Modular fusion power plant

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Original Research Article

Modular fusion power plant V.A. Chuyanov ∗ , M.P. Gryaznevich Tokamak Energy Ltd, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK

h i g h l i g h t s • A concept of the fusion power development based on a modular fusion plant is analyzed. • Specific requirements to fusion modules constituting such a plant are identified. • Economic characteristics of the plant based on high field spherical tokamaks and HT superconductivity are presented.

a r t i c l e

i n f o

Article history: Received 7 October 2016 Received in revised form 21 May 2017 Accepted 17 July 2017 Available online xxx Keywords: Spherical tokamak Fusion reactor Modular approach

a b s t r a c t The mainstream approach to the fusion power, following the JET/JT60-SA − ITER − DEMO route, assumes increase in the size and development of new physics and technologies at each step. In this paper, we present an alternative route, based on the modular approach, when the economically feasible fusion power plant will consist of several modules, where the physics and technology are developed for a single compact module, making the development path cheaper and quicker. The economics is based on that for a modular-based plant, where many auxiliaries are shared between modules, and a regular necessary maintenance is set in a module-to-module way, providing high availability of the power plant. An expression for the cost of the generation of electricity (CoE) and identification of cost factors important for the modular approach are presented. The minimal size of a single module is determined by the physics and technologies developed up to date. The presented cost example for a modular power plant is based on the selected set of parameters for a single module and the necessary number of the modules in an economically feasible power plant. The cost of electricity (CoE), dependence of the CoE on the neutron wall loading and the effect of a module reservation are examined in detail. While the final cost of the new approach may be not significantly cheaper than for the mainstream approach, the development path has many advantages and can be affordable and much faster, supported by the combination of private and public funding. © 2017 Elsevier B.V. All rights reserved.

1. Introduction From 2014, Tokamak Energy Ltd. is developing a modular design of a fusion power plant. The term “modular system” or “modular design” usually means that a product system can be decomposed into several components that may be mixed and matched in a variety of configurations [1,2]. In this article, we will use it in a narrower sense: a modular design is a design approach when several similar units (modules) of lower capacity fulfill together a function of a single unit of higher capacity. This design approach is widely used in different areas from windmills to space stations and now considered for fission power plants

∗ Corresponding author at: 216 Montée des cigales, Cartier Grand Hubac, 83560, Artigues, Var, France. E-mail address: [email protected] (V.A. Chuyanov).

[3] It permits to overcome technological and fundamental difficulties to build a bigger device, to save time and resources needed for the development, to improve reliability by the cheap reservation, to use economy of mass production and achieve some other benefits (like the load following) specific for each area of application. At the same time, it has some drawbacks and limitations specific for each application. Application of the modular approach to tokamaks looks counterintuitive. For many years starting from 1968 when the diffusive nature of the energy losses has been identified, bigger and bigger tokamaks have been built and this root of the development has been seen as a direct road to a fusion power plant. However, now the problems associated with this approach are becoming more and more clear [4]. High cost of a first prototype of a fusion power plant and a long duration of further technological development have been identified

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[4] as one of the main obstacles on the traditional mainstream tokamak path to the fusion power. Development of any new technology with big and expensive prototypes is inevitably slow and not efficient. A new concept of an exploration of the fusion power, based on a modular fusion plant consisting of several compact high field lowcost Spherical Tokamak (ST) modules, have been proposed in [4]. Economy of a mass production has been proposed as a substitute for the “economy of scale” expected for the traditional approach but difficult to be realized due to high cost and long timescale of development. The new proposed path can attract commercial cost sharing with reduced financial risks and lead to modular power plants that can realize “economy of mass production” during construction even of the first power plant prototype. At the same time, a system code analysis [5] has shown that for steady state tokamaks operating at fixed fractions of the density and beta limits, the fusion gain, Qfus , (where Qfus = Pfus /Paux , Pfus is the power produced by fusion reactions and Paux is total power needed to sustain reactor operations, i.e. power needed to sustain the plasma current) depends mainly on the absolute level of the fusion power and only weakly on the device size. Small size (with the major radius R ∼ 1.35 m) low fusion power devices (Pfus < 300 MW) with Qfus = 5 may be realized with very modest improvement of the ITER’s type (IPBy,2) [6] confinement or the beta independent scaling based on the ITER database [7]. A step in this direction − smaller size compared, i.e., with ITER but still using ITER-type geometry, higher power density, the use of high temperature superconducting (HTS) for magnets and so higher toroidal field − has been done in the ARC design proposed by the MIT team [8]. The results of studies on the first generation of spherical tokamaks [9–11] show that the scaling of the energy confinement time (␶E ) with the toroidal magnetic field and plasma current in an ST is significantly different from that the ITER confinement scaling (ITER98 IPB(y,2)) [6], having much stronger dependence on the magnetic field and weaker dependence on the plasma current. It will be shown that with an increase in the toroidal field up to 3–4 T, the energy confinement indicated by [9–11] is sufficient for realization of the modular approach to the Fusion power with the plasma current in a range of 5–8 MA. Of course, additional experiments in a proper domain of parameters are needed to confirm this. It may be also noted that lower plasma currents, Ip ∼ 3–4 MA, are also adequate for alpha-particle confinement in compact low aspect ratio tokamaks [12]. Further improvement of the confinement due to expected suppression of electron turbulence by high magnetic field in low aspect ratio tokamaks is also expected [13,14]. Minimum values of the aspect ratio (A = R/a, where “a: is the minor radius of the plasma) and the major radius R of a compact low aspect ratio fusion module are constrained by the thickness of the neutron radiation shield necessary to protect the superconducting toroidal magnet. The use of high temperature superconductors (HTS) in tokamak magnets permits significant reduction of the shielding requirements compared to that for the low temperature superconductor magnets [15]. A tungsten carbide (WC) shielding of ∼25–35 cm may be sufficient to decrease the neutron heating of the central post of HTS magnet with operational temperature >20 K to a level acceptable in a low aspect ratio fusion power device [16]. The shielding thickness around other parts of the magnet, as well as for poloidal field (PF) coils is not geometrically constrained and may and must be much thicker. The lifelong protection of the HTS in the central post from the neutron damage may require thicker shielding ∼40–60 cm [17] depending on the economically acceptable time of life of the central post which is a small and, in principle, a changeable part of the toroidal magnet. However, today there is not enough data on the irradiation properties of the industrially produced HTS tapes to make a conclusion. Such tests however are on-going [18].

Although the proposed line of the development is well supported by the current physics and technologies, for the realization of the new path some further advances in physics and engineering are necessary. Among other physics challenges, that are common to the whole development of the Fusion power, for the path based on compact STs it is necessary to evaluate whether such compact fusion power module can provide a sufficient energy gain. It is necessary to understand what are other requirements specific for the ST design, like high fraction of the bootstrap current, and are these requirements compatible with good energy confinement. All these new elements of burning plasma physics can only be checked in burning plasma devices and this is very important part of our development program. To attract an interest of researches and investors, the economic soundness of the modular approach must be demonstrated. The modular plant has not only advantages but also some disadvantages from the economic point of view: smaller plasma volume to plasma surface ratio, higher total machine mass to fusion power ratio, high total plasma current to be driven in many modules and so on. The goal of this paper is to address specific features of the modular plant economics, define possible ways to overcome these difficulties and present an example of a modular power plant based on a high field compact low aspect ratio tokamak. Many features of the burning plasma with a high bootstrap current are still unknown. Taking this into account we shall try to consider the most favorable conditions and find the low limit of the Cost of Electricity (CoE) produced by such a fusion plant to define a potential of the proposal and to guide the research in the directions most interesting from the point of view of economics. The analysis of the main cost factors is presented in Section 2. A general expression for CoE and identification of cost factors important for the modular approach are given. Specific features of a fusion modular plant are discussed in detail: the necessity of a high bootstrap and high beta operation; the synergy of a necessary fusion power and the wall loading limitations; the energy balance analysis. Section 3 shows a cost example for a modular power plant. The costing rules are determined, and parameters of a fusion module selected for costing are described. The results of the cost estimate are presented. The cost of electricity and the dependence of the CoE on the neutron wall loading and costing rules are discussed in Section 4. Section 5 compares results of this study with similar calculations done for classical fusion power plants. Section 6 presents conclusions. 2. Cost factors 2.1. A general expression for CoE and identification of cost factors important for the modular approach In the conventional approach to a fusion power plant design [19], the first step is to define a device with a reasonable confinement and after this to analyze its economics. In this paper, we will do this in an opposite order. We shall start with a definition of the economic requirements and consider the achievement of these requirements at first based on plasma equilibrium and stability constraints, which are known much better than the energy confinement physics of the burning plasmas, and only after that to define what confinement is needed and how it can be achieved. Cost of electricity (CoE) generated by any power plant may be calculated as CoE =  (Annual power plant expenses) / (8760 ∗ Penet ∗ PF)

