Canada Deuterium Uranium Reactor Research Articles

Best-Estimate Plus Uncertainty Analysis of CANDU Fuel Reliability Using Manufacturing and Simulated Core Data

Abstract A novel method of assessing the reliability of 37-element Canada deuterium uranium (reactor) (CANDU) fuel bundle was explored. The method implements a “best-estimate plus uncertainty” (BEPU) approach where a probabilistic treatment of manufacturing and operating inputs is used to predict fuel performance. The fuel performance was predicted using the Canadian industry standard codes for fuel performance, ELESTRESS and ELOCA, which, respectively, model fuel behaviors during normal and transient conditions. The outputs of the codes were compared against failure criteria from industry norms to determine the probability of failure. A Monte Carlo simulation method was applied to analyze this problem. Probability distributions of manufacturing input variables were estimated from real data, which were then randomly sampled. The inputs for fuel burnup and power were simulated using core-following data generated using a three-dimensional diffusion code, the Reactor Fuelling Simulation Program (RFSP), which were also then randomly sampled. The results of the simulations predict significant improvements in margins to limits for all performance parameters. An average improvement of 500 °C in centerline temperature, 10 °C in sheath temperature, 12 MPa in element internal pressure, and 0.8% in pellet end sheath hoop strain was predicted for the highest-powered region of the core, during normal operations, in comparison with the limit-of-envelope (LOE) benchmark. An 80% reactor overhead break (ROH) transient simulation was also simulated, and an average improvement of 500 °C in centerline temperature, 150 °C in sheath temperature, 6.5 MPa in internal pressure, and 2% in sheath hoop strain was predicted.

Tritium Research and Development Status at KAERI

Korea has 26 nuclear power plants (NPPs). Out of these 26 plants, 4 are Canada Deuterium Uranium (CANDU) reactors at the Wolsong nuclear power site. In CANDU reactors, deuterium oxide is used as a moderator/coolant, and tritium is produced whenever a deuterium oxide nucleus captures a neutron. The Wolsong Tritium Removal Facility was designed to remove tritium generated in CANDU reactors. We are introducing tritium environmental protection not only at the Wolsong NPP but also at the High-Flux Advanced Neutron Application Reactor (HANARO) and in high-temperature gas-cooled reactors (HTGRs). We present a tritium behavior analysis code and assess the concentration of tritium in combustible dry active waste. Advanced techniques are introduced to transfer tritium from tritiated water to the gaseous phase. In addition, research on the nuclear fusion tritium storage and delivery system, which is part of the fuel cycle, has been carried out. In this paper, we present the recent progress in the effort to develop tritium systems at the Korea Atomic Energy Research Institute.

The Impact of Fueling Operations on Full Core Uncertainty Analysis in CANDU Reactors

Abstract Within emerging best-estimate-plus-uncertainty (BEPU) approaches, code output uncertainties can be inferred from the propagation of fundamental or microscopic uncertainties. This paper examines the propagation of fundamental nuclear data uncertainties though the entire analysis framework to predict macroscopic reactor physics phenomena, which can be measured in Canada Deuterium Uranium (CANDU) reactors. In this work, 151 perturbed multigroup cross sections libraries, each based on a set of perturbed microscopic nuclear data, were generated. Subsequently, these data were processed into few-group cross sections and used to generate full-core diffusion models in PARCS. The impact of these nuclear data perturbations leads to changes in core reactivity for a fixed set of fuel compositions of 4.5 mk. The impact of online fueling operations was simulated using a series of fueling rules, which attempted to mimic operator actions during CANDU operations such as studying the assembly powers and selecting fueling sites, which would minimize the deviation in power from some desirable reference condition or increasing or decreasing fueling frequency to manage reactivity. An important feature of this analysis was to perform long-transients (1–3 years) starting with each one of the 151 perturbed full core models. It was found that the operational feedback reduced the standard deviation in core reactivity by 99% from 0.0045 to 2.8 × 10−5. Overall, the conclusions demonstrate that while microscopic nuclear data uncertainties may give rise to large macroscopic variability during simple propagation, when important macrolevel feedback are considered the variability is significantly reduced.

