2.1 Illustrations of symbols used in engineering diagrams. . . 7 2.2 Illustrations of power systems as engineering diagrams. . . 7 2.3 Plot to illustrate a simplified representation of Britain’s power system. 11 2.4 Plot to illustrate Britain’s power system with random generator
avail-abilities. . . 13 2.5 Graphs to demonstrate how mean constraint costs vary as simulator
inputs are varied. . . 17 2.6 Graphs to demonstrate how mean constraint costs vary as installed coal
capacity and coal availability probability are varied for three demand levels. . . 18 2.7 Graphs to demonstrate how mean constraint costs vary as simulator
inputs are varied for a larger curtailment cost. . . 21
3.1 Graph displaying the structure of zones and boundaries of the power system used in this thesis. . . 34 3.2 Plot to show how mean annual constraint costs vary with power system
background year. . . 47 3.3 Plots to show how mean annual constraint costs vary with power system
background year as the transmission capacity of particular boundaries is raised to infinity. . . 48 3.4 Plot to show how mean annual constraint costs vary with power system
background year when considering taking the transmission capacity of the B6 and B7a boundaries to infinity. . . 50
xxii
3.5 Graphs of how the sum of mean constraint costs across all 20 years varies as nuclear and new nuclear availability probabilities are varied for two different levels of peak demand. . . 52 3.6 Graphs of how mean constraint costs vary in year 1 as nuclear and CCGT
availability probabilities are varied for 2 different peak demand levels. . 54 3.7 Graphs of how mean constraint costs vary in year 6 as nuclear and CCGT
availability probabilities are varied for 2 different peak demand levels. . 55
4.1 Boxplots to show how evaluations of constraint costs from the simulator vary each time the simulator is run for a fixed power system background. 58 4.2 Graphs to illustrate a load duration curve. . . 61 4.3 Graphs to show how mean constraint costs vary by day for two power
system backgrounds. . . 62 4.4 Boxplots to illustrate the variation of work done (λ, in terms of
equiv-alent full simulator evaluations) to reach an estimate of mean annual constraint costs to an accuracy of 1% for a variety of choices of impor-tance sampling weights. . . 76 4.5 Plot to illustrate how error in the estimated response varies with work
done for one particular repetition of a year 6 power system background, using mean snapshot constraint costs as the basis of importance sam-pling weights. . . 78 4.6 Boxplots to illustrate the variation of work done (λ, in terms of
equiv-alent full simulator evaluations) to reach an estimate of mean annual constraint costs to an accuracy of 1% for a variety of choices of impor-tance sampling weights. . . 81 4.7 Boxplots to illustrate the variation of work done (λ, in terms of
equiv-alent full simulator evaluations) to reach an estimate of mean annual constraint costs with an error less than £100,000 for a variety of choices of importance sampling weights. . . 85
List of Figures xxiv
4.8 Plots to compare estimates of mean annual constraint costs from impor-tance sampling to ˆµ6 for a year 6 power system background, when it is required that the standard error of the estimate is less than 1% of the mean of the estimate. . . 89 4.9 Plots to compare estimates of mean annual constraint costs from
impor-tance sampling to ˆµ6 for a year 6 power system background, when it is required that the standard error of the estimate is less than £100,000. . 92 4.10 Boxplots to illustrate the variation in simulated constraint costs for 1000
simulations for 10 snapshots for a year 1 power system background. . . 97 4.11 Plot to compare importance sampling weights for a year 1 and year 6
power system background, when basing the weights on the mean con-straint costs of each snapshot. . . 98 4.12 Boxplots to illustrate the variation in simulated constraint costs for 1000
simulations for 10 snapshots for a year 6 power system background. . . 99 4.13 Boxplots to illustrate the variation of work done (λ, in terms of
equiv-alent full simulator evaluations) to reach an estimate of mean annual constraint costs to an accuracy of 1% for varying numbers of initial simulations to estimate weights. . . 103 4.14 Boxplots to illustrate the variation of work done (λ, in terms of
