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UPTEC F 15069 Examensarbete 30 hp 1 December 2015

Control Rod Effect at Partial SCRAM Upgrade of Plant Model for Forsmark 2 in BISON After Power Uprate

Daniel Constanda Abstract Effect at Partial SCRAM

Daniel Constanda

Teknisk- naturvetenskaplig fakultet UTH-enheten This study aims to improve the modeling of partial SCRAM in the BISON plant model for the Forsmark 2 after power uprate. Validation of the BISON Besöksadress: model against tests performed from March to May in 2013 have shown that this is Ångströmlaboratoriet Lägerhyddsvägen 1 one of the areas in which there is room for improvement. After partial SCRAM is Hus 4, Plan 0 performed, the model underestimates the reactor power, recirculation flow and steam flow when compared to the measurement data. Postadress: Box 536 751 21 Uppsala In BISON the partial SCRAM is modeled using a relative control rod effect vector (ASC vector). The aim is to replace the old values in this vector to improve the Telefon: model. The new model was shown to give an improved result for the reactor power, 018 – 471 30 03 recirculation flow and steam flow. The study gives recommendations on how to apply

Telefax: the new model and what values of the relative control rod effect vector that can be 018 – 471 30 00 used in the future.

Hemsida: http://www.teknat.uu.se/student

Handledare: Anna Aspman, David Palko Ämnesgranskare: Staffan Qvist Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F 15069 Popul¨arvetenskaplig sammanfattning

I kokvattenreaktorer (engelska: BWR, boiling reactors) kan det uppst˚ast¨orningar som kan uttrycka sig i exempelvis tryck¨okning,tryckminskning och temperatur¨okningi reaktorn. Ber¨akningsprogrammetBISON anv¨andsf¨oratt simulera dessa.

Anl¨aggningsmodellen i BISON f¨orForsmark reaktor 2 har under senaste ˚arengenomg˚att stora uppdateringar p˚agrund av anl¨aggningenseffekt¨okning.Modellen har nyligen j¨amf¨orts mot m¨atv¨ardenfr˚anreaktorn vid 120 % effekt. J¨amf¨orelsen har funnit skillnader mel- lan anl¨aggningsmodellens beteende och m¨atningarutf¨ordai anl¨aggningeni samband med effekth¨ojningav Forsmark 2. En av skillnaderna var i styrstavsinverkan vid delsnabb- stopp, vilket denna rapport fokuserar p˚a.Delsnabbstopp ¨arn¨arn˚agraav styrstavarna skjuts in i reaktorn, vilket g¨oratt effekten s¨anks. Detta kan j¨amf¨orasmed snabbstopp d¨aralla stavar skjuts in och effekten s¨ankshelt, reaktorn blir s.k. underkritisk och den sj¨alvst¨andigak¨arnreaktionenupph¨or.

Denna rapport syftar till att f¨orb¨attramodellen s˚aatt delsnabbstoppet hamnar n¨armare m¨atv¨ardenafr˚anj¨amf¨orelsen.Styrstavsinverkan i den tidigare anl¨aggningsmodellen i BI- SON ber¨aknasmed antagandet att den varierar proportionellt med styrstavar som skjuts in. Den nya styrstavsinverkan togs fram genom att BISON j¨amf¨ordesmed ett annat simuleringsprogram som simulerar h¨arden(br¨anslettill reaktorn) noggrannare. Detta simuleringsprogram anv¨andesallts˚asom referens f¨oratt unders¨oka vilken styrstavinverkan som BISON borde f˚a.Med en automatiserad j¨amf¨orelsemellan dessa program s˚atogs styrstavsinverkan fram f¨orflera olika fall.

D¨arefteranv¨andesden nya styrstavsinverkan i den tidigare anl¨aggningsmodellen i BISON och j¨amf¨ordesmed samma m¨atdatasom f¨orut.Den nya styrstavinverkan visade sig ge en f¨orb¨attringgentemot den tidigare, d¨armedhar m˚aletmed rapporten uppfyllts. Slutligen inneh˚allerrapporten en n¨armareanalys av resultaten samt rekommendationer kring hur metoden och resultaten b¨oranv¨andas. Det finns ¨aven m¨ojligheteratt forts¨attadenna studie vidare, diskussion kring detta ges ocks˚a.

3 Contents

0.1 Glossary...... 6

1 Introduction8 1.1 ...... 8 1.2 The Reactor, Safety and Auxiliary Systems...... 9 1.3 BISON - Analysis Program for BWR...... 12 1.4 Previous Results and Performed Tests...... 12 1.5 Aim of the Study...... 13

2 Theory 14 2.1 Transients...... 14 2.2 The SCRAM System...... 15 2.3 SCRAM Modeling in BISON...... 16 2.3.1 The Transport Equation...... 16 2.3.2 Defintion of the ASC vector...... 17 2.3.3 The ASC vector in the Previous Model...... 18

3 Model and Simulations 19 3.1 Simulation Software Selection...... 19 3.2 BISON Models...... 19 3.3 POLCA Models...... 20

4 Method 21 4.1 Computation of ASC vector...... 21 4.1.1 Calculate New ASC vector...... 22 4.2 Composition of the New Model...... 24

5 Results 25 5.1 ASC vector Results...... 25 5.2 Comparison with Previous Results...... 27 5.3 Comparison of BISON and Simulate-3K Calculations...... 30

6 Discussion 31 6.1 Future Work...... 32

7 Conclusions 34

A Comparison with Previous Results: Trip Case 36

B Comparison with Previous Results: House Load Operation Case 38

C Individual Control Rod Study 40 C.1 Introduction...... 40 C.2 POLCA Model...... 40 C.3 BISON Model...... 40 C.4 MATLAB Scripts...... 40 C.5 Results...... 40 C.6 Discussion and Conclusions...... 41

4 List of Figures

1 Overview of a BWR...... 8 2 The ...... 10 3 assembly example...... 11 4 Overview of a reactor core...... 11 5 Algorithm for calculating the ASC vector...... 22 6 Example of how the ASC and the reactor power given by BISON changes with each iteration...... 23 7 ASC vector for PSS during Cycle 32 and 34...... 25 8 ASC vector for PSS with and without runback...... 26 9 ASC vector for PSS and individual insertions of SCRAM bank 6 and 9.. 27 10 Loss of condensate pump: APRM...... 28 11 Loss of condensate pump: Recirculation flow...... 28 12 Loss of condensate pump: Steam flow from RPV...... 29 13 Comparison of the power given by BISON and Simulate-3K...... 30 14 Turbine trip: APRM...... 36 15 Turbine trip: Recirculation flow...... 37 16 Turbine trip: Steam flow from RPV...... 37 17 House load operation: APRM...... 38 18 House load operation: Recirculation flow...... 39 19 House load operation: Steam flow from RPV...... 39 20 Individual control rod study: APRM during loss of condensate pump... 41