Fate and transport of tritium in the Laurentian Great Lakes system

A mass balance modelling approach was used to help understand the movement and impacts of tritium discharged from Canada Deuterium Uranium (CANDU) reactor facilities into Lake Ontario. A concentration-time model, previously developed, is updated in this study. Historical and projected tritium concentrations for Lake Ontario waters are presented. A model calculated accident scenario (10 times highest accidental release) indicates the importance of dilution to the dispersion of tritium; a “modelled” release in 2016 has tritium levels declining by the year 2030 to “previous 2016 levels”. As part of the mass balance approach, lake-bottom sediments were considered as potential radionuclide “sinks”. Tritium porewater results were noted as perturbations at depth in both short (30–50 cm cores) and long sediment core profiles (to 300 cm). These change in tritium concentrations with depth may have been due to CANDU emissions (as the most likely source) over time, based on records of accidental releases of tritiated coolant water. However, the exact process (advection and/or diffusion) responsible for the penetration of tritium into the lake bottom requires additional physical and hydrogeological characterization of the lake bottom sediments.

Simulation of motion-dependent fluid forces in fuel bundles

Fuel bundles in CANDU (Canada Deuterium Uranium) reactors are subject to axial flow-induced vibrations. This could result in fretting wear in the inner surface of the pressure tube or in the integrity of the bundles being compromised. Various fluid excitations can affect the dynamic response of the fuel bundle. One of these excitations is the fluid forces produced by the motion of the fuel rods. These forces can be expressed in terms of inertia, stiffness, and damping components. A numerical approach was proposed to simulate these forces in a flexible fuel kernel. These forces were expressed in terms of coefficients and phases. In addition, an analytical model was developed utilizing the force coefficients to predict the dynamic response of the fuel bundle. The dynamic response was compared with the available experimental data. The numerical model developed in this work presents a step towards a realistic framework to simulate the dynamic response of CANDU fuel bundles.

Evaluation of the thermal strain of an NPP containment structure during leakage rate tests

A strategy for isolating the thermal strain due to solar radiation from the total measured strain during the Integrated Leak Rate Test of a nuclear containment structure is presented in this paper. In the proposed method, a series of numerical models are developed to estimate the temperature change on the prestressed concrete containment structure and to predict the strain due to temperature and applied pressure. The effects of the solar radiation, heat exchange with internal and external air and heat conduction are considered in the proposed method. The developed method is validated by reproducing the total strain profiles measured from the leak rate tests conducted at the Point Lepreau CANDU®11CANDU (CANada Deuterium Uranium Reactor) is a registered trademark of the Atomic Energy of Canada Limited (AECL), used under exclusive license by Candu Energy Inc. 6 containment structure.

Neutronic analysis of mixed thorium-uranium fuel bundle for CANDU reactors

The paper shows neutronic analysis of the CANDU (CANada Deuterium Uranium) nuclear reactor fuel channel with mixed thorium-uranium fuel bundles. The numerical model of the fuel channel was designed using The Monte Carlo Continuous Energy Burn-up Code – MCB developed at the AGH University of Science and Technology, Faculty of Energy and Fuels, Department of Nuclear Energy. The super-computer Prometheus available in the frame of the Pl-Grid Infrastructure at the Academic Computer Centre Cyfronet AGH was used for multi-scale calculations. The fuel bundles are composed of two clusters of fuel rods. The neutronic analysis considers detailed numerical simulation of neutron transport in fully heterogeneous geometry of the fuel channel. Moreover, burnup simulations were performed using Transmutation Trajectory Analysis method implemented in the MCB code. In the analysis we mainly consider time evolutions of neutron multiplication factor, fissile 233U, 235U and 239Pu and fertile 238U and 232Th. The simulations were performed for eight scenarios with various fuel composition.

Development of computer code IGDC for generation and depletion of fission products and actinides in pressure tube type heavy water reactors (PT-HWRs) using Klopfenstien-Shampine numerical differentiation formula (NDF)

Abstract Pressure tube type heavy water reactors (PT-HWRs) with Natural Uranium (NU) fuel, similar to conventional CANDU (CANada Deuterium Uranium) reactor of small and large sizes are operated successfully worldwide. Various nations have their nuclear program based on a closed fuel cycle. Therefore it is important to know the exact concentration, decay heat, radio-toxicity and radioactivity of all isotopes produced over the fuel cycle. To this end, a computer code is being developed with advanced capabilities on MATLAB-14 platform using Klopfenstien-Shampine family of Numerical Differentiation Formula (KS family of NDFs). In this paper, efforts are taken to highlight on the develop of an indigenous computer code IGDC: Isotopic Generation and Depletion Code” to perform the fuel cycle studies for fuel irradiated in 220 MWe class PT-HWRs.