(1)

Here in the denominator the value 8760 is the number of hours in one year, Penet − net electrical power produced by the plant and PF is a plant factor or plant availability equal to the ratio of the

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plant operation time (at the full power) per year, to 8760 h. In the nominator, we have the sum of all annual expenses of the power plant, consisting of the annual capital cost charge (CAC), the annual operation and maintenance cost (CO&M), the annual scheduled component replacement cost (CSRC), the annualized decontamination and decommissioning cost D&DC and the annual fuel cost (FC):  = CAC + CO&M + CSRC + D&DC + FC

(2)

The fuel cost is one of the most important cost component for thermal power stations. For fusion power stations consuming ∼90 kg of deuterium and ∼300 kg of Li6 per 1000 MW of the fusion power per year, the cost of the burned fuel (∼0.6 M$) is negligible. It must be noted that tritium is not strictly speaking, an externally supplied fuel for a fusion plant. It is produced and consumed inside the plant in the process of operation. Of course, an initial amount of tritium is needed for a startup, which is measurably high. This is a common problem for any fusion power plant. For a power plant considered below the total necessary initial amount of tritium is 4 kg. If all the modules will be started at the same time the cost of this tritium 140 M$. (<4% of the direct capital cost of a power plant considered below). This cost must be added to the capital cost of the plant. The lion share (∼85% of ) of the annual expenses is the annual capital cost charge (CAC) which is determined by the total capital cost, and by the amount of money which must be borrowed to build the plant and the interest. In the simplest case, it may be written as



 



CAC = Total capital cost/Top ∗ 1 + 0.5 ∗ y ∗ 1 + Tconstr + Top



(3) Here ‘y’ is the interest rate, Tconstr − duration of the plant construction, Top − duration of the plant operation which will be assumed to be equal to the time of repayment of the borrowing. Depending on financial conditions, the ratio CAC*Top /(Total capital cost) may be ∼ 2 (for example, for y = 5.5%, Tcosntr = 5 years, Top = 30 years) In this article we will be specifically focused on the determination of the plant total capital cost. Other terms in the expression (2) are much less important than the first one and together are responsible for less than 15% of the cost of electricity. In addition, the cost of fuel, the cost of operation and maintenance and the cost of decommissioning do not depend strongly on the type of the fusion plant and are determined mainly by its power. Values calculated in other reactor studies are directly applicable to the compact modular approach. The annual scheduled component replacement cost (CSRC) is specific for a modular plant and must be re-calculated as it will be different to one in the conventional approach. The net electrical power Pnet in (1) is Pnet = Pgross − Pinternal consumption = Pgross ∗ (1 − ␰)

(4)

Here Pgross is total electric power produced by the station, P internal consumptin is the electric power consumed for internal plant needs. The ratio ␰ ␰ = Pinternal consumption /Pgross

(5)

is a coefficient of the power recirculation. Qeng , a parameter often used to characterize reactor performance, is equal to 1/␰. Power recirculation is necessary for any power plant, for example to pump the cooling water, and for nuclear power plants it is usually <10%. For fusion power plants there are some other needs to use the electric power − plasma heating, plasma current drive, fuel processing,

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cryogenics. High values of ␰ may strongly increase the CoE. Decrease of ␰ is one of the main goals of the modular plant optimization. High availability is also very important to decrease the cost of the electricity. For fusion power plants, the availability may be strongly affected by scheduled periodic stops needed to change the first wall and parts of the blanket after neutron irradiation reaches a limiting fluency. Here the modular approach has significant advantages which may and must be used and will be examined in detail. The cost factors − capital cost, availability and power recirculation are important for any fusion power plant but economics of a modular plant has some specific features. For example, a small fusion power module will inevitably have smaller plasma volume to plasma surface ratio and lower ratio of fusion power to weight of the device. It increases the importance of achievement of a high beta and reasonably high neutron wall loading operation regime. At the same time the modular structure permits new solutions for achievement of high plant availability such as reservation of a module and change of the first wall/blanket structure of each module fully off line without negative impact on the plant availability as the rest of modules will be operational. The requirements for the remote handling and hot cell are also reduced as they will be used for one module at a time and so the hot cell will be used continuously, unlike in the case of a conventional high power plant. 2.2. Specific features of an ST modular power plant 2.2.1. Necessity for the high bootstrap current and high beta operation The physics of spherical tokamaks imposes some additional requirements and limitations on selection of physics and engineering parameters of a modular power plant. For any fusion power plant reduction of the power recirculation is an important requirement. To reduce heating power to a reasonable level, the fusion ignition (or at least operation very close to it with fusion Qfus 20) must be considered as a necessity. Taking into account that the difference in n*T*␶E needed for ignition and for Qfus = 20 is only ∼20% and the value is comparable with precision of any scaling law used for a system modeling, we shall consider ignition as a necessary condition for a fusion module of a power plant, at least at the first stage of the analysis. The need for a continuous plasma current drive (CD) during steady state operation is another reason for a power recirculation. For conceptual designs of fusion power plants, the usual limit for the recirculated power is ∼20% [20]. Even with optimistic assumptions on the efficiency of a non-inductive current drive methods (inductive methods can’t provide steady-state plasma current), such goal may be achieved only if less than 40% of the plasma current will be driven non-inductively [21]. The way to resolve the problem in a compact ST is to use the bootstrap current, generated by a high pressure plasma itself [22]. With the energy conversion efficiencies available today, more than 65–70% of the plasma current must be continuously self-driven by bootstrap. For spherical tokamaks which have limited means to induce and drive plasma current inductively, due to the lack of space in the central post for a central solenoid, conventionally used in high aspect ratio tokamaks, the external non-solenoid current drive is permissible during the start-up phase. However, external current drive cannot be used in a compact ST reactor continuously at high power because of economic considerations. The major radius of the plasma in a compact spherical tokamak is smaller, but the plasma density may be higher than in conventional reactors, and the current drive power requirement watt/ampere is not very different from a classical tokamak-reactor. The fusion power of a module is ∼10 times less than the power of the whole power plant and the electrical power available for the current drive in a module is also ∼10 times smaller. But the plasma current in a module (for example the plasma cur-

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rent value needed to confine alpha-particles) only weakly depends on the module power. Thus, the total summarized current to be driven in all modules and the total power needed to do it in a modular power plant would be several times higher than in a classical case (in a single device of the same fusion power) without a very high bootstrap. To achieve recirculation target mentioned above in a spherical tokamak fusion power module, the bootstrap fraction must be not ∼60–70%, but > 90% of the plasma current. It is prudent to try and optimize the design to achieve 100% bootstrap, allowing at the same time some (<10%) external current drive for the profile alignment and control purposes. Optimization of a classical fusion tokamak is often linked with a goal to increase the confinement efficiency of toroidal magnetic field, which can be characterized by the ␤tor ␤tor = Plasmapressure/Toroidalmagneticfieldpressure

(6)

