31:T6de,{ let cap='Modelled annual energy production from the Ngonye Falls hydroelectic plant over the last 100 years is used as an input to the Financial Model to determine the expected financial performance of the project and its resiliance under possible downside scenarios.' let mean = d3.mean(yearly_energy.map(yr=>yr.energy)) let movingAv = movingAverage(yearly_energy.map(yr=>yr.energy), 15) var data = [{ y: yearly_energy.map(yr=>yr.energy), x: yearly_energy.map(yr=>yr.waterYear), text: yearly_energy.map(yr=>d3.format('.0f')(yr.energy) + 'GWh'), showlegend:false, name: '', hoverinfo:'x+text', marker: {color: colr(1,0.6)}, type: 'bar' },{ y: movingAv, x: yearly_energy.map(yr=>yr.waterYear), name:'15yr Moving Average', mode: 'lines', hoverinfo:'none', line: {width:2,color: colr(3)}, type: 'scatter' },{ x: [1924, 2022], y: [mean, mean], mode: 'lines', line: {color: 'rgba(0,0,0,0.8)', width:1, dash:'dashdot'}, type: 'scatter', hoverinfo:'none', name: '1924-2022 Mean ' + d3.format(',.0f')(mean) + 'GWh' }]; var layout = { width: width-50, height:180, font: font, showlegend: false, margin: {t:5,b:40,l:70,r:5}, legend: {orientation: 'h', x: 0, xanchor: 'left', y: -0.15}, xaxis: { showgrid: false, zeroline: false, ticks:'outside', range:[1923,2022]}, yaxis: { range: [0,1200], title: 'Energy (GWh)', zeroline: false, tickformat: ',.0f', showline: false}, }; const div=container(cap, width-10) plt.newPlot(div, data, layout, {displayModeBar: false}); return div }32:T4b7,Statistical outputs from the P&E model are then used as inputs to the project Financial Model to answer two critical questions:

A] What will the base-case, long-term financial performance of the project be over the life of the offtake agreement (PPA) - used to determine:

1) the economic cost of the electricity (tariff),
2) the total amount of debt required to fund the project and,
3) the expected return to equity investors.

B] What are the project's sensitivities to reasonable, worst-case downside scenarios in energy variability - used to determine:

1) whether the project is robust under its financing structure and will meet its debt obligations under reasonable downside scenarios,
2) the impact of downside scenarios on the project equity investors and,
3) whether any other mitigations are required - eg debt sculpting, insurance.

The base-case and downside scenarios Financial Models require different inputs based on the statistics from the long-term P&E model that are described more fully below. 33:T6d9,

Model Error

Model error stems from errors and uncertainties in the collection of survey data (eg river flow, topogrphy) as well as errors and uncertainty in the application of physical models such as turbine performance. Model error is systematic and timescale invariant - it applies equaly at all timescales and it cannot be removed by longterm averaging.

For the Ngonye Falls project model error is expected to be very low - in contrast to eg Wind and Solar PV projects where it can be very significant. The major surveyed parameters such as river flow and topographic elevation can be measured to high precision and the power and energy modelling is straight forward and accurate - unlike eg wind measurements and CFD modelling for wind farms or irradiation measurement and PV yield modelling in solar PV.

The most significant parameter in the hydropower physical model, the efficiency of the turbines, is guarunteed by the EPC contractor and measured in advance by model testing.

Resource Variability

Resource variability is any natural variability in the underlying physical resource - in this case the flow of water in the river - and its implication on energy production and therefore revenue. Resource variability is timescale dependent, ie variability may be very different at daily, seasonal and interannual timescales, and mean-reverting - that is, its effects can be ignored by averaging over long timescales.