Mechanistic modelling of station blackout accidents for a generic 900 MW CANDU plant using the modified RELAP/SCDAPSIM/MOD3.6 code

CANDU (CANada Deuterium Uranium) reactors have many unique design features that play important roles during a severe accident, however analysis of such features using Light Water Reactor (LWR) specific computer codes is challenging. Severe accidents in CANDU involve complex thermo-mechanical deformation phenomena which differ from the phenomena present during LWR accidents. For example, during complete station blackout scenarios with a failure of all emergency measures, the pressure tubes may balloon or sag into contact with the surrounding calandria tubes (CTs) establishing a thermal conduction pathway for heat rejection to the large moderator water volume. As the moderator liquid evaporates or boils its level decreases until fuel channels become uncovered in the calandria vessel. The uncovered channels heat up quickly and the entire fuel channel assembly (fuel, pressure tube and calandria tube) will sag and possibly disassemble. During the disassembly process some channel components may fall to the bottom of the calandria while others may form a suspended debris bed supported by channels which are still submerged in moderator liquid. These phenomena impact event-timing, accident progression, hydrogen production and fission product release.In this work several mechanistic channel deformation models have been developed and integrated into RELAP/SCDAPSIM/MOD3.6 to provide a coupled treatment of the deformation phase for such postulated accidents. MOD3.6 is a new version of the RELAP/SCDAPSIM code being developed to support the analysis of Pressurized Heavy Water Reactors (PHWRs) under severe accident conditions. In this paper, the code system is used to simulate a postulated station blackout accident for a generic 900 MW CANDU plant. To reduce the uncertainty in the modeling of core disassembly and to overcome the memory constraints of the code, the simulation is broken into two phases with the first phase (i.e., from initiating event to the channel failure and depressurization) simulated using a full-plant RELAP5 model providing relatively high spatial fidelity of the entire heat transport system, and the second phase (i.e. continued from the end of the first phase until calandria vessel dryout) using alternative nodalization focusing on the calandria vessel and fuel channel components. The paper assesses the entire accident progression up to the point of calandria vessel dryout and performs sensitivity analysis on model parameters to assess their relative importance.

Improving the efficiency of shielding container design calculations

The reactor refurbishment project for a CANDU (Canada Deuterium Uranium) reactor is an example of a major reactor-maintenance-related project where significant amount of non-fuel radioactive materials are generated. The transportation of these radioactive materials to an off-site location for further processing must be done in a safe way following applicable International Atomic Energy Agency (IAEA) regulations. The shielding container used for transporting these radioactive materials must be designed such that the dose rate requirements in the vicinity of the container are met. In the process of designing such container, the designer needs to take into account various possible configurations of radioactive materials that will be loaded into the container. Even in a stylized fashion, the numbers of possible combinations of radioactive materials placement inside the container are simply astronomical and the evaluations of all possible combinations, given the current state of computational power, are simply prohibitive. In this paper, an alternative methodology is proposed where the principle of super-position is explored to aid the design process of the container. The overall dose rates in the area surrounding the radioactive materials are determined using the dose rates from the constituent radionuclides and the source strength of each type of the radioactive components being loaded into the container. Using this alternative approach, a significantly higher number of potential configurations can be evaluated in a relatively short period of time, which directly improves the robustness of the final container design.

Measurement and Analysis for Determination of PCR of the CANDU6 Core

The power coefficient of reactivity (PCR) needs to be negative to achieve the inherent safety of a reactor. However, the possibility that the PCR of CANada Deuterium Uranium (CANDU) reactors can be positive has been raised in recent studies. In such circumstances, there was an experimental approach on evaluating the PCR of CANDU in 2012 at an in-operation CANDU reactor, Wolsong Unit 2. In the evaluation, the PCR was indirectly measured by a method that required estimating the reactivity variation due to Xe, liquid zone controllers (LZCs), and fuel depletion based on the measurement data. In this study, the PCR of a CANDU was reevaluated by the same methodology with more proper and detailed methods to estimate all the factors in addition to some minor reactivity corrections. The estimation of Xe and LZC reactivity was performed by an in-house three-dimensional code and Serpent2 in addition to RFSP-IST. Furthermore, several short studies regarding the factors that result in uncertainty of the Xe/LZC reactivity estimation were done in detail. First, a method to determine 14 LZC levels at a certain time based on the measurement data was appropriately selected through determining the features of the measurement data. The influence of the power transient scheme and the impact of local refueling transients due to daily refueling of CANDU reactors on xenon reactivity estimation were also analyzed briefly. Finally, the PCR of the CANDU in operational conditions was evaluated to be ~0.5 pcm/%P on average at a measurement time of 5 to 20 min after the power perturbation.