High ␤tor decreases the cost of the toroidal field magnet − one of the most expensive parts of a tokamak fusion power plant. Optimization of a spherical tokamak modular power plant must at the same time target at an increase of the bootstrap fraction to decrease the power recirculation which affects the cost of all reactor systems. These two targets of optimization are driving design of a modular reactor based on compact ST modules in somewhat different to a traditional direction. To achieve highest ␤tor , a tokamak must operate at highest normalized plasma current IN = Ipl /(a*B) permitted by the plasma stability. To achieve high bootstrap fraction, an operation at a low normalized plasma current is desirable [21]. Let’s examine this dependence on normalized plasma current in detail. To achieve a significant improvement of power recirculation in a modular fusion plant the ratio of the bootstrap current to the plasma current “fbs ” must be close to unity: fbs = Ibootstrap /Iplasma ≥ 0.9 − 1.0

(7)

The bootstrap fraction depends on the plasma aspect ratio A and poloidal beta (␤pol ) − the ratio of the plasma pressure to the pressure of the poloidal magnetic field. Following [23–25] we may conservatively define the bootstrap fraction as fbs = 0.5 ∗ (1/A0.5 ) ∗ ␤pol

(8)

The latest works (such as [26]) give more optimistic expressions for the fbs . However, in this article the most pessimistic expression (8) will be used. It means that to achieve fbs ∼ 1 low aspect ratio (A = 1.7) fusion power modules must operate at ␤pol ∼ 2.6. For a tokamak with an elliptical cross section and an elongation K, ␤pol may be expressed approximately through ␤tor and normalized plasma current [27] ␤pol = 12.5 ∗ (1 + K2 ) ∗ ␤tor /IN2

(9)

From Eqs. (8)–(9) it is clear that achievement of a high bootstrap fraction fbs is connected with achievement of high ␤tor . So, both for minimization of the cost of the TF magnet and for minimization of the electric power recirculation, it is in our economic interests to operate at as high ␤tor as possible. The maximum achievable ␤tor max is defined by the ideal ballooning ␤ limit and for nearly 100% bootstrap current driven tokamak equilibria [27] may be conservatively (with ∼30% margin) written as [5] ␤tor

max





= 0.01 ∗ 9/A ∗ IN

(10)

To find optimum conditions of operation let us introduce a coefficient f␤ showing how close we are to the maximum permissible ␤tor f␤ = ␤tor /␤tor

max

≤1

To achieve fbs ∼ 1, the ratio fbs /f␤ must be also close to 1.

Fig. 1. Maximum value IN* of normalized plasma current IN permitting to achieve fbs = 1 as function of aspect ratio A for different elongations K. (Here and in all this article the triangularity ␦ = 0.5).

(11)

Fig. 2. Maximum value of toroidal beta achievable for IN = IN*. (The triangularity ␦ = 0.5).

The condition



 



fbs /f␤ = 0.56 ∗ 1 + K2 ∗ 1/A1.5 /IN ≥ 1 or





IN < IN∗ = 0.56 ∗ 1 + K2 /A1.5

(12)

(13)

is a necessary (but not sufficient) condition to achieve 100% bootstrap current. It means that to achieve ∼100% bootstrap fraction of the plasma current, a tokamak must operate in a narrow band of IN slightly below IN*. The condition (13) is independent on plasma parameters and a size of a tokamak. The dependence of IN* on aspect ratio A and elongation K are shown at Fig. 1. For K = 3 these values of the normalized current correspond to the safety factor q95% ∼10–11 [28] and are much smaller than normalized currents limited by the stability (q95% ∼3). Experimentally [29–31], IN ∼8 have been achieved for A∼ 1.5. Maximum values of ␤tor (Eq. (10)) that correspond to IN = IN* are shown in Fig. 2. The requirement to achieve 100% bootstrap leads to the maximum beta toroidal much below the limit determined by the stability (q95% ∼ 3). Both values: high beta toroidal and high bootstrap fraction are important for good economy of a fusion power plant. To find what is an optimum fraction of bootstrap current one needs to perform the full cost estimate. Such an analysis today would be premature. We will pursue a more limited goal: to consider an economy of a fusion power plant with 100% bootstrap and to find that the limited beta in this case will not be a factor limiting fusion economy at an unacceptably low level.

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Fig. 3. Major radius of the plasma R as function of neutron wall loading and fusion power for a module with A = 1.7, K = 3, ␦ = 0.5.

Indeed, loss of beta toroidal because of the fbs = 1 requirement is very significant for high aspect ratio reactors. For low aspect ratios, ␤tor is still sufficient to get practically any reasonable neutron wall loading without too high requirements on the magnetic field. Low aspect ratio highly elongated tokamaks are the natural candidates for an economical study of importance of 100% bootstrap current drive. 2.2.2. Fusion power and the wall loading To get significant cost reduction from the mass production during construction of the first fusion modular power plant the number of modules must be sufficiently high, say ∼ 10. (The cost reduction for the 10th module is ∼2). If the net electrical power of the plant must be not too high, say ∼ 1 GW (e), we must request the modules to have a fusion power Pfus ≤ 200 MW. Another fusion parameter, important for economics is the mean neutron wall loading NWL, NWL = 0.8 ∗ Pfus /Sw

(14)

Sw is the first wall surface, which can be approximated by expression [5]: Sw = (4 ∗ ␲2 ∗A’ ∗ K0.65 − 4 ∗ K ∗ ␦) ∗ (a + g)2

(15)

Here a − the plasma minor radius, g − the gap between plasma and the first wall, A’ = R/(a + g) − aspect ratio of the vacuum chamber, ␦ is triangularity of plasma cross section. Correspondingly the plasma volume Vpl is [5]: Vpl = (2 ∗ ␲2 ∗K ∗ (A−␦) + 16 ∗ ␲ ∗ K ∗ ␦/3) ∗ (R/A)3

(16)

For simplicity, we can neglect for a while the difference between the plasma and the wall surface and assume g = 0, and A’ = A because g/a « 1. Selection of the ratio of the module fusion power to the neutron wall loading (Pfus /NWL) immediately defines the plasma surface, plasma volume (Vpl ) and plasma dimensions (R, a) for given shaping parameters A, K, ␦. Plasma major radius as a function of the neutron wall loading is shown in Fig. 3 for different fusion powers of a single module and fixed shaping parameters (A = 1.7, K = 3, ␦ = 0.5). Fusion power can be calculated for a given plasma density and temperature profiles taking into account dilution of the plasma by impurities. For interesting for us conditions, these profiles are not known today. But the total fusion power is not very sensitive to details of the space distributions. Here it was calculated for a flat density profile (n/n0 = (1 + (r/a) 2 ) 0.5 ) and a peaked temperature profile (T/T0 = (1 + (r/a) 2 ) 1.5 [19]. Deuterium/tritium plasma is always diluted by impurities. High Z impurities must and can be minimized by high plasma density operation, lithium coating and so on. Following our approach

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Fig. 4. Parameters of a 200 MW fusion module with neutron wall loading = 1.35 MW/m2 , aspect ratio A = 1.7, triangularity ␦ = 0.5 and gap g = 0.05 m as functions of elongation. The minimum plasma current to achieve the needed density − called here Imin Greenwald is also shown for plasma temperature Tmean = 10keV. The toroidal field B and the plasma current Ipl are values determined by the balance of pressure (Eqs. (18) and (13)). For elongations K < 1.9, when Ipl < Imin Greenwald and plasma density must be higher than the Greenwald limit, the modules with selected parameters and Tmean = 10 keV cannot be realized.

to assume the most favorable conditions we will consider only unavoidable impurities produced by fusion reactions − alpha particles and protium (by DD reactions). The dilution of DT fuel by helium and protium leads to loss of fusion power and loss of revenue. The dilution may be decreased by stronger pumping which also costs money. We have estimated that the minimum CoE is achieved when the concentration of helium is <1%. However, to reach such small helium concentrations we need unrealistically high pumping speeds >300 m3 for a small module. For further consideration we selected maximum helium concentration ∼4% which leads to tolerable increase of CoE (∼12% in presence of 2% of protium) and is, probably, at the technical limit of pumping. Concentration of protium does not depend on pumping speed and is determined by the isotope separation capability. For the concentration of protium we assume 2% because it gives only a small additional to helium dilution and at the same time requires only a limited isotope separation capability assumed below for the cost estimate of a fueling system. Plasma may be diluted also by high Z metal impurities which mainly affect the value of effective charge Zeff . If values of Zeff 1.5–2 will be achieved and this is an important requirement, the dilution by high Z impurities may be neglected. Plasma fusion reactivity and pressure will be corrected in accordance with these assumptions. Very small increase of Zeff (Zeff = 1.077) will be not taken in account in calculations of Bremsstrahlung radiation. In the range of mean plasma temperatures (Tmean = /) between 8 and 19 keV, the dependence of the fusion power per unit of the plasma volume (Pfus /Vpl ) on the mean temperature may be approximated with an error less than <10% as Pfus /Vpl ∼(n ∗ T)2 ∼(␤tor ∗B2 )2