equiv-alent full simulator evaluations) to reach an estimate of mean annual constraint costs to an accuracy of 1% for varying amounts of initial simulations to estimate weights. . . 106 5.1 Plots to illustrate how to take Latin hypercube samples. . . 122 5.2 Plot to illustrate a poor Latin hypercube sample. . . 123 5.3 Plots to compare a Latin hypercube sample to a grid of points. . . 124 5.4 Plot to show how total costs calculated using the simulator vary with
peak demand magnification and B15 reinforcement magnitude. . . 128 5.5 Plots to show how estimates of total costs from the fitted emulator
models vary with peak demand magnification and B15 reinforcement magnitude. . . 133
5.6 Plots to show how differences between calculations from simulation and estimates from the fitted emulator models vary with peak demand mag-nification and B15 reinforcement magnitude. . . 135 5.7 Plot to show how estimated expected total costs vary with B15
rein-forcement magnitude. . . 137 5.8 Plots to compare credible intervals for estimates of total costs from the
fitted emulator to the calculation of total costs from the simulator. . . . 138 5.9 Plot to show how estimated expected total costs vary with B15
rein-forcement magnitude and the assumed cost to reinforce. . . 141 5.10 Plot to show how estimated expected total costs vary with B15
rein-forcement magnitude and prior beliefs. . . 143 5.11 Plot to show how estimated expected total costs vary with B15
rein-forcement magnitude and prior beliefs. . . 144 5.12 Plot to illustrate how the loss function varies with power, p, and the
value it is applied to. . . 146 5.13 Plot to show how inverted expected losses vary with B15 reinforcement
magnitude and attitude to risk. . . 148
6.1 Plots to show how calculations of mean constraint costs from the simu-lator vary with nuclear and CCGT availability probabilities. . . 153 6.2 Plots to show how calculations of total costs from the simulator vary
with reinforcement magnitude of the B6 and B7a boundaries. These plots assume a CCGT availability probability of 0.875 and the year 6 peak demand level projected by [69]. . . 154 6.3 Plot to show how calculations of mean constraint costs from the
simu-lator vary with reinforcement magnitude of the B6 and B7a boundaries.
Nuclear and CCGT availability probabilities were assumed to be 0.7 and 0.875 respectively, whilst peak demand level was fixed at the year 6 projection from [69]. . . 155
List of Figures xxvi
6.4 Plots to show how total costs vary with B6 and B7a reinforcement mag-nitude. Nuclear and CCGT availability probabilities were assumed to be 0.7 and 0.875 respectively, whilst peak demand level was fixed at the year 6 projection from [69] (i.e. the central values for the variables containing uncertainty). . . 157 6.5 Plots to show how the difference between total cost calculations from
the simulator and estimates from the emulator vary with B6 and B7a reinforcement magnitude. Nuclear and CCGT availability probabilities were assumed to be 0.7 and 0.875 respectively, whilst peak demand level was fixed at the year 6 projection from [69]. . . 157 6.6 Plots to show how credible bounds of the estimates of total costs from
the emulator vary with B6 and B7a reinforcement magnitude, when assuming central values for the variables containing uncertainty. . . 158 6.7 Plots to show how total costs vary with B6 and B7a reinforcement
mag-nitude, for reinforcement magnitudes greater than 2000 MW on both boundaries. Nuclear and CCGT availability probabilities were assumed to be 0.7 and 0.875 respectively, whilst peak demand level was fixed at the year 6 projection from [69]. . . 159 6.8 Plots to show how the difference between total cost calculations from the
simulator and estimates from the emulator vary with B6 and B7a rein-forcement magnitude, for reinrein-forcement magnitudes greater than 2000 MW on both boundaries. Nuclear and CCGT availability probabilities were assumed to be 0.7 and 0.875 respectively, whilst peak demand level was fixed at the year 6 projection from [69]. . . 160 6.9 Plots to show how estimates of mean constraint costs from the emulator
vary with nuclear and CCGT availability probabilities, assuming no B6 or B7a reinforcement has been made. . . 160 6.10 Plots to show how estimates of total costs from the emulator vary with
the reinforcement magnitudes of the B6 and B7a boundaries. These plots assume a CCGT availability probability of 0.875 and the year 6 peak demand level projected by [69]. . . 162
6.11 Plots to show how calculations of total costs from the simulator vary with the reinforcement magnitudes of the B6 and B7a boundaries, for reinforcement magnitudes greater than 2000 MW on both boundaries.