List of Tables

1 Values of the ASC vector...... 24 2 Values of the ASC vector: Individual control rod study...... 40

5 0.1 Glossary English term Swedish term Explanation APRM (Average Power APRM Equivalent to the average neutron Range Monitor) flux in the core. Usually mea- sured in % and equivalent to the reactor power in %. ASC vector ASC vektor Relative control rod effect vector. Channel power Kanaleffekt The power generated by the fuel in a certain channel. Control rod Styrstav A neutron absorbing rod which can be inserted/withdrawn into/from the reactor. CPR (Critical Power Ra- Torrkokningskvot Ratio of the critical channel tio) power needed for dryout and ac- tual channel power. Dryout Torrkokning Boiling on a heated surface dur- ing high steam concentration in a coolant channel, loss of water layer on the heated surface. EFPH (Effective Full- Ekvivalent fulleffekttid Ratio of extracted energy during Power Hour) a given time and the rated output of the reactor. F1, F2, F3 F1, F2, F3 Forsmark Reactor 1, 2 and 3 re- spectively. Feedwater Matarvatten (Mava) The water entering the RPV from the primary coolant loop. Operating cycle Driftcykel The period from start of the reac- tor to refueling. Operation point Driftpunkt The point on a coordinate system with power and RF on the y- and x-axis respectively. Partial SCRAM (PSS) Delsnabbstopp Insertion of one or a few control rods groups, decreasing the reac- tor power. Reactor Pressure Vessel Reaktortank See Figure 2 for an illustration. (RPV) Reactor Coolant Pres- RCPB/Prim¨arsystemet Self explanatory. sure Boundary (RCPB) Recirculation flow (RF) Huvudcirkulationsfl¨ode Water being circulated in the re- (HC-fl¨ode) actor for cooling. Powered by re- circulation pumps. Runback (of recircula- Nedstyrning (av HC- Reduction of speed of the reactor tion pumps) pumpar) recirculation pumps. Safety System Logic S¨akerhetskedjor Shows which safety systems are activated and in what order, de- pends on certain conditions such as reactivity and pressure etc.

6 SCRAM Snabbstopp Rapid insertion of all control rods into the reactor, shutting down the reactor quickly. SCRAM bank Snabbstoppsgrupp A group of control rods used for SCRAM. Void Void/Anginneh˚all˚ The amount of vapor in the mod- erator. Higher void implies less effective moderation (thus lower reactivity).

7 1 INTRODUCTION 8

1 Introduction

This section introduces the basics of the nuclear reactor and gives background to the study. It concludes with the aim of the study.

1.1 Boiling Water Reactor A Boiling Water Reactor (BWR) is one of the two most common water reactor types, the other is Pressurized Water Reactor (PWR). The most salient difference is what gives them their names, the water inside the reactor of a BWR is boiling but in a PWR it is not due to the higher pressure in the reactor. Another difference is that BWR is a single-loop system, see Figure 1, while PWR is a double-loop system. This is because the PWR needs a steam generator in order to get steam to drive the turbine. The BWR already generates steam due to the boiling. All the reactors of the Forsmark Plant (hereby simply referred to as Forsmark) are of BWR type. Thus this report will only focus on this reactor type. Figure 1 shows an overview of a typical BWR with internal recirculation pumps.

Figure 1: Overview of a BWR with internal recirculation pumps.[1] 1 INTRODUCTION 9

1.2 The Reactor, Safety and Auxiliary Systems The reactor pressure vessel is shown in Figure 2. This shows the most important parts of the reactor and the vessel. In the feedwater inlet the water enters from the primary coolant loop, as shown in the overview in Figure 1. The same overview shows where the steam leaves the RPV through the steam outlet.

The feedwater flows through the downcomer and is pumped through the reactor core, composed of fuel assemblies, by recirculation pumps. The recirculation pumps shown in Figure 2 are of the internal type. After the water has been heated enough to boil, the steam generated will pass through the steam separators and steam dryers in order to remove water. This is done since liquid water can damage the . Then the steam leaves through the steam outlet.

Figure 2 also shows the control rods. They are located at the bottom of the RPV and are inserted/withdrawn vertically upwards/downwards in order to regulate the reactivity in the core. In Figure 2, the illustrated control rod is inserted. The control rods are of a cross shape when viewed from above, as shown in Figure 3.

In the case of loss of off-site power the recirculation pumps can be powered by energy stored in a flywheel, enabling runback. There is one flywheel for each of the four pairs of recirculation pumps. Each one of these is placed so that a failure of flywheel will not affect another flywheel or other safety systems. [2] If the flywheels are not accounted for in the safety analysis, then the design of the core (i.e. the placement of the fuel assemblies, see Figure 3 and Figure 4) needs to be different in order to fulfill the safety criteria. The core needs to have more new fuel in order to keep the power level evenly spread out over the core. There cannot be large local variations in power. This is less economical since not all fuel can be used optimally.

Before cycle 34, which started with the refueling of F2 in July 2015, The Swedish Safety Authority (Str˚als¨akerhetsmyndigheten (SSM)) have not allowed the inclusion of the flywheels in the safety analysis. Thus the core designs have had different design conditions until this cycle. 1 INTRODUCTION 10

Figure 2: The reactor pressure vessel.[3] 1 INTRODUCTION 11

Figure 3: Fuel assembly example. [4] The Forsmark reactors contain a similar but not identical design for their fuel assemblies.

Figure 4: Overview of a reactor core.[5] 1 INTRODUCTION 12

1.3 BISON - Analysis Program for BWR BISON is a one dimensional dynamic analysis program for BWR originally developed by ASEA-ATOM, and is currently distributed and developed by Westinghouse. BISON is used for analyzing transients in the reactor. The modeling in BISON starts with with the feedwater pumps and ends with the turbine control valves. It contains models for the different parts in the RPV, such as steam separators etc. (see Figure 2), and detailed models of various safety relief valve systems and control systems. It also has a simple model for the feedwater system. BISON models both thermal-hydraulics and neutron kinetics. The neutron kinetics are explained further under 2.3.1 The Equation.[6]

Some of the important physical processes which take place in these systems are thermal- ization, diffusion and absorption of in the core, with or without the influence of control rods. The flow of the steam from the RPV, through the turbine system or the safety and relief valves. The flow of the condensate and feedwater in their respective systems. These processes can be modeled in BISON.