(17)

The exact selection of the plasma density and temperature is not important from the point of view of fusion power production, but may be optimized later for better confinement or more effective current drive. For selected profiles and dilution Pfus /Vpl (MW/m3 ) = 0.53 ∗ (␤tor ∗B2 )2

(18)

Using formulas (7–18) we can calculate fusion module parameters for selected fusion power, neutron wall loading, aspect ratio, elongation, triangularity and gap for IN = IN* and therefore for fbs /f␤ = 1. The main parameters of a 200 MW fusion module with

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neutron wall loading = 1.35 MW/m2 , aspect ratio A = 1.7, triangularity ␦ = 0.5 and gap g = 0.05 m are shown in Fig. 4 as functions of elongation. Increase of the elongation dramatically increases ␤tor and decreases the required magnetic field B because the plasma pressure is fixed by the selection of the fusion power and the neutron wall loading. The required plasma current Ipl increases with the elongation. For elongations below 1.5 the plasma current is too small for confinement of alpha-particles [12]. For K < 1.9 the plasma current is too small for achievement of the needed plasma density limited by the Greenwald limit [32]. The minimum current needed to achieve the needed density is also shown in Fig. 4 for operations at Tmean = 10 keV. The module parameters do not depend on plasma temperature in the range of 8 keV < Tmean < 19 keV and the Greenwald limit may be overcoming by operations at higher plasma temperature (Tmean ∼14 keV). It is obvious that the highest elongation achievable without significant increase of cost for its control is preferable. The probable range of elongations for a fusion power module is 2.5 < K < 3.5. The minimum fusion power of a module with 100% bootstrap current and the highest achievable toroidal beta (fbs = 1, f␤ = 1) is determined by technical limitations: by the thickness  of the radiation shield, protecting HTS in the central post, by the maximum current density J in the HTS cable and by permissible maximum mechanical stresses ␴ in the steel part of the central post. The deformations in the HTS may be not compatible with stresses in steel, but this limitation may be avoided by a proper design decoupling of deformation of steel and deformation of conductors, for exam-

Fig. 5. (Continued)

Fig. 5. Parameters of fusion modules with elongation K = 3, neutron wall loading 2 MW/m2 , beta toroidal = maximum toroidal beta (f␤ = 1), 100% bootstrap current (fbs = 1, IN = IN*) with shield thickness in the central post  = 0.4 m, current density in HTS cable J = 500 MA/m2 and mean tensile stress in steel structure ␴ = 600 MN/m2 as functions of aspect ratio. a) Major radius, b) Fusion power, c) Toroidal magnetic field at the major radius and its maximum value at the surface of HTS conductor, d) Plasma current and toroidal beta, e) Radius of toroidal magnet inside the central post and fraction of total cross section occupied by HTS.

ple using sliding contacts [33]. The maximum value of the toroidal field is determined usually not only by characteristics of HTS but by mechanical stresses in the steel as well. To determine the achievable fusion powers and wall loadings, let us consider a simplest model of the central post. The space near the center of a tokamak (in the central post) is distributed between the radiation shield, the steel mechanical structure (vertical tie-rod) and the superconducting cable. Radiation (neutron) shielding has 2 functions: to limit nuclear heating of the TF coil and to decrease the cooling power to an acceptable level, and to protect HT superconductor from the by neutrons during all lifetime of the magnet. Shielding with the thickness of the order of  = 25 cm may be sufficient to decrease the power needed for cooling of HT superconductor in the central post operating at 20 K to acceptable level [17] but much thicker shield will be needed to protect the conductor from the damage by neutrons during all lifetime of the magnet. Some internal part of this shielding may be kept at low temperature and be in fact a part of toroidal coil structure participating in taking radial mechanical loads. The space inside the shielding with the radius Rmagn R magn = R − R/A − 

(19)

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Fig. 6. Dependences of plasma major radius R, toroidal magnetic field at the major radius and fusion power on maximum permissible tensile stress ␴ in the central post steel structure for a module with elongation K = 3, aspect ratio A = 1.7, f␤ = 1, bootstrap fraction 100%, IN = IN*.

7

Fig. 7. Dependences of plasma major radius R and fusion power on maximum current density in the HTS cable for two values (300 and 600 MPa) of permissible tensile stress ␴ in the central post steel structure for a module with elongation K = 3, aspect ratio A = 1.7, f␤ = 1, bootstrap fraction 100%, IN = IN*.

is divided between a steel mechanical structure and a HTS cable carrying a current ITF ITF (MA) = 5 ∗ B ∗ R

(20)

Some part of the total cross section S inside the shield S = ␲ ∗ R magn 2

(21)

is occupied by the HTS. This part SHTS is determined by the achievable current density J SHTS = 5 ∗ B ∗ R/J

(22)

The rest of the cross section is occupied by the steel structure. Stresses in the structure depend on the design. To avoid detailed consideration of the design we shall characterize the module by mean tension stress created in the structure by vertical force Fvert = ␲ ∗ ((B ∗ R)2 /(2 ∗ ␮0 )) ∗ ln(R max /R magn )

(23)

Here Rmax = R + R/A+ 1.5 is the external radius of the TF coil (assuming the thickness of external blanket and shield and space for remote handling equal to 1.5 m). In bending free magnet configuration 50% of this force will be applied to the central post and will create a tensile stress ␴ = ␲∗((B ∗ R)2 /(4 ∗ ␮0 )) ∗ ln(R max /R magn )/(␲ ∗ R magn 2 − SHTS ) (24) The module must produce necessary values of the fusion power and the neutron wall loading. The value of the neutron wall loading cannot be too small if we want to achieve a reasonable cost of electricity. The wall loading can be calculated with Eqs. (15)–(18) and provides an additional equation needed to close the system written above. The results of calculations for different aspect ratios and neutron wall loading equal 2 MW/m2 are shown in Fig. 5. To get minimum fusion powers, values of , ␴ and J are selected at the probable limits of achievable performance. For aspect ratios A ∼1.7–1.8 these parameters are acceptable for a modular plant. For smaller aspect ratios, fusion power is too high. For higher aspect ratios, beta and plasma current are too low. Maximum magnetic field is in a reasonable range for the HTS. The magnet parameters J and ␴ may look too extreme. However, they may be significantly relaxed with only minor changes in parameters of a module. The dependences on J and ␴ for a module with aspect ratio A = 1.7 are shown at Fig. 6. Dependences on current density are weak if the permissible stress level is high (see Fig. 7 and Fig. 8). Tensile stresses ␴ > 400 MPa are desirable and probably achievable. It is important to note that maximum value of the toroidal magnetic field is determined not

Fig. 8. Maximum toroidal field as function of current density in HTS cable for two values of maximum permissible tensile stress for fusion modules with following parameters: A = 1.7, K = 3, ␦ = 0.5, neutron wall loading 2 MW/m2 , f␤ = 1.