Because resource variability is timescale dependent it is important to analyse its impact at the correct timescales which, when considering the financing of a power station, depends on the cash flows required to service debt and provide adequate equity returns.34:T776,{ let cap=`Annual energy exceedance of the Ngonye Falls plant over the historic hydrology period. Median (P50) annual energy generation - exceeded in 50% of all years - is ${d3.format(',.0f')(annualExceedance.find(v=>v.exceedance==0.5).energy)} GWh. P90 annual energy (exceeded 9 years in 10) is ${d3.format(',.0f')(annualExceedance.find(v=>v.exceedance==0.9).energy)} GWh` var data = [ { y: annualExceedance.map(f => f.energy), text: annualExceedance.map(f => d3.format(',.0f')(f.energy) + 'GWh'), x: annualExceedance.map(f => f.exceedance), mode: 'lines', name:'Energy', hoverinfo:'x+text', line: { shape: 'spline' }, marker: { color: colr(1) }, type: 'scatter' }, { y: [d3.format(',.0f')(annualExceedance.find(v=>v.exceedance==0.5).energy)], x: [0.5], mode: 'markers+text', hoverinfo:'none', marker: { color: colr(5), size: 10 }, type: 'scatter' } ]; var layout = { height: 300, width:width>900?870:width-30, font: font, showlegend: false, margin: {t:5,b:50,l:70,r:5}, legend: { x: 1, xanchor: 'right', y: 1 }, annotations: [ { y: annualExceedance.find(v=>v.exceedance==0.5).energy, x: 0.5, text: "P50 Median Annual Energy - " + d3.format(',.0f')(annualExceedance.find(v=>v.exceedance==0.5).energy) + ' GWh', showarrow: true, arrowhead: 2, ax: -90, ay: 50, xref: 'x', yref: 'y' } ], xaxis: { showgrid: true, zeroline: false, dtick: 0.10, tickformat:',.0%', ticks: 'outside', title: 'Exceedance Probability', range:[0,1.015] }, yaxis: { title: 'Energy (GWh)', zeroline: false, showline: false, separatethousands: true } }; const div=container(cap, width>900?890:width-10,) plt.newPlot(div, data, layout, {displayModeBar: false}); return div } 35:T664,{ let cap = `Probability distribution (histogram) of annual energy generation for the Ngonye Falls project over the 100 years of historic data. Similarity of the mean and the median (P50) show that the distribution is relatively symetrical. High central peak shows stability of annual generation.` let mean = d3.mean(yearly_energy.map(yr=>yr.energy)) let median = annualExceedance.find(v=>v.exceedance==0.5).energy var data = [{ x: yearly_energy.map(yr=>yr.energy), showlegend:false, histnorm: "count", marker: {color: colr(1,0.6)}, name: '', hoverinfo:'y', type: 'histogram', xbins: { size: 20, }},{ x: [median, median], y: [0,25], hoverinfo:'x', showlegend:true, line: {color: '#777'}, mode:'lines', name: `P50 Median - ${d3.format(',.0f')(median)}GWh`, type: 'scatter', },{ x: [mean, mean], y: [0,25], showlegend:true, hoverinfo:'x', line: {dash:'dot',color: '#777'}, mode:'lines', name: `Mean - ${d3.format(',.0f')(mean)}GWh`, type: 'scatter', }] var layout = { width: width>700?670:width-30, height:220, font: font, showlegend: true, margin: {t:5,b:60,l:70,r:5}, legend: {orientation: 'h', x: 0, xanchor: 'left', y: 1.1}, xaxis: { range:[600,1100], showgrid: false,zeroline: false, ticks:'outside', title:'Annual Energy GWh' }, yaxis: { title: '% of Years', zeroline: false, tickformat: ',.0f', showline: false}, }; const div=container(cap, width>800?790:width-10,) plt.newPlot(div, data, layout, {displayModeBar: false}); return div }36:T48c,The main drawbacks in using the mean or median (P50) annual energy when analysing the future performance of the plant stem from the fact that these measures ignore the timeseries nature of the data. When considering the financial performance and possible downside risks in particular, using the longterm average has the following issues:

Drift - The recent past may be a better guide to the near future than the whole series.
Timeframe - Using the variability of the long-term data does not properly capture the variability over the shorter timescales of project financing.
Autocorrelation - Timeseries data may exhibit autocorrelation - where the value in one period depends to some extent on values in previous periods, ie the energy values from year to year are not independent of each other.
Time value of money - In the analysis of financial performance, the time value of money means that the order of particular values becomes important, with lower revenue in earlier periods having a higher impact than if they occur in later periods.