These plots assume a CCGT availability probability of 0.875 and the year 6 peak demand level projected by [69]. . . 162 6.12 Plots to show how estimated total costs from the emulator vary with the
reinforcement magnitudes of the B6 and B7a boundaries, for reinforce-ment magnitudes greater than 2000 MW on both boundaries. These plots assume a CCGT availability probability of 0.875 and the year 6 peak demand level projected by [69]. . . 163 6.13 Plots to show how estimated expected total costs from the emulator vary
with reinforcement magnitudes of the B6 and B7a boundaries. . . 164 6.14 Plots to illustrate how credible bounds for the estimated expected total
costs vary with reinforcement magnitudes of the B6 and B7a boundaries. 165 6.15 Plots to illustrate decisions considered in the second wave, and the region
uniformly sampled when acquiring input values for the decision variables for training runs for the second wave. . . 169 6.16 Plots to illustrate decisions sampled as potential input decisions for the
second wave, and which decisions of the sample were not rejected. . . . 169 6.17 Plots to illustrate a decision space which it would be inefficient to
uni-formly sample second wave decisions from. . . 171 6.18 Plot of credible bounds for estimates of expected total costs from the
emulator model fitted in the first wave. . . 172 6.19 Plots to illustrate decisions not eliminated as the wave process progresses.173 6.20 Plots to show how estimated expected total costs vary with
reinforce-ment decisions and fitted emulator model. . . 174 6.21 Plots to compare the fitted emulator models in wave 1 and wave 3. . . 176 6.22 Plots to show how decisions considered in the third wave vary with the
assumed cost to reinforce. . . 179
List of Figures xxviii
6.23 Plots to show how estimated expected total costs vary with B6 and B7a reinforcement magnitude for the emulator models fitted in the third wave for three different values of the assumed cost to reinforce. . . 180 6.24 Plots to illustrate risk profiles for various decisions. . . 183 6.25 Plot to illustrate the loss function as attitude to risk is varied. . . 186 6.26 Plots to illustrate the ranges of decisions considered in each wave when
using 50 training runs in each wave to construct the emulator model. . 188 6.27 Plots to show how estimates of expected total costs vary with
reinforce-ment decisions and fitted emulator model, when using 50 training runs per wave to construct the emulator. . . 189 6.28 Plots to show how estimates of expected total costs vary with
reinforce-ment decisions and fitted emulator model over the range of decisions considered in wave 4, when using 50 training runs per wave to construct the emulator. . . 190 6.29 Plots to compare the estimated expected total costs from the final wave
emulator models when using 50 or 300 training runs to fit the emulator model. . . 191 6.30 Plots of the differences in estimated expected total costs as decision is
varied when using 50 or 300 training runs per wave to fit the emulator model. . . 192 6.31 Plot to compare estimates of expected total costs and corresponding
credible intervals for the final wave emulator models fitted using 50 and 300 training runs. B7a reinforcement magnitude fixed at 2890 MW (the estimated optimal when using 300 training runs). . . 192
7.1 Boxplots to show how simulations of constraint costs vary with month and assumed seasonal model for a year 1 power system background. . . 196 7.2 Boxplots to show how simulations of constraint costs vary with month
and assumed seasonal model for a year 6 power system background. . . 198
7.3 Plots to show how estimates of expected total costs vary with B6 and B7a reinforcement magnitude and assumed seasonal model. . . 203 7.4 Plots to show how estimates of expected total costs vary with B6 and
B7a reinforcement magnitude and assumed seasonal model, after two waves of eliminating decisions which have evidence against them being optimal. . . 205 7.5 Plots to show how estimated expected total costs vary with
reinforce-ment decisions and fitted emulator model. . . 210 7.6 Plot to compare observed LDCs from different years. . . 213 7.7 Boxplots to show how simulations of constraint costs vary with assumed