by characteristics of HTS, but by mechanical characteristics of the central post, aspect ratio and the wall loading. These considerations up to now are based only on our knowledge of the plasma equilibrium and stability. They do not take in account the energy balance. It is assumed that heating is always sufficient to achieve the maximum beta limited by stability. 2.3. Energy balance The energy confinement in low aspect ratio high magnetic field high beta tokamaks has not been experimentally studied. We have only some hints from current low field ST experiments and need further experimental research. The most currently used for conventional aspect ratio tokamaks ITER scaling (IPB98y2) for energy confinement may be not applicable for high betas. We know that this scaling is in contradiction with direct measurements of dependences of confinement on beta in STs. For low beta STs, ITER scaling gives confinement time quantitively close to experimental results from existing spherical tokamaks, but observed on these tokamaks dependences on major parameters like magnetic field and plasma current are in contradiction with the ITER scaling. At present, probably the most appropriate confinement data to be used for prediction of performance in STs is based on results of MAST and NSTX studies [9,11]: ␶E MAST ∼ 0.252 ∗ Ipl 0.59 ∗B1.4 ∗G ∗ Pheating −0.73

(25)

(geometrical parameter G = R1.97 *K0.78 *(1/A)0.58 *Mi 0.19 (Mi = 2.5 mean ion mass) is selected in accordance with ITER scaling), however, the density dependence has not been evaluated on MAST,

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Fig. 9. Ratio of the energy confinement time needed to achieve ignition in fusion modules. described in Fig. 5 to the energy confinement time described by MAST confinement data. (K = 3, neutron wall loading = 2 MW/m2 ,  = 0.4m, J = 5000MA/m2 , ␴ = 600MPA).

or NTSX scaling: ␶E NTSX = 0.262 ∗ Ipl 0.57 ∗B1.08 ∗n0.44 ∗G ∗ Pheating −0.73

Fig. 10. Wall loading and fusion power for modules with ignition (when the needed energy confinement time ␶E = W/(0.2*Pfus ) is equal to confinement time calculated in accordance with (22): ␶E = ␶E MAST ) as functions of aspect ratio. The module technical parameters selected at maximum to get the modules of minimum size and power. (fbs = 1, f␤ = 1, K = 3, ␦ = 0.5, ␴ = 600MPA, J = 500MA/m2 ).

(26)

(the difference between the mean electron density and the chordaveraged electron density is ignored here.) Both scalings have similar dependences on plasma current and magnetic field. However, it must be noted, that NTSX experimental data can be equally well described by expression similar to the MAST scaling (25) (i.e. without including plasma density) but with weaker degradation of confinement with loss power which corresponds to direct experimental power scans [10]. ␶E

NTSX

∼ Ipl 0.56 ∗B0.94 ∗Pheating −0.4

(26a)

For ignition conditions, when at constant temperature Pheating ∼ n2 , expressions (26) and (26a) lead to different results. So, comparison of these two scalings must be done with care. In fact, a scaling combining all experimental results (START, MAST and NTSX) is possible [34]. In accordance with this scaling the magnetic field needed to get ignition is less than 10% higher than the field calculated with MAST scaling for the same parameters. The experimental results have been obtained at low magnetic field and there is no sufficient data-base to define better dependences of confinement on plasma size (G (R, K, A)). However, we can try to apply extrapolation of MAST results to modules described above. The requirement of ignition, in accordance with this extrapolation modifies but not very significantly the parameters of modules determined above without taking into account confinement consideration. For aspect ratio A > 1.7 the MAST data predicts ignition and achievement of needed plasma parameters, Fig. 9. For A <1.7 the confinement is not sufficient. However, the energy confinement time ␶E MAST has a strong dependence on the toroidal magnetic field and a weak dependence on the plasma current. This permits an easy adjustment of a module parameters to get the energy balance without destroying high bootstrap fraction. (For other parameters fixed, fbs does not depend on magnetic field.) There is nothing specific with A = 1.7, where the needed confinement is exactly equal to the scaling prediction. For a lower wall loading this would happen for a higher aspect ratio. At Figs. 10–12 parameters of minimum size modules with ignition in accordance with the MAST data (eq.25) [9] are shown as functions of aspect ratio. For aspect ratios <1.6 the fusion power is too high for modular plants (∼1.4 GW for A = 1.5). For aspect ratios A > 2 the neutron wall loading is too small (<0.5 MW/m2 ) for a reactor to be economically attractive. For A > 1.9 the plasma current is too low ( < 5 MA) for the alpha-particles confinement.

Fig. 11. Main parameters of the minimum size module with ignition in accordance with the scaling (25) (fbs = 1, f␤ = 1, K = 3, ␦ = 0.5, ␴ = 600MPA, J = 500MA/m2 ).

Fig. 12. Maximum toroidal field at conductors and toroidal beta for minimum size modules with ignition in accordance with the scaling (25) (fbs = 1, f␤ = 1, K = 3, ␦ = 0.5, ␴ = 600 MPA, J = 500 MA/m2 ).

Similar considerations apply if the NTSX scaling (26a) is used which is slightly less optimistic at selected parameters. The difference in confinement may be compensated by operation at higher field, higher density and lower temperature. The unit power will be slightly higher. We can make the following conclusion: The aspect ratio of modules for a modular plant must be in a narrow range 1.6 < A < 1.9 when ignition, 100% bootstrap, high beta (10–15%), a reasonably small fusion power (150–500 MW) and

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sufficiently high neutron wall loading (between 1 and 3 MW/m2 ) may be achieved at the same time. The maximum toroidal field at conductors is acceptable for HTS (∼20 T). The major radius R is ∼1.45–1.8 m and a small increase of it would permit to relax requirements for ␴ and J.

3. Cost example for a modular power plant 3.1. Costing rules The approach and results of a preliminary cost estimate of a ∼2 GW (thermal) modular fusion power plant with the analysis of advantages and disadvantages of the modular design is described in this Section. The 1st of a kind fusion plant consisting of 11 small fusion modules (10 working and one under the scheduled maintenance) with the thermal power ∼200 MW for each module is considered. Reservation of one module permits off line service and planned changes of the first wall and diverter. It gives effectively 100% availability for 10 modules of the power plant. It also decreases the cost of the hot cell and the cost of remote handling equipment due to a continuous work load for the hot cell and remote handling tools. All modules have a common thermal conversion power plant and common fusion technology equipment (i.e. heating/CD system, vacuum pumping, fuel processing, cryogenic plant, remote handling, hot cell, building) to take advantage of the economy of scale. These elements of fusion power plants have been already analyzed in previous studies of conventional power plants and an advantage of these studies can be taken. To be able to compare the modular approach with a conventional one everywhere when it is possible, the cost algorithms accepted in previous reactor studies such as ARIES project and in particular costing recommendation proposed by L.M. Waganer [35] will be followed. The estimates will be done in 2009 USD, using Gross Domestic Product Implicit Price Deflator given also in [35]. The cost of the fusion technology equipment (pumping, fueling, cryogenic, tritium cycle) has been estimated on the basis of the ITER costing taking in account the difference between an experimental machine and a commercial power plant. It was assumed that modules can operate without continuous additional heating and external current drive during the plasma burn. The operational point with the stable fusion burn and a stable plasma current may be achieved with external heating and CD at low density applied sequentially to each module. In this case the cost of heating and CD equipment is negligible in comparison with the power plant cost and it was not taken into account. The costs of the module components were calculated based on their dimensions and weights. The dimensions are determined mainly by the neutronics − by the thicknesses of shields and breeding blankets − and may be estimated without a detail design based on existing calculations. The following costing rules were used. The cost of materials is calculated in accordance with weights and current material market prices advertised at the Internet. The cost of manufacturing for the parts of the first module is assumed to be 8 times higher than the cost of materials. There is no direct justification for this rule. As an indication, we can only consider two limiting examples. In automobile industry − the ultimate example of mass production − raw materials contribute about 47% to the cost of a vehicle. Other 53% is in labor including R&D, administration and advertising. Direct labor is only 21% [36]. In civil aviation [37] (Boing and Airbus) in 2013 the total production value was ∼99E9 $. They have bought 0.44E9 kg of raw materials for the mean price ∼16.8 $/kg. The cost of 1 kg of finished

Fig. 13. Learning curves used for cost calculations. Normalized cost of the Nth unit and the mean cost for N units as functions of number of units N.

