame the tradeoffs mentioned in Sec. 3.2.

In a feasibility study considering profit, the present study enables estimating the revenue requirements, but revenues will require estimating the expected output and the expected price of electrical power. These two elements will also depend on several factors, such as energy policies, and the state of an economy (Sec. 3.2). In addition, utilities strategically consider the age of existing fleet of power plants in earning power generation revenue in different market segments. For example, utilities may “depend to an increasing extent on aging power plants for both base load and spinning reserve” [177]—Olds refers to aging as units over 30 years old.

Corrosion assessments in an industry are done under a policy, strategy or planning (includes designing) context [46,178–181], e.g., pro-active planning. Nuclear corrosion studies that discuss pro-activity [2,73,172,173] indicate that “[h]istorical plant behavior is most informative in reminding us what we did not anticipate correctly, and that despite major efforts to resolve materials degradation issues, existing and new categories of degradation have continued to occur” [173].

As the SCWR concept is an evolution of water-cooled reactors (WCRs), it is expected to have corrosion-related challenges that affect existing WCRs [7,74]. Similar corrosion-related issues are, for instance, IGSCC, corrosion product activation and deposition, and fuel cladding corrosion [7]. Based on the past experience of PWRs and BWRs, costs due to corrosion product activation and deposition ranks first, and fuel cladding corrosion ranks sixth amongst the ten largest corrosion cost problems in the electric power industry (see Table 2).

When corrosion costs in power generation are subdivided in terms of the main cost categories, the bulk of corrosion costs for the nuclear sector are associated with O&M costs (Table 3). Amongst these corrosion costs, the main topics discussed in the present study, fuel cladding, and corrosion product rank in the top ten for LWRs (Table 4).

More recently, EdF shut down twelve reactors in 2022 to inspect for stress corrosion cracking, reducing its nuclear power output [183]. This reactor outage is estimated to reduce EdF's earnings by approximately €18.5 billion [183]. In addition, the cost of capitalized work relating to the stress corrosion phenomenon amounts to €78 million as of 30 June 2022 [184].

As experienced in LWRs, corrosion and SCC increase with temperature, so it is expected that temperature will aggravate both degradation modes [74], though not the only factor [7,73–75,173]. It is also anticipated that the plant age and operational desire for longer fuel cycles will pose challenges to minimizing corrosion [73,74,173].

To address corrosion challenges, there are successful examples of controlling corrosion in power plants, such as: choosing the optimum materials of construction; applying coatings to materials that are otherwise susceptible to corrosion; removing aggressive species from the chemistry environment; changing the corrosion potential; modifying the distribution of residual stresses near welds; and using on-line corrosion monitoring [99]. Collectively, these corrosion control options saved the electric power utility industry approximately US $570 million [99].

While experience is important for anticipating future challenges, the pro-active view warns to avoid “too much confidence and optimism” [173]. Laboratory studies can indicate many corrosion modes or mechanisms, yet operating conditions often throw up surprises in terms of unexpected corrosion events and this is especially true of new reactor designs [2,73,172,173]. The pro-active view also encourages having “sustained expertise, laboratory capability and plant inspection so that future degradation can be identified and managed” [173].

A CNL project and the ECC-SMART project aim to understand the corrosion behavior of the most promising candidate materials for a future SCW-SMR. To support this aim and the requirements of these projects, a power generation cost model that incorporates the impact of corrosion, in particular, the impact of fuel cladding corrosion was developed. Developing the power generation cost model required overcoming several limitations, such as no corrosion economics studies for any SCWR concept, the SCW-SMR concept remains under development, and past nuclear economics studies on corrosion are fragmented. The broader aim of developing a power generation cost model that incorporates corrosion was to provide a framework for understanding the kind of information available and needed for taking economical preventive corrosion measures for the current generation of WCRs and advanced reactors, such as the SCWR. Elements of the model that could be further developed were, therefore, discussed to enable future use as the SCW-SMR concept evolves, and current WCRs continue operating.

Overcoming the limitations required understanding:

• fuel cladding corrosion phenomena in an SCWR concept

• the revenue requirement method for nuclear power generation

To understand the relevance of fuel cladding corrosion phenomena for economics, the in-core environment, fuel cladding, corrosion product transport, and modeling fuel failure were discussed to identify the key control parameters that enable performance and corrosion control, with temperature as the primary control variable, and a model for fuel reliability to incorporate the key control factors.