Investigation of Fuel Burnup Impacts on Nuclear Reactor Safety Parameters in the Canadian Pressure Tube Supercritical Water-Cool Reactor

The Canadian pressure-tube super critical water-cooled reactor (PT-SCWR) is an advanced generation IV reactor concept which is considered as an evolution of the conventional Canada Deuterium Uranium (CANDU) reactor that includes both pressure tubes and a low temperature and pressure heavy water moderator. The Canadian PT-SCWR fuel assembly utilizes a plutonium and thorium fuel mixture with supercritical light water coolant flowing through the high-efficiency re-entrance channel (HERC). In this work, the impact of fuel depletion on the evolution of lattice physics phenomena was investigated starting from fresh fuel to burnup conditions (25 MW d kg−1 [HM]) through sensitivity and uncertainty analyses using the lattice physics modules in standardized computer analysis for licensing evaluation (SCALE). Given the evolution of key phenomena such as void reactivity in traditional CANDU reactors with burnup, this study focuses on the impact of fission products, 233U breeding, and minor actinides on fuel performance. The work shows that the most significant change in fuel properties with burnup is the depletion of fission isotopes of Pu and the buildup of high-neutron cross section fission products, resulting in a decrease in cell k∞ with burnup as expected. Other impacts such as the presence of protactinium and uranium-233 are also discussed. When the feedback coefficients are assessed in terms of reactivity, there is considerable variation as a function of fuel depletion; however, when assessed as Δk (without normalization to the reference reactivity which changes with burnup), the net changes are almost invariant with depletion.

Thermal hydraulic design and analysis of Thorium-based Advanced CANDU Reactor (TACR)

The application of thorium fuel in advanced CANDU (CANadian Deuterium Uranium) reactors is an economic solution to sustainable development of nuclear energy. In this paper, the thermal hydraulic design and transient analysis of Thorium-based Advanced CANDU Reactor (TACR) heat transport system are conducted using self-programmed code CANTHAC (CANDU Thermal Hydraulic Analysis Code). In the code CANTHAC, the single channel model and the point kinetics model are adopted to simulate the core thermal hydraulic and neutronic behaviors. A distributed parameter model is adopted in the simulation of integral economizer U-tube steam generator (IEUTSG). Multi-region non-equilibrium model is applied to simulate the pressurizer. The characteristics of heat transport pumps were simulated by four-quadrant analogy curves. The code is verified with ACR-700 steady-state analysis. The calculation results agreed well with ACR-700 design values and the relative errors are all within 2%. With CANTHAC steady-state module, the preliminary thermal hydraulic design of TACR is conducted. A 1000MW double loop reactor with 520 fuel channels is designed. Single loop mass flow rate was designed to be 3202kg/s. Under the design scheme, the maximum cladding temperature and the maximum temperature are 340.2°C and 1369.9°C, which satisfied the design requirements. The transient analysis of TACR heat transport system was performed with CANTHAC transient module. SG feed water temperature reduction accident, and complete loss of Class IV accident were conducted. In the SG feed water temperature reduction accident, a new balance of neutron and thermal hydraulic is reached and the core power increases by 5% at last. In the complete loss of Class IV accident, the maximum fuel temperature reaches peak value at 1172°C and still has some safety margin. The ROH (Reactor Outlet Header) pressure does not exceed 105% of the design value, which meet the safety criterion demand.