37:T5d1,{ let mean = d3.mean(yearly_energy.map(yr=>yr.energy)) let cap=`Expanding mean annual energy - mean from the most current year expanding back in time. The mean for the most recent 25 years is ${d3.format(',.0f')(yearlyExpandingMean[99-24].energyMean/1000)}GWh compared with the full series 100 year mean of ${d3.format(',.0f')(mean)}GWh.` var data = [{ y: yearlyExpandingMean.map(yr=>yr.energyMean), x: yearlyExpandingMean.map(yr=>yr.years), text: yearlyExpandingMean.map(yr=>d3.format(',.0f')(yr.energyMean)), showlegend:false, mode:'lines', name: '', hoverinfo:'x+text', line: {width:3,color: colr(1,0.6), shape:'spline'}, type: 'scatter' },{ y: [mean,mean], x: [1,yearlyExpandingMean.length-1], showlegend:false, mode:'lines', name: '', hoverinfo:'none', line: {color: 'rgba(0,0,0,0.8)', width:2, dash:'dashdot'}, type: 'scatter' }]; var layout = { width: width>800?770:width-30, height:220, font: font, showlegend: true, margin: {t:5,b:40,l:70,r:5}, legend: {orientation: 'h', x: 0, xanchor: 'left', y: -0.15}, xaxis: { showgrid: false,zeroline: false, ticks:'outside', range: [0, 101], title:'Years' }, yaxis: { title: 'Annual Energy GWh', zeroline: false, tickformat: ',.0f', range:[650,950], showline: false}, }; const div=container(cap, width>800?770:width-10,) plt.newPlot(div, data, layout, {displayModeBar: false}); return div }38:T45d,Because the timeframe of the project debt financing (eg 18 years) is lower than the project life and the full timeseries of available data (100 years) it is neccassary to scale the variability in energy generation and therefore, revenue to the shorter timeframes when analysing the level of downside risk for the financing.

The standard method for timescaling of risk is the square-root-of-time rule. This method generates new distributions showing the probability that, over a particular timeframe - say 15 or 20 years - that the median energy will meet or exceed a certain value. For example, for Ngonye Falls there is a 75% probability that the average annual energy over any 15 year period will exceed ${ formatEnergy(durationVariability.find(v=>v.durationYears==15)['0.75']) }.