There are of course several ways in which BISON performs approximations in order to perform accurate computations at a reasonable time. One example is that all flow paths are considered to be strictly one dimensional. This approximation is motivated because the flow is close to being one dimensional in most sections of the coolant loop, with exception of the bottom of the lower plenum and in the liquid portion of the upper plenum. [6]

1.4 Previous Results and Performed Tests This section covers the relevant tests performed from March to May in 2013 and the re- sults of the previous model. There were eleven tests performed which were relevant for the testing of the BISON model. These tests show that there is room for improvement in the modeling of partial SCRAM in BISON. Out of these eleven test there were three in which partial SCRAM occurred in both the actual test and the previous model. Thus these are the only measurements that can be used to verify the results of the BISON model for partial SCRAM with the uprated reactor. The tests were loss of condensate pump, turbine trip, and house load operation. [7]

A comparison of the previous model and the measurements can be seen under5 Results for the loss of condensate pump case. The other two cases are very similar and the results are analogous, see Appendix A and Appendix B. In each of these three cases the previous model underestimates the reactor power after partial SCRAM. As a consequence, the model also underestimates the recirculation flow and steam flow. It is clear that the partial SCRAM modeling needs improvement. 1 INTRODUCTION 13

1.5 Aim of the Study The aim is to improve the modeling of partial SCRAM in the plant model in BISON for the Forsmark 2 nuclear reactor after power uprate. Validation of the BISON model against tests performed from March to May in 2013 have shown that this is one of the areas in which the model has room for improvement when compared to the tests. After partial SCRAM is performed, the model underestimates the reactor power, recirculation flow and steam flow when compared to the measurements. In BISON the partial SCRAM is modeled using a relative control rod effect vector. The aim is to replace the old values in this vector to improve the model. 2 THEORY 14

2 Theory

This section explains several types of transient, the SCRAM system and how SCRAM is modeled in BISON. For the SCRAM modeling in BISON it covers the neutron transport equation and the ASC vector.

2.1 Transients In general the term transient is used to indicate time dependence or fast change. Here, however, the term takes on a more specific meaning. It is defined as an event which leads to an imbalance in supplied and removed heat from the reactor core (with the exception of loss of coolant accident). Usually these are in the order of a few seconds to a few minutes.

Transients can be grouped into the following categories: [8][9]

1. Pressure increase.

2. Pressure decrease.

3. Feedwater flow increase or feedwater temperature decrease.

4. Feedwater flow decrease.

5. RF increase.

6. RF decrease.

However, it is worth to note that these categories are not precise because a transient can sometimes fit into more than one of these categories at the same time. Transients also change over time since there will be a response from the safety and control systems. The effect of the transient can also depend on the time scale involved. Some of the slower transients might take minutes while some faster take seconds. Thus a transient can for example be viewed as pressure increase if the time frame is only a few seconds. But it might at the same time lead to a reduction of feedwater temperature in a few minutes.

The test case of loss of condensate pump falls into the 4th category of feedwater flow decrease because it will eventually lead to a decrease in feedwater. This case is actually sometimes referred to as ”loss feedwater or condensate pump” since they are very similar.

The general description is as follows, feedwater flow decrease leads to a lower water level in the RPV. Since the feedwater is subcooled, the feedwater flow decrease implies less subcooling of the moderator. Thus there will be more water closer to the saturation tem- perature. The void increases and reactivity decreases.

This type of transient can sometimes be handled by the water level controller. However, if it is not able to counteract transient by changing the water level reaching a stable op- eration point, then runback of the recirculation pumps is activated. This is followed by partial SCRAM. The decrease in reactivity and the decrease in the speed of water level reduction can be enough to reach a stable operation point. If it does not, then other sys- tems would pump water into the RPV and SCRAM would be activated. This would lead to completely shutting down the reactor. For the test case runback is activated directly before the water level is affected by the loss of condensate pump, then partial SCRAM 2 THEORY 15 follows.

Turbine trip falls into 1st category of pressure increase and the 3rd of feedwater flow increase or feedwater temperature reduction. That is because it takes on both character- istics, in the short term (in order of seconds) it will see a pressure increase. But in the longer term (in order of minutes) it will have a feedwater temperature reduction as well.

Pressure increase transients are caused by a decrease in the extraction of steam from the RPV. During turbine trip the steam extraction is prevented by the closure of turbine control valves. The increase in pressure will cause increase in subcooling and decrease in void. Thus the reactivity increases.

The feedwater temperature reduction occurs since the turbine trip will stop the heating of the feedwater. The feedwater is heated in pre-heaters before entering the RPV. It takes a few minutes for the colder water to enter the RPV however, which is why this is a slow transient. The lower temperature gives a higher subcooling in the core, decreasing the void and increasing the reactivity. The increase in reactivity can be stopped by partial SCRAM or SCRAM depending on the severity of the transient.

House load operation falls into the 6th case of pressure increase. When the house load operation occurs, the reactor only supplies power to the plant itself. Therefore the power (i.e. the reactivity) must be decreased, in order to protect the turbines. Because if the connection to the grid is lost and the reactor produces steam at the level of full power, then the turbine will speed up leading to turbine trip. The power decrease is achieved by runback of the recirculation pumps and partial SCRAM. Runback causes increase in void and thus decrease in power. Partial SCRAM reduces the power through the neutron absorption of the control rods.