LDC. . . 214 7.8 Plots to show how estimates of expected total costs vary with B6 and
B7a reinforcement magnitude and assumed LDC, when assuming it costs
£1000 per MW per km to reinforce. . . 218 7.9 Plots to show how estimated expected total costs vary with
reinforce-ment decisions and fitted emulator model. . . 221 7.10 Graphs to compare how estimates of mean annual constraint costs vary
with the change in the projected increase in installed wind generating capacity. . . 224 7.11 Plots to show how estimates of expected total costs vary with B6 and
B7a reinforcement magnitude and assumed wind generating capacity projection. . . 226 7.12 Plots to show how estimated expected total costs vary with
reinforce-ment decisions and fitted emulator model. . . 229 7.13 Boxplots to show how simulations of constraint costs vary with
assump-tions made. . . 232 7.14 Plots to show how estimated expected total costs vary with
reinforce-ment decisions and fitted emulator model, when assuming a cost of
£1000 per MW per km to reinforce. . . 239
List of Figures xxx
7.15 Plots to illustrate decisions not eliminated as the wave process pro-gresses, when assuming a cost of £1000 per MW per km to reinforce. . 240
8.1 Plots to illustrate values of d1 and dT2 considered in the first wave. . . 272 8.2 Plots to illustrate values of d1 and dT2 considered in the second wave. . 273 8.3 Plot to illustrate how estimates of expected total costs across both stages
vary with B6 reinforcement magnitude and B7a reinforcement magni-tude in the first wave of stage 1. . . 281 8.4 Plots to show credible bounds for the estimates of expected total costs
from the emulator models fitted in the first wave. . . 282 8.5 Plots to show decisions not eliminated from consideration in the second,
third and fourth waves. . . 283 8.6 Plots to show how estimates of expected total costs across both stages
vary with stage 1 reinforcement decisions and fitted emulator model. . . 284 8.7 Plots to show how estimates of expected total costs across both stages
vary with stage 1 reinforcement decisions and fitted emulator model. . . 285 8.8 Plots to show credible bounds for the estimates of expected total costs
from the emulator models fitted in the first wave over the range of deci-sions considered in the fourth wave. . . 286 8.9 Plots to show credible bounds for the estimates of expected total costs
from the emulator models fitted in the fourth wave. . . 287 8.10 Plot to illustrate how credible intervals for the estimated expected total
costs vary as B6 reinforcement magnitude is varied. . . 287 8.11 Plots to show decisions considered in each wave for the second stage. . 291 8.12 Graph to show how estimated expected total costs in stage 2 vary with
total B6 and B7a reinforcement for the emulator model fitted in the first wave of the second stage. . . 292 8.13 Plots to show how credible bounds for the estimated expected total costs
in stage 2 vary with total reinforcement decisions for the emulator model fitted in the first wave of the second stage. . . 292
8.14 Plots to show how estimated expected total costs in stage 2 vary with total B6 and B7a reinforcement decisions and fitted emulator model in the second stage. . . 294 8.15 Plots to show how estimated expected total costs vary with total B6 and
B7a reinforcement magnitudes and fitted emulator model in the second stage, over the range of values for the decision variables considered in the third wave. . . 295 8.16 Plots to show how credible bounds for the estimated expected total costs
vary with total B6 and B7a reinforcement decisions for the emulator model fitted in the first wave of the second stage, over the range of values for the decision variables considered in the third wave. . . 295 8.17 Plots to show how credible bounds for the estimated expected total costs
vary with total B6 and B7a reinforcement decisions for the emulator model fitted in the third wave of the second stage. . . 296 8.18 Plot to illustrate how expected total cost estimates vary with total B6