Note, what is commonly called "15-year P90" (or erronously just "P90") could more properly be called "P90 of average annual energy over 15 year periods" where P90 simply means a value from a probability distribution and the probability distribution in question is the average energy over any 15-year period.39:T5e7,{ let cap = `Exceedance probability of mean annual energy scaled to various timeframes for the Ngonye Falls project. For eaxmaple, the chart shows that with 95% confidence, the mean annual energy delivered by the project would 829GWh or more over any 15 year period and 837GWh or more over any 25 year period. ` const exceedances = [...Array(21).keys()].map(v=> v/20).slice(1,-1) function series(yearsNo) { return { y: exceedances.map(e=>durationVariability[yearsNo][e]), text: exceedances.map(e=>formatEnergy(durationVariability[yearsNo][e])), x: exceedances, mode: 'lines+markers', hoverinfo:'x+text', name: durationVariability[yearsNo].durationYears + '-year', marker:{size: 4}, line: { shape: 'spline', color: colr(yearsNo-1), width: 2.5 }, type: 'scatter' } } var data = [series(1 ),series(2),series(4),series(5)] var layout = { width: width>800?770:width-30, height:300, margin: {t:10, b:60, l:80, r:10}, font: font, showlegend: true, legend: {orientation:'h', x: 1, xanchor: 'right', y: 1 }, xaxis: { showgrid: true, zeroline: false, dtick:0.1, ticks: 'outside', tickformat:',.0%', title: 'Exceedance Probability' }, yaxis: { title: 'Mean Annual Energy
over Period', zeroline: false,showline: false,separatethousands: true, } } const div=container(cap, width>800?790:width-10) plt.newPlot(div, data, layout,{displayModeBar: false}); return div }3a:T5b5,{ let cap=`Autocorrelation plot of annual energy with 95% confidence interval. Annual energy correlates strongly with the previous year value and weakly with the second previous year at the 95% confidence level.` var data = [{ y: acf.map(v=>v.upper), text: acf.map(v=>d3.format('.3f')(v['0'])), showlegend:false, mode:'lines', name: '', hoverinfo:'none', line:{width:0, shape: 'spline'}, type: 'scatter' },{ y: acf.map(v=>v.lower), text: acf.map(v=>d3.format('.3f')(v['0'])), showlegend:false, mode:'lines', name: '', hoverinfo:'none', line:{width:0, color:'#CCC',shape: 'spline'}, fill:'tonexty', type: 'scatter' },{ y: acf.map(v=>v['0']), text: acf.map(v=>d3.format('.3f')(v['0'])), showlegend:false, mode:'markers', name: '', hoverinfo:'text', marker: {size:10,color: colr(1,0.6)}, type: 'scatter' }] var layout = { width: width-50, height:220, font: font, showlegend: true, margin: {t:5,b:40,l:70,r:5}, legend: {orientation: 'h', x: 0, xanchor: 'left', y: -0.15}, xaxis: { showgrid: false,zeroline: false, ticks:'outside', title:'Years Lag',range:[-0.2,7.2] }, yaxis: { title: 'R\u00B2 Correlation', zeroline: false, tickformat: ',.0%', range:[-0.05,1.05], showline: false}, }; const div=container(cap, width-10) plt.newPlot(div, data, layout, {displayModeBar: false}); return div }3b:T460,{ let cap = `Moving average annual energy over various timeframes.` function series(yearsNo) { return { y: rollingMean[yearsNo].means.map(v=>v/1000), text: rollingMean[yearsNo].means.map(v=>formatEnergy(v/1000)), x: rollingMean[yearsNo].means.map((v,i)=>i+1924), mode: 'lines+markers', hoverinfo:'x+text', name: rollingMean[yearsNo].duration + '-year', marker:{size: 4}, line: { shape: 'spline', color: colr(yearsNo-1), width: 2.5 }, type: 'scatter' } } var data = [series(2),series(4),series(5)] var layout = { width: width>800?770:width-30, height:300, margin: {t:10, b:60, l:80, r:10}, font: font, showlegend: true, legend: {orientation:'h', x: 1, xanchor: 'right', y: 1 }, xaxis: { showgrid: true, zeroline: false, ticks: 'outside', tickformat:'.0f' }, yaxis: { title: 'Mean Annual Energy GWh', zeroline: false,showline: false,separatethousands: true, } } const div=container(cap, width>800?790:width-10) plt.newPlot(div, data, layout,{displayModeBar: false}); return div }3c:T672,{ let cap = `Exceedance probability for the moving average of mean annual energy over various periods.