2.2 The SCRAM System The SCRAM system is used for complete shutdown (SCRAM) or for a power reduction called partial SCRAM. The systems consists of controls rods which can be hydraulically inserted into the core of the reactor. [10]

The control rods affect the reactivity in the core by absorbing neutrons. Thus the in- sertion of control rods leads to a reduction in neutrons in the core that are causing the , decreasing the reactivity. Withdrawal will in the same manner increase the reactivity. The material used is mostly carbide (B4C), containing the highly neutron absorbing B-10. [11]

The control rods are divided into control rod groups referred to as SCRAM groups or SCRAM banks. The insertion of these groups is how the system is operated. For SCRAM all groups (and consequently all rods) are inserted completely in a few seconds. For partial SCRAM only one or a few of the groups are inserted, as the name implies. This leads to a fast and large reduction in reactivity, and thus in power, without shutting down the reactor. Thus if the reactivity does not need to be reduced as much it has the advantage over SCRAM that the rector can remain in operation. Then a stable operation point can be reached. 2 THEORY 16

In F2 the SCRAM system consists of 161 control rods which are divided into 18 groups consisting of 9 rods each, except for one group of 8 rods. Partial SCRAM consists of two groups, one of 8 rods and one group of 9, making the total 17 rods. [12]

The SCRAM system should not be confused with the fine motion control rod drive system. This system uses the same control rods but it is physically separated from the SCRAM system and it is of another design and has another purpose. The rods are inserted and withdrawn using an electric system. [10] This system is used to control the power level, the power distribution, and to compensate for the long term reactivity changes due to burn up of the fuel. [11] This system can also shutdown the reactor, however the minimum insertion time is 4 minutes rather than a few seconds as is the case of the SCRAM system.[10]

2.3 SCRAM Modeling in BISON 2.3.1 The Neutron Transport Equation To understand how SCRAM is modeled, the neutron transport modeling needs to be shown first. In BISON it is modeled using the 2-group diffusion equations with time dependence and one dimension in space. 2-group means that the neutrons are divided into the two groups: group 1 (fast) and 2 (thermal) depending on whether their energy is high or low. Please note that it is not essential to understand this complicated set of equations in detail in order to follow this study. It is included in order to give a formal definition of the ASC vector and to give a complete picture of the SCRAM modeling. The equations have the following form [6]

m 1 ∂φ1 1 X = ∇·D∇φ −D B2φ −Σ φ + (νΣ (1−β )φ +νΣ (1−β )φ )+ λ C −Σ φ v ∂t 1 1 r 1 A1 1 k F 1 1 1 F 2 2 2 i i R1 1 1 eff i=1 (1)

1 ∂φ2 2 = ∇ · D∇φ2 − D2Br φ2 − ΣA2φ2 + ΣR1φ1 (2) v2 ∂t where the equations for the delayed neutrons are

∂Ci 1 = −λiCi + (νΣF 1β1iφ1 + νΣF 2β2iφ2) (3) ∂t keff (i = 1, ..., m) 2 THEORY 17

The variables in the equations are the following

φ1 = Neutronflux in group 1

φ2 = Neutronflux in group 2

Ci = Precursor density for group i of delayed neutrons

λi = decay constant for group i of delayed neutrons m = number of groups of delayed neutrons (6 groups are included)

β1i = fraction of delayed neutrons in group 1 from fast fission

β2i = fraction of delayed neutrons in group 2 from thermal fission m X β1 = β1i i=1 m X β2 = β2i i=1 2 Br = equivalent radial buckling

1/keff = first eigenvalue for equations (1) through (3) in the steady state (time derivatives = 0).

The code iterates to find 1/keff in the steady state and holds it constant in the transient.

v1, v2 = average velocities for the neutrons in groups 1 and 2.

D1,D2, ΣA1, ΣA2, ΣF 1, ΣF 2 and ΣR1 are diffusion coefficients and neutrons cross sections, respectively. ν = average number of neutrons released per fission.

2.3.2 Defintion of the ASC vector

The coefficients (macroscopic neutron cross sections D1,ΣA1 etc.) in the Equations1 to3 given in the previous section are calculated by 2-dimensional nodal code (softwares such as CASMO or PHEONIX) for two cases i) Cross sections in every axial node in BISON with control rods withdrawn (relatively low absorption cross section), Σunrodded. ii) Cross sections in every axial node in BISON with control rods completely inserted (high absorption cross section), Σrodded.

The insertion of a control rod group (such as a SCRAM bank) is then modeled by using the relative control rod effect vector (ASC vector) in the following way for each coefficient

Σ = (1 − ASC) · Σunrodded + ASC ∗ Σrodded (4) where Σ is a macroscopic cross section such as D1,ΣA1 etc. The ASC in the equation represents the element in the ASC vector corresponding to the control rod group being 2 THEORY 18 inserted.

In BISON the number of elements in the ASC vector is determined by the creator of the model. The number can be between 1 to 25. For the models used in this study it contains 5 elements representing SCRAM banks number 9, 6, 12, 18, and the remaining. Like this:

ASC = [A9,A6,A12,A18,Aremaining] (5) The first two are the banks being inserted during partial SCRAM, the second two can also be used for partial SCRAM but are not currently in use, the last element represents the rest of the SCRAM banks. The sum of the vector must be 1 since the sum of all elements corresponds to all control rods being inserted (SCRAM), as described in Equa- tion 4. Thus the individual elements in the vector can have values between 0 and 1, given the constraint that the sum of all elements is 1.

Lastly, it is important to note that if a SCRAM bank is inserted, all of its rods are completely withdrawn beforehand. This is the case for the two banks used in partial SCRAM for both the tests and the simulations. However, if this was not the case then it would be possible for BISON to take it into account.

2.3.3 The ASC vector in the Previous Model In the previous model the ASC vector is assumed to be proportional to the number of control rods inserted into the core. This gives the following vector

ASC = [8/161, 9/161, 9/161, 9/161, 126/161] (6) where each element contains the number of control rods in the group or groups it corre- sponds to, divided by the total number of control rods in order to get the sum of 1.

This assumption is equivalent to assuming that all control rods are the same and affect the core same. This is only approximative due to three main reasons. First of all the placement of the control rods matter. The neutron flux is not uniform in the core, it can be higher in the center of the core compared to its boundary for example. The fuel assemblies also have different burn up, affecting the neutron flux.

Secondly, there will be diminishing returns for each rod being inserted. Take for example the action of inserting one control rod alone. This rod will have a lot of neutrons to absorb. Now take the same rod and insert all the other rods except this one. Then after a while insert the rod. This time it will have a lot fewer neutrons to absorb since all the other rods already have absorbed many of the neutrons. Every time the number of rods inserted is increased the returns will diminish since the amount of available neutrons de- creases. However, by assuming that all rods have the same effect, the diminishing returns is not taken into account.

Lastly, there is also the fact that all the control rods ability to absorb neutrons changes with time. It depends on how many they have already absorbed. Control rods need to be changed with time just like the fuel of the reactor [11]. This and the other two factors are not considered with the assumption in the previous model. 3 MODEL AND SIMULATIONS 19

3 Model and Simulations

This section concerns the selection of the software for the simulations used to calculate the ASC vector. It also briefly covers the models in the software used for the simulations.