The Effect of Alkali Metal and Alkaline Earth Metal Impurities on the Iodine-Induced Corrosion of CANDU Fuel Sheathing

During normal operation in Canada deuterium uranium (CANDU®) reactors, the stress corrosion cracking (SCC) of fuel sheathing is mitigated effectively, in part, using a thin graphite-based coating known as CANDU lubricant (CANLUB). Mechanisms typically proposed for the demonstrated SCC mitigation offered by CANLUB include lubrication and/or chemical interactions. An additional possibility, that was recently suggested, involves the sequestering of iodine through its interaction with alkali metal and/or alkaline earth metal impurities in the CANLUB coating. This possibility is supported by the systematic analysis and testing in this paper, wherein three prevalent impurities (Na, Ca, and Mg) found in CANLUB were incorporated into SCC slotted ring experiments as metal oxides. When the amount of metal oxide (Na2O, CaO, or MgO) matched or exceeded the amount of iodine (6 mmol = 16 mg/cm3), Na2O and CaO protected the rings from corrosion whereas MgO enhanced their corrosion. When Zircaloy-4 sheathing is subjected to mechanical stress, high temperature, and high concentrations of iodine vapor, it is better protected by siloxane coatings than by graphite-CANLUB coatings. Consequently, since metal impurities (Na, Ca, and Mg) are found more abundantly in siloxane coatings than in graphite-CANLUB coatings, Zircaloy-4 slotted rings were coated with graphite-CANLUB containing Na, Ca, and/or Mg at those more abundant concentrations. Since these concentrations remain below 6 mmol, SCC test results suggest that the siloxane's superior adhesion is an essential first step in preventing corrosion induced by 6 mmol of iodine.

Utilization of alternating conditional expectation method for pin power reconstruction in CANDU reactor physics analyses

This paper presents a new methodology using the Alternating Conditional Expectation (ACE) algorithm for reconstructing fuel pin powers in the Canada Deuterium Uranium (CANDU) reactor. Variations in assembly powers in neighboring fuel bundles are used as predictor variables to determine the response, e.g. power in each of the fuel pin. When a sufficient number of cases have been evaluated, the relation between the predictor variables and the response can be treated as a multi-variable regression problem. The ACE algorithm which is a non-parametric approach (i.e., no predetermined functional relationship between the predictors and the response is assumed) is then utilized for estimating optimal transformations of the predictors and the response. The geometry used is a two-dimensional, 3-by-3 fuel bundles configuration. Three hundred different configurations, which represent various power distributions, were evaluated to develop these optimal transformations. The optimal transformation is determined for each of 37 pins in a CANDU natural uranium fuel bundle. The quality of these transformations for predicting fuel pin power has been verified against an independent set of 50 two-dimensional, 3-by-3 fuel bundle configurations. The results indicate that the proposed pin power reconstruction algorithm is sufficiently accurate, giving an RMS (root-mean-squared) error of around 0.59% for all pins. The RMS error in predicting the maximum pin power is 0.60%. The development of the proposed methodology continued with the examination of its robustness. The impact of employing different training sets on the overall relative error in predicting the pin power is examined. In addition, the impact of employing different sizes of training sets is also investigated. For each training set size, a bias in predicting the maximum pin power is determined statistically based on the numerical results observed during the study. The results indicate that the methodology is likely to under-predict the maximum pin power; to circumvent this issue a multiplicative bias of 1.0118 needs to be applied, based on evaluating results from various sizes of the training sets.

Results of Air Barbotage Experiments Simulating Two-Phase Flow in a CANDU End Shield During In-Vessel Retention

This paper presents the results of experimental investigations into two-phase mass transport in a coarse packed bed representing the Canada Deuterium Uranium (CANDU) end shield. This work contributes to understanding of phenomena impacting in-vessel retention (IVR) during postulated severe accidents in CANDU reactors. The air barbotage technique was used to represent boiling at the calandria tubesheet surface facing the inner cavity of the end shield. Qualitative observations of the near-wall two-phase region were made during air injection. In addition, flow visualization was carried out through the addition of dye to the water. Air flow rate, shielding ball diameter, and cavity dimensions were varied within relevant ranges; and the impact of these parameters on the near-wall region was identified. A brief review of the relevant knowledge base is presented, allowing demonstration of the applicability of the test parameters. The observed phenomena are compared to published results involving similar geometries with capillary porous media.