reinforcement magnitude and ψ2 (stage 2 scenario). . . 297 8.19 Plots to illustrate how estimates of optimal total reinforcement
magni-tude vary with ψ2 (stage 2 scenario). . . 298 8.20 Plots to show how estimated expected total costs vary with total B6 and
B7a reinforcement decisions for each of the stage 2 scenarios considered in Table 8.3. . . 301 8.21 Plot to illustrate the prior beliefs about peak demand level in stage 2 at
year 1. . . 305 8.22 Plot to illustrate how expected total cost estimates vary with B6
re-inforcement magnitude and B7a rere-inforcement magnitude in the first wave. . . 305 8.23 Plots to show how credible bounds for the estimates of expected total
costs vary with B6 and B7a reinforcement magnitude. . . 306 8.24 Plots to show values of decision variables considered in the first and final
waves. . . 307
List of Figures xxxii
8.25 Plots to show how estimated expected total costs vary with reinforce-ment decisions and fitted emulator model. . . 307 8.26 Plots to illustrate how estimates of expected total costs vary with B6
reinforcement magnitude and B7a reinforcement magnitude in the final wave. . . 308 8.27 Plots to show how estimates of expected total costs across both stages
vary with stage 1 reinforcement decisions and fitted emulator model. . . 311 8.28 Plots to show credible bounds for the estimates of expected total costs
for the emulator models fitted in the first wave. . . 311 8.29 Plots to compare the values of decision variables considered in the first
and fourth wave. . . 312 8.30 Plot to show how estimates of expected total costs across both stages
vary with stage 1 B6 and B7a reinforcement magnitude for the emulator models fitted in the fourth wave. . . 313 8.31 Plots to show how estimated expected total costs vary with total
rein-forcement decisions and fitted emulator model in the second stage. . . . 314 8.32 Plot to compare risk profiles for different methods of decision making. . 316 8.33 Plots to compare the estimates of expected stage 2 costs for θ1. . . 324 8.34 Plots to compare the estimates of expected stage 2 costs for θ2. . . 324 8.35 Plots to compare the estimates of expected stage 2 costs for θ3. . . 325 8.36 Plots to compare the estimates of expected stage 2 costs for θ4. . . 325 8.37 Plots to compare the estimates of expected stage 2 costs for θ5. . . 326
9.1 Plot to illustrate how estimates of expected total costs across all three stages vary with B6 reinforcement magnitude and B7a reinforcement magnitude in the first wave of stage 1. . . 360 9.2 Plots to show credible bounds for the estimates of expected total costs
across all three stages from the emulator models fitted in the first wave of stage 1. . . 361
9.3 Plots to show decisions not eliminated from consideration in the second, third and fourth waves of Stage 1. . . 362 9.4 Plots to show how estimates of expected total costs across all three
stages vary with reinforcement decisions and fitted emulator model. . . 363 9.5 Plots to show how estimates of expected total costs across all three
stages vary with stage 1 reinforcement decisions and fitted emulator model, over the range of decisions considered in the fourth wave. . . 364 9.6 Plots to show credible bounds for the estimates of expected total costs for
the emulator models fitted in the first wave, over the range of decisions considered in the fourth wave. . . 365 9.7 Plots to show credible bounds for the estimates of expected total costs
for the emulator models fitted in the fourth wave. . . 365 9.8 Plots to illustrate the ranges of possible total reinforcement considered
in each stage in Section 9.3.1. . . 368 9.9 Plots to illustrate the smaller ranges of possible total reinforcement
con-sidered in each stage in this section. . . 368 9.10 Plot to illustrate how estimates of expected total costs across all three
stages vary with B6 reinforcement magnitude and B7a reinforcement magnitude in the first wave of stage 1. . . 369 9.11 Plots to show credible bounds for the estimates of expected total costs
stages vary with B6 reinforcement magnitude and B7a reinforcement magnitude in the first wave of stage 1. . . 369 9.11 Plots to show credible bounds for the estimates of expected total costs