3.1 Simulation Software Selection There are several kinds of software available for simulating the changes in the core of the reactor. As described in 2.1 Transients there are several things going on in the core, the reactivity, pressure and temperature and other quantities changes and affect each other. Safety and control systems will also interact with the core. Thus software can often sim- ulates different aspects such as thermo-hydraulics and neutron kinetics in more detail than the other, it might also choose to include more details about the safety and control systems or exclude them completely.

BISON performs dynamic simulations of the core in 1D and includes many safety and control systems. It does not model the neutron kinetics in detail, the focus is more on thermo-hydraulics. Thus it will not give a very detailed picture of the core but it will give good estimates of pressure changes in pipes and similar types of changes.

POLCA performs steady state 3D simulations of the core, with focus on neutron kinetics, but does not include other systems in the analysis. It does not perform dynamic simula- tions.

Simulate-3K (Simulate 3D Kinetics) performs dynamic 3D simulations of the core. Sum- marily it can be described as: it models the core in the same amount of detail as POLCA but it models systems outside of the core in less detail than BISON does.

The choice of software for comparison with BISON was POLCA. The idea of using POLCA is that when it’s modeling partial SCRAM it could get a better estimate of the reactivity changes (and power changes) in the core.

However a brief comparison between the reactor powers given by BISON and Simulate-3K was also done, in order to see whether the choice of POLCA would have a huge impact on the results or not. Because POLCA will decide what power BISON ends up with this choice is very important.

The versions of the softwares used are BISON 6.9.4.1, POLCA 4.13.1, Simulate-3K 2.06.00.

3.2 BISON Models The BISON models used are the same as the ones which were used for the validation with some minor modifications. Instead of modeling the exact events which took place during the measurements, it only models the event of interest: partial SCRAM. The model sim- ulates what happens when the reactor goes from normal operation to partial SCRAM.

However, the most important difference is that another control rod model is used. Instead of using a proportional relative control rod density for each SCRAM group (based on the number of control rods in the group only), a calculated control rod density is used. 3 MODEL AND SIMULATIONS 20

The vector with these values can be changed. The models also includes the options of individual insertion of SCRAM bank 6 and 9 (which replaces PSS in such a case), changing the steam separator model, keeping the main circulation flow constant (no runback), and changing the core data (the configuration of the core changes every time the core is refueled).

3.3 POLCA Models POLCA is used in the following way: First the model takes the RF, feedwater temperature and reactor power as input given by the run of the BISON model. Then it performs an option called POWERSEARCH where the given RF and the feedwater temperature are held constant, and the given reactor power is an initial guess at what the power should be after PSS (or individual SCRAM bank insertion). POLCA then iterates on reactor power to keep the same effective multiplication factor as prior to the insertion. The given power is the output of the model and the value which the BISON model should correspond to.

The POLCA models need to be able to simulate the same cases as the BISON models if they are to be compared. Thus it can perform PSS, individual insertion of SCRAM bank 6 and 9, keeping the main circulation flow constant (no runback), and changing the core data. 4 METHOD 21

4 Method

This section describes the study in more detail. First how POLCA is used together with BISON in order to compute the new ASC vector, including both how the algorithm itself works and how the actual calculation of the ASC vector is done. Second, how the new BISON model is composed.

4.1 Computation of ASC vector The computation of the ASC vector is done with an algorithm described in Figure 5. This algorithm calculates the ASC vector for each point of burn up in the operating cycle. It is done in the same way for every point.

This is algorithm falls under the general optimization algorithm of the form: [13]

1. Specify some initial guess of the solution x0. 2. For k = 0, 1, ···

i) If xk is optimal, stop.

ii) Determine xk+1, a new estimate of the solution. But in Figure 5 it is described in more detail.

The first step 1 simply includes an initial guess of the ASC vector. If this is done for the first burn up point then the guess will be the proportional assumption, see Equation 6. But for all subsequent points the guess will be the ASC vector of the previous point. This is done on the assumption that this guess is better and thus will save some computation time since fewer iterations will be needed if the guess is good. 1 In step 2 the iterations start with a loop-structure, described by roman numerals.

The next step i) is to insert the ASC vector into the BISON model, i.e. the BISON script. This model is described under 3.2 BISON Models. Then the model runs in step ii) and returns results in step iii). The results for the RF, feedwater temperature and reactor power are extracted.

In step iv) POLCA runs with the RF and feedwater temperature given by BISON. This is needed in order to make them simulate the same scenario. The POLCA model is described under 3.3 POLCA Models. In step v) the extracted power from POLCA is compared with that of BISON. If they are the same within a tolerance of 0.1 % then the algorithm ter- minates and we have calculated the ASC vector. If on the other hand the power from POLCA and BISON are not within the tolerance then the algorithm continues to step vi).

In step vi) a new guess for the ASC vector is calculated based on the previous ASC vector guesses and the power given by POLCA and BISON. How this is done is described in detail under 4.1.1 Calculate New ASC vector. It then precedes to the beginning of the loop and the same procedure starts again but with a different ASC vector from last time.

1This also turned out to be the case. The number of iterations was about 3-5 with the proportional assumption as the initial guess, but with the ASC vector of the previous point it was about 1-3. 4 METHOD 22

Figure 5: Algorithm for calculating the ASC vector. This is performed for each point of burn up in the operating cycle.

4.1.1 Calculate New ASC vector This is done with a MATLAB script which extrapolates or interpolates based on the val- ues of the ASC vector and the power given by the POLCA and BISON runs.

When the first guess is done the script only has one data point for the ASC vector value, the initial guess against the difference between power from POLCA and BISON. Thus it is not possible to interpolate to guess the next answer. Instead the relation between the reactor powers given by POLCA and BISON is used. If for example the power given by POLCA is higher than that given by BISON, then that means that BISON underesti- mates the power. Thus the ASC vector needs to be lower, since a lower value implies a higher power. The script then divides the value by two.

If the relation between the power from POLCA and BISON would be the other way around, then the value should be increased instead. This is done by multiplying by two. The reason that the value increases (or decreases) by a factor of 2 is to give a an overes- timation or underestimation of the ASC vector. Then there will be values to interpolate between. If the factor was smaller (or a small number was added or subtracted) then there would be the risk of having to iterate many times before interpolation could replace extrapolation. 4 METHOD 23

In Figure 6 there is an example of how the values of the reactor power and the ASC changes as iteration proceeds. The process described in the previous paragraph is shown. First the value underestimates the reactor power, then in the second iteration it overestimates it a lot. However, after the interpolation starts in the third iteration, the changes to the value of the ASC (and thus the reactor power) is much smaller and the values is fine-tuned.