Effects of Filtered Containment Venting on Fission Product Releases During CANDU Reactor Severe Accidents

Severe accidents are of increasing concern in the nuclear industry worldwide since the accidents at Fukushima Daiichi (March 2011). These events have significant consequences that must be mitigated to ensure public and employee safety. Filtered containment venting (FCV) systems are beneficial in this context as they would help to maintain containment integrity while also reducing radionuclide releases to the environment. This paper explores the degree to which filtered containment venting would reduce fission product releases during two Canada Deuterium Uranium (CANDU) 6 severe accident scenarios, namely a station blackout (SBO) and a large loss of coolant accident (LLOCA) (with limited emergency cooling). The effects on the progression of the severe accident and radionuclide releases to the environment are explored using the Modular Accident Analysis Program (MAAP)–CANDU integrated severe accident analysis code. The stylized filtered containment venting system model employed in this study avoids containment failure and significantly reduces radionuclide releases by 95–97% for non-noble gas fission products. Filtered containment venting is shown to be a suitable technology for the mitigation of severe accidents in CANDU, maintaining containment integrity and reducing radionuclide releases to the environment.

A Study on the Application of CRUDTRAN Code in Primary Systems of Domestic Pressurized Heavy-Water Reactors for Prediction of Radiation Source Term

Abstract The importance of developing a source-term assessment technology has been emphasized owing to the decommissioning of Kori nuclear power plant (NPP) Unit 1 and the increase of deteriorated NPPs. We analyzed the behavioral mechanism of corrosion products in the primary system of a pressurized heavy-water reactor-type NPP. In addition, to check the possibility of applying the CRUDTRAN code to a Canadian Deuterium Uranium Reactor (CANDU)-type NPP, the type was assessed using collected domestic onsite data. With the assessment results, it was possible to predict trends according to operating cycles. Values estimated using the code were similar to the measured values. The results of this study are expected to be used to manage the radiation exposures of operators in high-radiation areas and to predict decommissioning processes in the primary system.

MEASUREMENT OF CRITICAL HEAT FLUX IN A CANDU END SHIELD CONSISTING OF A VERTICAL SURFACE ABUTTING A PACKED BED OF STEEL SHIELDING BALLS

The critical heat flux (CHF) was measured in an experimental apparatus representative of a Canada Deuterium Uranium (CANDU) reactor end-shield porous medium heated by a vertical surface. The understanding of peak heat transfer rates in this geometry is crucial to the demonstration of in-vessel retention of corium during a postulated severe accident in a CANDU reactor. In this paper, a large-scale experimental apparatus is described, which is capable of measuring CHF from a vertical surface to a packed bed of steel balls representative of a CANDU end shield. Data are presented from a large matrix of tests investigating the impact of heated surface height on CHF. Preliminary analysis of a subset of these data is also presented as well as the derivation of an empirical correlation for CHF as a function of elevation along the vertical wall. Qualitative comparisons are made between the results of the tests and predictions of peripherally related work carried out in the past, with reasonable agreement being observed in vapour-phase transport in the porous medium and in trends of CHF. It is expected that future work including further analysis of the current data will allow for the development of an improved mechanistic understanding of the phenomenon, leading to a more broadly applicable correlation for CHF that is subject to smaller uncertainties and could be incorporated into severe accident codes such as MAAP-CANDU.

SARAPAN—A Simulated-Annealing-Based Tool to Generate Random Patterned-Channel-Age in CANDU Fuel Management Analyses

In any reactor physics analysis, the instantaneous power distribution in the core can be calculated when the actual bundle-wise burnup distribution is known. Considering the fact that CANDU (Canada Deuterium Uranium) utilizes on-power refueling to compensate for the reduction of reactivity due to fuel burnup, in the CANDU fuel management analysis, snapshots of power and burnup distributions can be obtained by simulating and tracking the reactor operation over an extended period using various tools such as the *SIMULATE module of the Reactor Fueling Simulation Program (RFSP) code. However, for some studies, such as an evaluation of a conceptual design of a next-generation CANDU reactor, the preferred approach to obtain a snapshot of the power distribution in the core is based on the patterned-channel-age model implemented in the *INSTANTAN module of the RFSP code. The objective of this approach is to obtain a representative snapshot of core conditions quickly. At present, such patterns could be generated by using a program called RANDIS, which is implemented within the *INSTANTAN module. In this work, we present an alternative approach to derive the patterned-channel-age model where a simulated-annealing-based algorithm is used to find such patterns, which produce reasonable power distributions.