product is ∼226.51 $/kg. It means that cost of labor 12.48 times higher than the cost of materials. A significant part of this labor cost is the cost of R&D. So, our allocation for labor cost of the first of a kind product is in a middle of this range and is reasonably generous. To find sensitivity of the cost of electricity to this costing rules, calculations with different ratios of labor cost to cost of raw materials have been performed To find sensitivity of the cost of electricity to this costing rules, calculations with different ratios of labor cost to cost of raw materials have been performed. To take into account learning during manufacturing, it was assumed that material costs are the same for all modules, but manufacturing cost goes down by factor of 2 between the first of a kind and 10th of a kind component (See Fig. 13). The total manufacturing cost of 10 components is equal 6.25x (manufacturing cost of the first component). The manufacturing cost of the 11th component (and for all after it for changeable components) is 50% of the manufacturing cost of the first one. This rule was slightly corrected only in one case. For HTS magnets all the cost has been considered as cost of manufacturing, because the cost of raw materials is low. 3.2. Parameters of a fusion module selected for costing Main parameters of the fusion module (see Fig. 14), shown in Table 1, have been selected on the basis of 0-d modeling and cylindrical approximation of profiles described in the Section 2 of this article. These parameters are not optimal but just an example of a possible selection. For selected wall loading (Pwall < 2 MW/m2 ) and fusion power, the toroidal magnetic field needed to get maximum stable beta is not sufficient to get ignition when A = 1.7. The magnetic field was increased up to ignition level in accordance with scaling (22) without changing other parameters. As a result, for the module under consideration, f␤ = 0.806 < 1, and to achieve 100% bootstrap (fbs = 1) fraction the IN = IN*/(fbs /f␤ ). For spherical tokamaks, the neoclassical effects are more important than for high aspect ratio devices. To be sure that extrapolating of MAST scaling to high temperatures and low collisionality we will not be limited by neoclassical transport as an ultimate limit, the 1-d self-consistent transport modeling has been performed [38] for a module with aspect ratio A = 1.6, elongation K = 3, triangularity ␦ = 0.5, major radius R = 1.6 m, magnetic field B(R) = 3.17 T and plasma current Ipl = 6.7 MA. The mean coefficient of neoclassical ion thermal conductivity ␹i nc was calculated with NCLASS code and was found to be equal to ∼0.04 m2 /s. It corresponds to the energy confinement time ∼a2 /␹i nc = 22s, more than 10 times higher than that needed for the module ignition. 1-d modeling has also

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Fig. 14. Geometry of in-vessel components of the ST fusion power module. All dimensions in meters. Sh - Shafranov’s shift.

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Table 1 Parameters of a fusion power module for cost estimate. Parameter

Units

Value

Impurities Tmean Aspect ratio Fusion power Blanket energy multiplication factor Thermal power Total thermal efficiency Electrical power (gross) Plasma major radius Plasma minor radius Elongation Triangularity Plasma volume Wall surface Plasma density Plasma energy Toroidal magnetic field Plasma current Radiative losses (Bremsstrahlung + Synchrotron radiation) Energy confinement time ␤ toroidal ␤ poloidal Mean neutron wall loading Neutron shield thickness in the central post Maximum value of TF field Total current in TF coil

% keV

4 of He, 2 of H2 14 1.7 172 1.34 218.4 35.4 77.45 1.6 0.94 3 0.5 80 116 0.97 51.13 3.17 6.7 2.2 1.6 10.7 2.6 1.18 0.4 19.6 25.4

shown that bootstrap current is somewhat higher than assumed in 0-dimensional modeling. In formula (8) the ratio fbs /(␤pol/ A0.5 ) was selected to be equal to 0.5. 1-d modeling has given corresponding value of 0.57. Selected for 0-d modeling profiles of density and temperature could not be reproduced in 1-d modeling. However, when the mean density and mean plasma pressure have been matched, the results of 1-d modeling of fusion power and plasma energy coincide well with 0-d modeling.

3.3. Results of the cost estimate The results of the cost estimate are presented in Table 2.

4. Cost of electricity

MW MW % MW m m

m3 m2 1E20/m3 MJ T MA MW s % MW/m2 m T MA*turn

(assuming 30 years of operations, 10 years of decommissioning, 5% interest rate and 300 M$ cost of decommissioning). The accuracy of this estimate is not very high but even an error of 300% will add only 2% to the cost of electricity. So, the total value of the nominator in expression (1) is ∼373 M$. The total recirculation of electrical power for cryogenic or other uses is ∼5.2% (40 MW). The modular plant has no stationary additional heating or CD systems that are required during production of electricity (although some will be needed for start-up and control). Thanks to reservation of a module, the modular plant has no loss of availability caused by the fusion core systems. The average plant capacity factor (PF) is determined by the thermal part of the plant and is ∼ 0.95. Substituting all these numbers in expression (1) for CoE we have CoE = 0.06 $/kWh

4.1. Estimate of the cost of electricity Now the cost of electricity CoE given by formula (1) can be calculated. For 30 years of operation, 5 years of construction and “standard” financial conditions (interest rate = 5%), the total capital that must be borrowed is 9930 M$ and the annual capital cost charge CAC (3) is equal to 331 M$/year during 30 years. The scheduled component replacement cost (CSRC) for a modular plant was estimated as 598 M$ total or ∼20 M$/year. The cost includes costs (see Table 2) of changeable parts of central posts and neutron shields, costs of the first wall/blanket structures and costs of breeder (Li6 ) and multiplier (Be) recycling for 5 changes in 10 modules. The costs are calculated taking in account industrial learning. Other terms in the formula (2) do not depend strongly on the type of the fusion plant and are determined mainly by its power; so, results of other reactor studies are directly applicable. The annual operations and maintenance cost are: (CO&M) ∼ 10%*Total capital cost/Top = 17.4 M$/year. The annual fuel costs are: FC ∼ 1 M$/year. The annualized decontamination and decommissioning cost is: (D&DC) ∼ 3.3 M$/year

(27)

This is the CoE in 2009 US dollars and for the first plant of a kind. 4.2. Dependence of CoE on the neutron wall loading The cost calculated above corresponds to rather low mean neutron wall loading: Pwall ∼1.17 MW/m2 . May the plant economic be improved if modules would operate with higher wall loading? The data presented above permit easily to estimate the maximum possible effect of operation with a higher wall loading. Let us assume that we can increase the wall loading (and the fusion power) of each module without any increase of the module cost and at the same time to decrease the number of modules to keep the total fusion power constant. A decrease in the number of modules will permit a decrease in the size and cost of the reactor building, but the mean cost of modules will rise because of a weaker learning effect for a smaller number of modules. All other costs that depend on the fusion power value will not be changed. The plant factor and availability will stay also unchanged. To achieve that, one module will be kept in reserved in all cases. At this condition, the CoE is almost exactly proportional to the total plant cost and may be easily recalculated from Table 2. The change of the relative CoE normalized by (27) is shown in Fig. 15 as a function of the wall

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Table 2 Results of the cost estimate for 2 GW (thermal) modular fusion power plant. Main component

Sub component

Material

Weight T

Unit cost of material $/kg

Cost of the first of a kind M$

Cost of 11 modules M$

Changeable parta Permanent part

SS

17.9 76.3 94.2

17.0 45.0

2.7 30.9 33.6

19.8 223.0 242.8

Steel structure Breeder Multiplier

SS Li4 SiO4 (90% Li6 ) Be

30.7 33.6 25.8 90.1

17.0 200.0 1100.0

4.7 8.6 28.4 41.7

33.9 94.6 312.4 440.7

Changeable parta Permanent part

SS SS

62.5 562.9

4.0 4.0

2.3 20.3 22.5

16.3 146.3 162.6

SS SS

153.0 343.0

8.7 5.0

HTSC HTSC

I*L = 438MA*m I*L = 396MA*m

12.0 15.4 3.6 90.0 43.0 261.8

86.3 111.5 26.0 650.0 315.0 2034.9

Central post

Central post total Breeding blanketa

Blanket total Shield

Shield total Vacuum Vessel Cryostat Divertor TF coils Poloidal system Total cost of modules Heating & CD Vacuum pumping Fuel processing Cryogenic plant Main Heat Transfer loop Total Reactor Plant Equipment