Figure 6: Example of how the ASC and the reactor power given by BISON changes with each iteration.

Because then there will be two values between which interpolation can be done. If the extrapolation is done in too small steps a lot of computations can be wasted in order to get to the other side of the sought value. It is when the sought value lies between data points which interpolation can be used. This value is assumed to be where the difference between the reactor power of POLCA and BISON is zero, they have the same value. The value of the ASC vector for which this is true is chosen to be the next guess.

The interpolation is done using the spline function in MATLAB since the shape of the curve is unknown and thus a polynomial of a certain degree cannot be chosen. The ex- trapolation is done by using a simple first order polynomial since the spline function can give strange values outside the data points, it relies on piecewise polynomials.

Lastly, it is important to note that the interpolation or extrapolation of the ASC values is done on a single element for the case of individual SCRAM bank insertion, since only one element is used. However, for partial SCRAM two elements are used. Thus the interpolation or extrapolation is done on the sum of the two elements, since BISON uses the sum of them in this case. 4 METHOD 24

4.2 Composition of the New Model The new model is simply the previous model with a new ASC vector. Thus when evalu- ating this model against the three test cases the ASC vector is the only thing that needs to be changed. The models are kept the same in all other aspects, in contrast to the ASC calculations where several things such as point of burn up etc. were changed.

The important question is how the ASC vector should be chosen. Due to the large number of different conditions for which the ASC was calculated there are hundreds of different choices. The conditions which clearly should apply are PSS and runback, since both of these occurred during the tests. Cycle 32 is chosen because it is more similar to cycle 31, when the tests were performed, than cycle 34. The core designs of cycle 31 and 32 do not take the flywheels into account, while 34 does. The mean of all points of burn up is chosen, so that no one point will have a large effect on the result. This gives the ASC vector shown in Table 1.

Table 1: Values of the ASC vector. The new value is the calculated value, the previous value is the value with the proportional assumption. ASC groups 9 6 12 18 remaining New value 0.0187913 0.0440809 0.0440809 0.0440809 0.848966 Previous value 0.0497 0.0559 0.0559 0.0559 0.7826 5 RESULTS 25

5 Results

5.1 ASC vector Results In this section the results of the ASC vector calculations are presented. If the ASC vec- tor is given by the proportional assumption, see Equation 6, then it is shown as dotted line. The steam separator type is AS01, as in the previous model, unless stated otherwise.

At PSS the ASC vector is presented as the sum of the two first elements. For individual SCRAM bank insertion the element corresponding to the SCRAM bank is presented. In some of the points of burn up the BISON calculations did not converge2, therefore they are missing in the figures.

Figure 7: The ASC vector for PSS during Cycle 32 and 34 at the burn up of 0 to 7000 EFPH.

2This convergence issue was due to the default initial guess of the burn up which does not work for some points of burn up. While this could easily be changed manually for individual cases of convergence issues. It wasn’t deemed important enough to correct since the number of points affected is low and the scripts would need to be changed for this to be automated. The points affect are in cycle 32 2500 and 3500 EFPH, and in cycle 34 it is 0 EFPH. 5 RESULTS 26

Figure 8: The ASC vector for PSS with and without runback during cycle 34 at the burn up of 0 to 7000 EFPH. 5 RESULTS 27

Figure 9: The ASC vector for individual insertion of SCRAM bank 9 and 6, the sum of the ASC vectors for the individual insertions, and PSS. The values are for cycle 34 at the burn up of 500 to 7000 EFPH. Note that there are three black lines representing the proportional assumption for PSS, individual insertion for SCRAM bank 6, and for 9.

5.2 Comparison with Previous Results The most interesting results for evaluation are APRM, RF and steam flow. These are shown for the case of loss of condensate pump in figure 10, 11, and 12. In these figures the measurements are shown together with the previous and the new model. The results for the turbine trip case and the house load operation case are very similar. For this reason they aren’t included here. But they can be viewed in Appendix A where the same figures are included for turbine trip, and in Appendix B for house load operation.

All three figures show that the new model produces results closer to the measurements. In figure 10 the APRM for the new model is much closer than in the previous model, although it is still a bit below the measurements. The same is true for the RF, figure 11, while the steam flow is a bit higher than the measurements, figure 12. 5 RESULTS 28

Figure 10: APRM (power) in % during loss of condensate pump. The red lines are the measured APRMs (four different measurements), black is the previous model, and blue is the new model.

Figure 11: Recirculation flow in kg/s during loss of condensate pump. The red line is the measured RF, black is the previous model, and blue is the new model. 5 RESULTS 29

Figure 12: Steam flow from reactor pressure vessel in kg/s during loss of condensate pump. The red line is the measured steam flow, black is the previous model, and blue is the new model. 5 RESULTS 30

5.3 Comparison of BISON and Simulate-3K Calculations The BISON calculations are compared with some calculations done in Simulate-3K. The case used for comparison was PSS with runback in cycle 32, and with steam separator type AA. The Simulate-3K model uses a correlation equivalent to the AA type, so BISON needs to use AA in this case for the comparison to be valid. In these calculations the core data from POLCA has been converted for use in Simulate-3K, thus the programs use the same core data (since BISON uses the same as POLCA). The points of burn up differ a bit however, since they are shifted by +250 EFPH relative to the points of burn up in POLCA and BISON (except the last step where it is only +105 EFPH). In figure 13 the comparison is shown.

Figure 13: Comparison of the power given by BISON and Simulate-3K, for the case PSS with runback during cycle 32, and with steam separator type AA 6 DISCUSSION 31

6 Discussion

It is very clear from Figure7,8 and9 that all results for the ASC vector are smaller than the previous values of the proportional assumption. This means that the reactor power after insertion of the control rods will be higher than in the previous model. Therefore the new model gives improved results when evaluated against the measurements in Figure 10, 11 and 12. The same results for the other test cases can be seen in Appendix A and Appendix B.

The results from Figure7,8 and9 also show that the proportional assumption of the ASC vector is flawed. It is not possible to assume that the relative effect of the control rod insertion changes linearly with the number of control rods inserted. This is apparent in Figure 9 where it is shown that the individual insertion of SCRAM bank 6 has a much larger effect than the insertion of 9, although it only contains one more control rod. Fur- thermore, the sum of the individual insertion of SCRAM bank 6 and 9 is not the same as the value given for their simultaneous insertion, i.e. PSS, even though the number of rods is exactly the same.

It is therefore clear that the proportional assumption will not yield accurate results for the relative control rod effect. Instead it needs to be calculated from core simulations in order to give a better result, as shown when compared to measurements in figure 10, 11 and 12.