This understanding was incorporated in a power generation cost model that applied the revenue requirements approach to LCC. To make the LCC approach useful for corrosion in an SCWR concept required specifying the LCC in terms of the main corrosion cost categories and a reliability factor used in assessing power generation costs, the costing of chemical species for controlling corrosion, and the present worth of revenue requirements. In addition, a pro-active planning perspective provided context for the revenue requirements approach to understand more broadly existing and new corrosion issues expected for a SCW-SMR. The main cost categories for power generation (with emphasis on nuclear) that are affected by corrosion were used to identify the major types of corrosion costs for LWRs. The costs associated with fuel cladding, and corrosion product related issues are amongst the top three of the ten largest issues for LWRs. A lesson of the pro-active approach is that over time not only do existing issues continue (e.g., SCC-related issues), but new issues are also expected to arise for both existing and new reactors. The latter type of reactor has the additional challenge of addressing the implications of a new design. A way to mitigate these surprises is to conduct research and development on an ongoing basis.

The author is grateful for the support of B. Haley, T. Cameron, A. Turcotte, D. Hoover, M. Sayler, J. Tilson, S. Luk, E. Hachey, D. Nauss, P. Sanongboon, K. Khumsa-Ang, M. Edwards, A. Nava-Dominguez, T. Wilson, A. V. Colton, A. Siddiqui, and G. Strati. The author is also grateful to the editors and the anonymous reviewers for their reviews, comments, and suggestions. This study was funded by Atomic Energy of Canada Limited, under the auspices of the Federal Nuclear Science and Technology Program. The research was conducted at the Canadian Nuclear Laboratories.

• Atomic Energy of Canada Limited (Funder ID: 10.13039/501100004953).

Notation

b_e =

coefficient defining the effect of corrosion on fuel element failure in the data series observed

cf =

capacity factor

C_F =

annual fuel cost

COE =

unit cost of generating electricity

C_OM =

annual O&M cost

C_pur =

purification equipment or annual resin purchased

C_LCC =

estimate of LCC

C_tpc =

total plant cost

C_wccor =

water chemistry costs

E =

expected annual electric power to be produced, MWh

ΔE =

replaced energy, MWh

E_a =

activation energy of the rate controlling phenomenon (assuming that it is thermally activated)

fcr =

annual fixed charge rate

G =

number of parameters considered (corrosion included)

hr =

hours

H =

maximum annual operating hours

k =

corrosion rate constant

k_B =

Boltzmann constant, 1.380662$×$10^-23 J/K

LF =

fraction of rated capacity

MAR =

minimum acceptable return

n₁ =

actual number of failures for LWRs

n₀ =

nominal number of failures for LWRs

P =

net maximum power plant output rating (MW_el), i.e., output capacity

P_c =

critical pressure

ΔP =

replaced capacity (MW_el)

q_pur =

a physical quantity, which represents either the resin or the installed purification equipment

q_wc,m =

quantities consumed required for each mth species used for corrosion control

R =

reliability factor

t =

oxidation time

T =

oxidation temperature

T_c =

critical temperature

sh =

scheduled outage hours

S_rcap =

grid system replacement capacity cost

S_rene =

grid system replacement energy cost

uc_pur =

unit cost of either the resin or the installed purification equipment

uc_wc,m =

chemical species unit costs

V_e =

variation of the corrosion parameter during a year of observation

WG =

weight gain (mg/dm²)

WL =

weight loss (mg/dm²)

Greek Symbols

φ =

is the fraction of operation and maintenance cost associated with water chemistry (or consumables) from a reference reactor

Abbreviations

BWRs =

boiling water reactors

CF =

corrosion fatigue

CNL =

Canadian Nuclear Laboratories

DO =

dissolved oxygen

EAC =

environmentally assisted cracking

ECC-SMART =

European Canadian Chinese development of small modular reactor technology

EdF =

Électricité de France

EPRI =

Electric Power Research Institute

FERC =

Federal Energy Regulatory Commission

FPPs =

fossil power plants

GIF =

generation IV International forum

M&C PMB =

materials and chemistry project management board

MW_el =

megawatt of electricity

MWh =

megawatt-hours of electricity

HTS =

heat transport system

HWC =

hydrogen water chemistry

IGSCC =

intergranular stress corrosion cracking

LCC =

life cycle cost or costing

LWR =

light water reactor

NMCA =

noble metal chemical addition

O&M, OM =

operation and maintenance

PHWR =

pressurized heavy water reactor

PWRs =

pressurized water reactors

R&D =

research and development

RBMK =

reaktor bolshoy moshchnosty kanalny

SC =

supercritical

SCC =

stress corrosion cracking

SCFPP =

supercritical fossil power plants

SCW =

supercritical water

SCWR =

supercritical water-cooled reactor

SHS =

superheated steam

SMR =

small modular reactor

USC =

ultra-supercritical

WCRs =

water-cooled reactors

WWER =

water-cooled and water-moderated energy reactor