60.0 87.8 49.3 205.6 52.5 2490.0

Conventional energy conversion plant and buildings Reactor building and Hot cell Cost of land, other buildings and site structures Turbine Generator Plant Equipment. Electrical Plant Equipment Heat rejection equipment Miscellaneous Plant Equipment Total cost of conventional energy conversion plant and buildings Initial tritium load to start the plant (∼4 kg) Total direct capital cost

206.0 187.6 308.1 150.0 52.4 70.0 974 140 3604

Indirect capital costs Construction Services and Equipment (assumed 10% of direct capital cost (DCC)) Home Office Engineering and Services (assumed 10% of DCC) Field Office Engineering and Services (assumed 10% of DCC) Owner’s Cost (assumed 5% of DCC) Project Contingency (assumed 10% of DCC) Total Indirect Capital cost

360.4 360.4 360.4 180.2 360.4 1621.8

Total capital cost

5226

a

Taking in account peaking of the neutron wall loading, the first wall and divertor must be changed every 5 years. (Maximum dpa< 100). Together with the first wall the total breeding blanket and parts of shielding damaged up to the limit of weldability must be also changed.

loading. As we keep the total fusion power constant, in this case the neutron wall loading is ∼ 1/N where N − number of modules. With the increase of the neutron wall loading the share of fusion modules in the total cost goes down from 66% at 1 MW/m2 (11 modules) to 30% at 10 MW/m2 (2 modules) and at the same time the cost of the reservation of 1 module goes up from 4.7% of the total plant cost to ∼ 15%. Relative effect of the increase of the neutron wall load goes down for neutron wall loads Pwall > 3 MW/m2 . The data in Fig. 15 have been calculated with an assumption that the module reservation fully eliminates the change of the plant factor with the change in the neutron wall loading. However, this is not always the case and must be considered in more detail. The first wall and the blanket can survive only limited fluency, limited number of displacement per atom (dpa) and must be changed after reaching the limit. In Fig. 16, the duration of operation between changes for selected neutron wall loadings is shown as a function of the limiting dpa. The need for a change is determined by limiting fluency in the most loaded part of the first wall. The peaking factor for the neutron

wall loading is assumed to be equal to 1.5. (This factor depends on the elongation, the aspect ratio and radial distributions of the plasma temperature and density). The Plant Factors as functions of the limiting dpa depend on the duration of the break needed to make each change of the first wall. The estimation of this time is very speculative. A significant effort of the design and remote handling development is needed to make a proper estimate. In any case, achievement of fast changes will demand significant additional investment. The Plant Factors corresponding to characteristic durations of changes of 6 and 12 months for a standard non-modular design (without any reservation of equipment) are shown in Fig. 17. Without the reservation, even for a moderate neutron wall load Pwall ∼ 3 MW/m2 and expected maximum dpa ∼ 75, the decrease of Plant Factor is significant for 6 months’ change and unacceptable for the 12 months’ change. For a modular design, the reservation of one module permits to achieve plant factor = 1 for the limiting dpa >75 at the 6 months’ change time. It must be underlined, that in this case the change

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Table 3 Comparison of Model A with the modular plant. Fractions of the total capital cost per system. System

Model A

Modular plant

Magnets + Cryostat% Buildings, site% Heating/CD% Divertor% Blanket+ FW%

41 19 8 2 3

31.1 11.4 1.7 0.7 15.2

4.3. Example of a module with a higher wall loading

Fig. 15. Relative cost of electricity as a function of the neutron wall loading. The total fusion power is kept constant Pfus ∼ 2 GW (thermal). The number of modules is ∼1/NWL.

The module described in Section 3.2 would operate at ␤tor = 10.7% < ␤max = 13.27%. The ␤ value could be increased to the limit by operation at density ne = 1.26 × 1020 m−3 and plasma current Ipl = 10 MA. The fusion power will reach 290 MW at the wall loading of 2 MW/m2 . In this case the modular plant will need 6 modules +1 reserved to produce ∼ 1 GWe. The CoE for this plant will be ∼82% of the 10-module plant and is ∼ 0.05 $/kWh. 4.4. Dependence of CoE on the costing rules The costing rules used here are obviously simplified. It is possible, however, to check how much a change of the rules will influence the CoE. Let us increase the labor cost for all module components by factor of 2 and make the labor cost 16 times higher than the cost of materials for the first module. This labor/material cost ratio is higher than that used in, for example, civil aviation. However, the CoE for the 7 modules plant described above will be CoE = 0.07 $/kWh − still very attractive.

Fig. 16. Time between changes of the first wall and the blanket as a function of the maximum fluence (dpa) which the selected materials can withstand for different neutron wall loadings. Peaking factor of neutron wall loading is assumed to be 1.5 [17].

Fig. 17. Plant Factors as functions of the limiting dpa for a standard design without reservation and for modular deigns with 11 and 5 modules (correspondingly with the neutron wall load Pwall = 1 and 2 MW/m2 . (Curves for the standard design with 2 MW/m2 and 6-month duration of change and 1 MW/m2 and 12 months coincide one with another).

time is the change time per module, not the total change time. To increase the plant factor of a modular plant to 1 for the limiting fluency 50 dpa, the change time must be decreased to 4 months. The cost of the 11th module is only 5% of the total direct cost. It adds 4.4% to CoE. At the same time the change of availability by 10% decreases the cost of electricity by 10%. 11th module gives directly 5.6% decrease of CoE. On top of that it decreases the cost of the hot cell, remote handling equipment and man power for the operation of a hot cell.

5. Comparison of the results with other economic studies The economics of fusion has been extensively studied. It is sufficient to name the fundamental “ARIES” project in USA fully documented in [39]. Different fusion systems have been studied with a variety of physics and technological assumptions. In our work we widely used the costing recommendations suggested by “ARIES” [35]. Significant work has been done in Europe in the frame of the European Power Plant conceptual study (PPCS) [40–42] and in Japan [43–45]. Given the same assumptions, all results are broadly consistent. For us the most interesting and sufficient is a comparison with the so-called “Model A” of the EU studies, which has the technology and physics very similar to assumptions used by us: the main construction material − steel, cooling by water with thermal efficiency ∼35%, the same assumptions for equilibrium and bootstrap physics, energy confinement described by ITER scaling with a minor improvement (H = 1.1). The Model A is much bigger than modules of the modular plant and has much higher fusion power. The comparison of absolute values is not useful. However, the comparison of relative values can be very productive. The European study has used exactly the same method of CoE calculations as “ARIES“ and us with the same conclusion: the lion share of CoE is determined by the capital cost: ∼75% for the “Model A” and 88% for the modular Plant. This difference is a result of lower costs of changeable parts in case of the modular Plant thanks to the mass production and the learning during production. The relative distribution of the total capital cost between different systems (see Table 3) is also broadly consistent: The Model A has an additional element of magnet system − central solenoid − and operates at lower beta (4% in comparison with 10.7% for the modular plant). The higher relative cost of magnet may be expected. For the breeding blanket, the modular plant design uses expensive materials: Beryllium and Lithium enriched up to 90% of Li6

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Table 4 Power and economic characteristics of two designs. Characteristic

Model A

Modular plant

Fusion power MW Neutron wall loading MW/m2 Blanket gain Thermal power MW Thermal efficiency% Gross electric power MW Recirculation fraction of electric power% Net electrical power MW Specific cost of fusion power=Dir. cap. cost/Pfusion $/W Specific cost of net electric power =Direct capital cost / Pe net ) $/W . Direct capital cost M$ Total capital cost M$ Availability % CoE (1st of a kind) $/kWh