It should also be noted that the results for the RF and steam flow can be improved further by changing the steam separator model. This has been shown in a study performed at Forsmark [14]. Although this falls outside the scope of this study and is therefore not explored further.

The comparison of the power given by the computations in BISON and Simulate-3K, in Figure 13, shows that the results are close to each other. Although the points of burn up aren’t exactly the same, they follow the same trend where the difference between the programs is about the same for the whole cycle. Simulate-3K gives a higher value for the power during the whole cycle, however only about 2 %.

This is good result since this indicates that the choice of POLCA as a reference for the computation of the ASC vector shouldn’t have a too large effect on the results, compared to if Simulate-3K had been used. While the simulation tools will never precisely model reality, it is much less likely that they are incorrect if two of them give similar results. Then the calculations does not solely rely on one of the softwares. It is also notable that the average difference in power is only about 2 %. Because during safety analysis, the APRM values should be assumed to be either 2 % higher or 2 % lower than full power, with the choice being the alternative which would have the most negative effect on safety. This is done in order to account for instrument errors etc. [15] Thus the result of 2 % difference between the softwares can be said to close to each other.

The choice of the ASC vector for future simulations should probably be different from the ASC vector used during the evaluation in this study. This is because ASC vector varies between cycles, as shown in Figure 7. Furthermore the flywheel is accounted for in the safety analysis from cycle 34 and onward. This will make the core design look a bit 6 DISCUSSION 32 different. Besides, there are many other reasons the design might change. Such as change in various safety criteria, such as CPR, or change in operation length of the cycle.

However, if no new value is calculated, it is recommended that the minimum value of the ASC vector for cycle 34 is used. Cycle 34 should be used instead of 32 due to the flywheel being included, and the minimum value results in a higher power after PSS, which is a more conservative assumption. However, what is conservative might depend on the scenario analyzed, in general PSS is used to significantly reduce power. Therefore it would in general be better from a safety perspective to have a higher control rod effect, and a lower would be more conservative.Thus the vector in Equation 7 could be used in the future. 3

ASC = [0.0187913, 0.0440809, 0.0440809, 0.0440809, 0.848966] (7) Note that the elements for SCRAM bank 12 and 18 are the same as for 6, this is only because they need a value. These values have not been evaluated and cannot be assumed to be correct.

6.1 Future Work There are several possibilities for future studies on this topic. One would be to perform the same or similar calculations of the ASC vector for other operating cycles. Then the conclusions in this work could be confirmed and expanded on. It might shine more light on how important it would be to perform these calculations for each cycle. There is a lot of time that can be saved if the amount of ”cycle specific” analyses are limited.

There is also the possibility of doing the same study for other reactors, such as Forsmark reactor 3. The design of this reactor is similar but still a bit different from F2. F1 is very similar to F2, so a comparison between them might not be as interesting. It would be interesting to see if the same conclusions can be drawn for F3 as for F2. The design of the reactor could be important for the ASC vector. It is very likely that the proportional assumption will yield disappointing results in this case too. However, it might be the case that the difference between the proportional assumption and the calculated ASC vactor is much larger or much smaller for F3 than for F2. If so, then there is room for a lot more to study in order to find the cause for this.

Finally there is also the possibility of improving the algorithm and method itself. There might be some improvements which could give results even closer to the measurements or there might be some way of making it more efficient. The latter could improve the ease of usage in implementing the method as a standard, if progress is possible in that area.

Westinghouse has announced that BISON 6.10.0 will include an additional feature for the SCRAM modeling, called FCR. [16] This would change Equation 4 to Equation 8.

Σ = (1−ASC0)·Σunrodded +ASC0 ·Σrodded +FCR·(ASC −ASC0)(Σrodded −Σunrodded) (8)

3The distribution of the ASC value on the elements for SCRAM bank 9 and 6 was chosen as:

ASCPSS · ASCindiv9 ASCPSS · ASCindiv6 ASC9 = , ASC6 = ASCindiv9 + ASCindiv6 ASCindiv9 + ASCindiv6 6 DISCUSSION 33

where ASC0 is the steady state ASC. FCR is the nodal SCRAM reactivity multiplier. If FCR = 1 then Equation 8 turns back into Equation 4. This feature would be interesting to explore since it wasn’t available at the time of this study. 7 CONCLUSIONS 34

7 Conclusions

The assumption that the ASC vector is proportional to the number of inserted control rods is flawed. It does not vary linearly with the number of rods. To get a more accurate Partial SCRAM the control rod effect should instead be calculated by core simulations.

The flywheels will be taken into account starting with cycle 34. Thus it is recommended that new ASC values should be calculated for future cycles, or that the values for cycle 34 are used. This is important since the core design influences the values.

Instead of the mean value, the smallest value should be used. It is more conservative since a smaller value results in a higher power after Partial SCRAM.

There are a few avenues available for future study. The same or similar studies to this one can be performed for more cycles or for other plants. The algorithm used in this study might have room for improvements, future studies could look for these. The FCR feature, explained in 6.1 Future Work, could also be studied. It was not available during this study and thus could not be taken into consideration. References

[1] K¨arnkrafts¨akerhet och Utbildning AB (KSU). Gemensam reaktorutbildning, del 1, 2010. Image source. In Swedish, translation of image performed by the thesis author.

[2] Forsmarks Kraftgrupp. Forsmark 2 - S¨akerhetsrapport, System 649 - Frekvensom- riktare flr HC-pumpar inkl transformatorer. Technical report, 2013. In Swedish.

[3] K¨arnkrafts¨akerhet och Utbildning AB (KSU). GK3 Reaktorsystem, 2013. Image source. In Swedish, translation of image performed by the thesis author.

[4] MIT OpenCourseWare. 2D view of four bundle module, 8 x 8 lattice, with control blade. http://ocw.mit.edu. Image source.

[5] Nejdet Erkan. Typical Control Cell Core Lay. Lecture in Nuclear Plant Engineering at The University of Tokyo. Image source.

[6] H. Svensson. BISON - A One Dimensional Dynamic Analysis Code for Boiling Water Reactors. Technical report, ABB Atom, 1991.

[7] Marielle Perup Lars Furedal. Validering av anl¨aggningsmodell f¨orBISON av Fors- mark 2, 3253 MWt. Technical report, Westinghouse Electric Sweden AB, 2014. In Swedish.

[8] Michael Timm Dag Ribbing. Forsmark 1/2, Underlag till referensrapport i s¨akerhet- sredovisningen, Transienter och haverier, Huvudmetodikrapport, Metodik. Technical report, Westinghouse Atom AB, 2000. In Swedish.

[9] Anna Aspman. Forsmark 3 - 3775 MWt, Huvudmetodikrapport f¨orverifiering av br¨ansletsoch RCPB:s integritet vid transienter. Technical report, Westinghouse Electric Sweden AB, 2011. In Swedish.

[10] Swedish Nuclear Power Inspectorate (SKI). St¨orningshandboken - BWR. 2003. In Swedish.

[11] Sten Lundberg. Praktisk reaktorfysik. Lund University, 1986. In Swedish.

[12] Forsmarks Kraftgrupp. Forsmark 2 - S¨akerhetsrapport, System 532 - Man¨ovrering och indiktering av styrstavar. Technical report, 2014. In Swedish.

[13] Ariela Sofer Igor Griva, Stephen G. Nash. Linear and Nonlinear Optimization. So- ciety for Industrial and Applied Mathematics, second edition, 2009.

[14] Anghel Ionut. Forsmark 2 - Anl¨aggningsspecifika ber¨akniningsf¨oruts¨attningarvid best¨amning av torrkokningsgr¨ansv¨ardeninf¨orcykel 34 (RA15). Technical report, Forsmarks Kraftgrupp, 2014. In Swedish.

[15] Forsmarks Kraftgrupp. Forsmark 2 - S¨akerhetsrapport, Allm¨andel kapitel 4 - S¨aker- hetskrav. Avsnitt 4.4 - Reaktors¨akerhetsstyrda konstruktionskrav. 4.4.5 USNRC Reg- ulatory Guides, Division 1. Technical report, 2015. In Swedish.

[16] Carl Hals. BISON 6.10.0, News in coming release. In BISON Users Group 2015, 2015. A Comparison with Previous Results: Turbine Trip Case

The case of turbine trip is shown in figure 14, 15, and 16. In these figures the measure- ments are shown together with the previous and the new model. All three figures show that the new model produces results closer to the measurements. In figure 14 the APRM for the new model is much closer than in the previous model, although it is still a bit below the measurements. The same is true for the RF, figure 15, while the steam flow is a bit higher than the measurements, figure 16.

Figure 14: APRM (power) in % during turbine trip. The red lines are the measured APRMs (four different measurements), black is the previous model, and blue is the new model. Figure 15: Recirculation flow in kg/s during turbine trip. The red line is the measured RF, black is the previous model, and blue is the new model.

Figure 16: Steam flow from reactor pressure vessel in kg/s during turbine trip. The red line is the measured steam flow, black is the previous model, and blue is the new model. B Comparison with Previous Results: House Load Operation Case

The case of house load operation is shown in figure 17, 18, and 19. In these figures the measurements are shown together with the previous and the new model. All three figures show that the new model produces results closer to the measurements. In figure 17 the APRM for the new model is much closer than in the previous model, although it is still a bit below the measurements. The same is true for the RF, figure 18, while the steam flow is a bit higher than the measurements, figure 19.

Figure 17: APRM (power) in % during house load operation. The red lines are the measured APRMs (four different measurements), black is the previous model, and blue is the new model. Figure 18: Recirculation flow in kg/s during house load operation. The red line is the measured RF, black is the previous model, and blue is the new model.

Figure 19: Steam flow from reactor pressure vessel in kg/s during house load operation. The red line is the measured steam flow, black is the previous model, and blue is the new model. C Individual Control Rod Study

C.1 Introduction This study was conducted in a very different fashion than the main study. The attempt was to compute the ASC vector independently of BISON by only using POLCA, instead of iteratively comparing BISON with POLCA. However these efforts were abandoned after unsatisfactory results. But it is presented here as it might gain some further insight into the workings of the ASC vector and why the main study was needed instead of this one.

C.2 POLCA Model The POLCA model used the POWER option instead of the POWSEARCH option. The control rods were inserted individually and not by group basis. Everything else in the model is kept constant: the power, recirculation flow, pressure, carry under, and feedwater temperature. The idea being that since we are only interested in the relative change of reactivity for the individual rods, and not the absolute, any error due to the simplification of the model will be negated.

C.3 BISON Model The same as in the main study, see 3.2 BISON Models.

C.4 MATLAB Scripts There were three different MATLAB-scripts, one for creating the POLCA-scripts, one for running them, and one for calculating the ASC vector and plottning the results.

The first script creates a POLCA-script with the specifications given under C.2 POLCA Model, only inserting one rod for each run. After the POLCA scripts have been exe- cuted using the second script, the third script calculates the ASC vector. This is done by extracting the difference in reactivity before and after the insertion of the rod. These reactivity changes are then added together by SCRAM bank basis, to get the contribution from each SCRAM bank. Each bank is then normalized with respect to the sum of the reactivity change for all rods.

Even if the absolute numbers for each rod contain some error, we are only interested in the values of the rods relative to each other. Thus with the normalization we should expect the errors of the POLCA computations not to be a big issue.

C.5 Results In table Table 2 the results are displayed. One comparison with the previous model is done in figure Figure 20.

Table 2: Values of the ASC vector. The new value is the calculated value of the individual control rod study, the previous value is the value with the proportional assumption. ASC groups 9 6 12 18 the rest New value 0.04722 0.06000 0.05139 0.06342 0.77798 Previous value 0.0497 0.0559 0.0559 0.0559 0.7826 Figure 20: APRM (power) in % during loss of condensate pump. The red lines are the measured APRMs (four different measurements), black is the previous model, and blue is the new model with the new values in Table 2.

C.6 Discussion and Conclusions The results are very similar to that of the previous model, there is no improvement. Note that the sum of the values in group 6 and 9 for the new values is 0.10722 compared to 0.10560 for the proportional assumption. In contrast to the same value in the main study used for evaluation which is 0.06287, see Table 1. In the main study the difference is much larger.

However there are some interesting observations from these results. It appears that the assumption that the contributions of the rods could be added up independently to form a bank is not accurate. The ASC vector does not appear to change linearly with the increase or decrease of number of rods. Thus adding the individual reactivity decrease contributions cannot be done to get the reactivity decrease contributed by a whole bank. These calculations gave the same results as simply assuming that the ASC vector can be determined by simply adding the number of rods by the total number of rods. This is essentially the same assumption, they both assume linearity.

This study gives good grounds for taking another approach to calculating the ASC vector. It gave rise to the main study of this problem because this simpler approach did not work.