5000 2.2 1.18 5900 35 2000 25 1500 1.9 6.25 9375 13594 75 0.12

172*10 = 1720 1.2 1.34 218.8*10 = 2188 35.4 774.5 5.16 735.77 2.01 4.7 3464 5023 95 0.06

to decrease the thickness of the blanket and decrease the magnet cost. This blanket is relatively more expensive (15.2% in comparison with 3% for Model A) but together blanket and magnet have the same relative cost for both designs: 44% and 46.3% of the capital cost. Buildings and site have similar relative costs for both designs. For the modular plant, buildings are cheaper because of the cheaper hot cell and lifting equipment (Table 3). Magnet, cryostat, building, site, first wall and blanket together represent 63% of the capital cost for model A and 57.7% for the modular plant. There are also some differences. For model A, the heating/current drive is expensive. It represents 6% of total capital cost. For the modular plant this cost is insignificant (∼1.7%). There is also a factor 3 difference in relative divertor costs (2%, 0.7%). But the heat loads on diverters are very different. Even with assumption of a high radiated fraction of heating power (60%) in the Model A case, the specific thermal load on divertor of Model A is Pdiv /(2*␲*R) = 8 MW/m in comparison with 3.2 MW/m in the case of the modular plant. For both machines the cost of divertor is relatively small and does not influence the CoE. Let us compare specific costs of these two designs (See Table 4). For Model A, the specific cost of Fusion power − the ratio Capital cost/Pfusion − is ∼1.9 $/W. For the modular plant the same ratio is 2 $/W. It is interesting to note that this specific cost is close to 2 $/W for all models considered by PPCS [41] despite of big differences in dimensions, used technologies and the fusion power. However, if we compare the ratio of the capital cost to the net electric power (CC/Penet ) − we shall find that these values are very different for the Model A and for the modular plant: 6.25 $/W and 4.7$/W. This is a result of high recirculation fraction for the Model A (25%) in comparison with 5.2% for the modular plant and lower power amplification in the blanket. The cost of electricity CoE is proportional to (CC/Pelnet )/availability. Reservation of a module permits to achieve very high reliability of the Modular plant (95%) in comparison with 75% claimed for model A. This lead to CoE = 0.06$/kWh in comparison with 0.12/kwh for the first of the kind Model A. The CoE of the Model A can be decreased to a level ∼0.06 $/W for 10th of the kind with total spending ∼100B$. This comparison shows that relative values for both designs are very similar and all differences between them are understandable in spite of the very big difference in parameter and dimensions and different methods of the cost estimation. The modular design does not increase significantly the specific cost of the fusion energy − this effect could be, in principle, expected. At the same time, high bootstrap decreases the electrical power recirculation and cheap reservation increases the availability. These two factors lead to very attractive CoE. However, the factor 2 difference in CoE is not very important. Future developments can modify the value. What

is more important, is the difference in the cost of the first prototype, which must be built and tested, before the commercial power plant can be built. In the case of the Model A, a prototype (let us call it DEMO) and, may be, two prototypes will cost ∼10B$ each and demand 10–20 years for construction and development. For the modular plant, all development can be done with, at least, 10 times cheaper module and at least 2–3 times faster. The calculated CoE for the modular Plant is very competitive and some increases even by a factor of 2 will not damage the competitiveness today and even more tomorrow when stricter environmental regulations like carbon tax will increase the CoE for competitors. What is the possible showstopper? To build a modular power plant we must confirm that MAST/NTSX scaling can be extrapolated to lower collisionality and higher magnetic fields, that good confinement is possible in high beta and high bootstrap regimes. To proof it we do not need to build a GW power DEMO with multy-B$ price as we must test only one module, not the whole GW plant.

6. Conclusions A modular fusion power plant consisting of several compact high field Spherical Tokamak modules and common conventional high power energy conversion systems have been proposed [4] as a way to decrease the cost and the risk of the fusion development. The modular system combines advantages of the “economy of scale” for the conventional part of the plant and the “economy of mass production” for the fusion core part of the plant, and as shown above does not lead to an increase of the specific cost of the fusion energy. At the same time, it provides not only an option to progress with small low cost prototypes and faster and cheaper development path, but a possibility to achieve high availability of the fusion core part of the plant. The large number of similar modules to be built will also help to create and sustain a supply chain. The capital cost, the electric power recirculation for internal power plant needs and the plant availability define the cost of electricity. The capital cost in its turn depends on the plasma beta and the neutron wall loading. Low aspect ratio fusion power modules can be optimized to simultaneously achieve high beta and high bootstrap fraction and so minimize capital cost and electrical power recirculation. To achieve this, the modules must be operated with IN ≤ IN* ∼ 0.6* (1 + K2 )/A1.5 (q95% > 10). HTS technology already permits generation of strong toroidal magnetic fields that will be sufficient to obtain reasonable (>1 MW/m2 ) neutron wall loading even in small low aspect ratio power modules. The rapid progress of the technology promises the HTS cost reduction in the near future.

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The modular structure of a power plant permits cost effective reservation of modules and off-line periodic change of the first wall/blanket components after irradiation up to the limiting fluence. It decreases the cost of the hot cell and remote handling equipment. It practically eliminates the dependence of the plant availability on the duration of the first wall/blanket changes and increases the plant factor up to the level typical to that of conventional power plants. The cost estimates performed, despite of their obvious limitations, insufficiency and preliminary nature, show that the economics of a modular plant is not only no worse than that of a conventional fusion plant, but in some aspects, it is superior. The first of a kind modular fusion plant with a thermal power Ptherm ∼ 2 GW (∼10 working fusion modules) and a high field HTS magnet system can produce electricity with a very competitive cost CoE <6 cents/kWh. This low cost of electricity can be achieved mainly due to the low recirculation of the electrical power for internal needs and a very high availability of the modular system. The main condition to realize this approach is to get sufficiently good energy confinement in a small low aspect ratio tokamak with IN ≤ 2.5 (q95%  10). “Sufficiently good” means confinement which will not strongly modify requirements for the fusion module parameters determined on the basis of the equilibrium and stability assumptions and constraints. It looks like the “MAST/NSTX type” energy confinement [9–11] is sufficient to satisfy the modular fusion reactor design requirements. The main task for R&D today is to confirm these results for high field low aspect ratio tokamaks with fusion parameters. References [1] M.A. Schilling, Towards a general modular systems theory and its application to inter-firm product modularity, Acad. Manage. Rev. 25 (2000) 312–334. [2] C.Y. Baldwin, K.B. Clark, Design rules The Power of Modularity, vol. 1, MIT Press, Cambridge, MA, 2000. [3] G. Locatelli, A. Fiordaliso, S. Boarin, M.E. Ricotti, Cogeneration: An option to facilitate load following in Small Modular Reactors, Prog. Nucl. Energy 97 (2017) 153–161 (2017-05-01); A. Glaser, Small Modular Reactors − Technology and Deployment Choices (presentation), NRC, (5 November 2014). [4] M.P. Gryaznevich, V.A. Chuyanov, D. Kingham, A. Sykes, Tokamak Energy Ltd, Advancing fusion by innovations: smaller quicker, cheaper, J. Phys.: Conf. Ser. 591 (2015) 012005. [5] A.E. Costley, J. Hugill, P.F. Buxton, On the power and size of tokamak fusion pilot plants and reactors, Nuclear. Fusion 55 (2015) 033001. [6] ITER physics basis, Nucl. Fusion 39 (2175) (1999). [7] C.C. Petty, Sizing up of plasmas using dimensionless parameters, Phys. Plasmas 15 (2008) 080501. [8] B.N. Sorbom, et al., ARC: A compact, high-field, fusion nuclear science facility and demonstration power plant with demountable magnets, Fusion Eng. Des. 100 (2015) 378. [9] M. Valoviˆc, et al., Scaling of H-mode energy confinement with Ip and BT in the MAST spherical Tokamak, Nucl. Fusion 49 (2009) 075016. [10] S.M. Kaye, et al., Energy confinement scaling in the low aspect ratio National Spherical Torus Experiment (NSTX), Nucl. Fusion 46 (2006) 848–857. [11] S. Kaye, et al., Scaling of electron and ion transport in the High- Power Spherical Torus NSTX, Phys. Rev. Lett. 98 (2007) 175002. [12] M.P. Gryaznevich, A. Nicolai, P. Buxton, Fast particles in a steady-state compact FNS and compact ST reactor, Nucl. Fusion 54 (2014) 104005. [13] W. Guttenfelder, et al., Simulation of micro tearing turbulence in national spherical torus experiment, Phys. Plasmas 19 (2012) 056119.

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