` function series(yearsNo) { return { y: rollingMeanExceedance [yearsNo].exceedance.slice(1, 20).map(v=>v.meanEnergy/1000), text: rollingMeanExceedance [yearsNo].exceedance.slice(1, 20).map(v=>formatEnergy(v.meanEnergy/1000)), x: rollingMeanExceedance [yearsNo].exceedance.slice(1, 20).map(v=>v.exceedance), mode: 'lines+markers', hoverinfo:'x+text', name: rollingMeanExceedance[yearsNo].duration + '-year', marker:{size: 4}, line: { shape: 'spline', color: colr(yearsNo-1), width: 2.5 }, type: 'scatter' } } var data = [series(2),series(4),series(5)] var layout = { width: width>800?770:width-30, height:300, margin: {t:10, b:60, l:80, r:10}, font: font, showlegend: true, legend: {orientation:'h', x: 1, xanchor: 'right', y: 1 }, annotations: [ { y: fifteenyearp75RollingMean.meanEnergy/1000, x: 0.74, text: `15-year P75 Energy
${ formatEnergy(fifteenyearp75RollingMean.meanEnergy/1000) }`, showarrow: true, arrowhead: 2,xanchor: 'left', ax: -120, ay: 60, xref: 'x', yref: 'y' } ], xaxis: { showgrid: true, zeroline: false, dtick:0.1, ticks: 'outside', tickformat:',.0%', title: 'Exceedance Probability' }, yaxis: { title: 'Mean Annual Energy
over Period', zeroline: false,showline: false,separatethousands: true, } } const div=container(cap, width>800?790:width-10) plt.newPlot(div, data, layout,{displayModeBar: false}); return div }3d:T61e,function normalinv(p, mean, std) { return -1.41421356237309505 * std * erfcinv(2 * p) + mean; function erfcinv(p) { var j = 0; var x, err, t, pp; if (p >= 2) return -100; if (p <= 0) return 100; pp = (p < 1) ? p : 2 - p; t = Math.sqrt(-2 * Math.log(pp / 2)); x = -0.70711 * ((2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t); for (; j < 2; j++) { err = erfc(x) - pp; x += err / (1.12837916709551257 * Math.exp(-x * x) - x * err); } return (p < 1) ? x : -x; } function erfc(x) { return 1 - erf(x); } function erf(x) { var cof = [-1.3026537197817094, 6.4196979235649026e-1, 1.9476473204185836e-2, -9.561514786808631e-3, -9.46595344482036e-4, 3.66839497852761e-4, 4.2523324806907e-5, -2.0278578112534e-5, -1.624290004647e-6, 1.303655835580e-6, 1.5626441722e-8, -8.5238095915e-8, 6.529054439e-9, 5.059343495e-9, -9.91364156e-10, -2.27365122e-10, 9.6467911e-11, 2.394038e-12, -6.886027e-12, 8.94487e-13, 3.13092e-13, -1.12708e-13, 3.81e-16, 7.106e-15, -1.523e-15, -9.4e-17, 1.21e-16, -2.8e-17 ]; var j = cof.length - 1; var isneg = false; var d = 0; var dd = 0; var t, ty, tmp, res; if (x < 0) { x = -x; isneg = true; } t = 2 / (2 + x); ty = 4 * t - 2; for (; j > 0; j--) { tmp = d; d = ty * d - dd + cof[j]; dd = tmp; } res = t * Math.exp(-x * x + 0.5 * (cof[0] + ty * d) - dd); return isneg ? res - 1 : 1 - res; } }3e:[{"id":"fc975a42ed0c5625d463585386c4e29940561a554fe085115d70abd5be99b7bcf9a3becfe46f71b2fd742807a7d876507da125964b8fe3d73df8d0862a6910a9","href":"https://static.observableusercontent.com/files/fc975a42ed0c5625d463585386c4e29940561a554fe085115d70abd5be99b7bcf9a3becfe46f71b2fd742807a7d876507da125964b8fe3d73df8d0862a6910a9","name":"ngonye_models.png","type":"image/png","lastModified":1738057220033.681,"size":48595},{"id":"1fce78c96c0d15a6630cb5b14d13c3c53ba00913fe30732bfd7a28bf12281150a8cf180821fa154e4ea9939de73f88db9bb8c6d1f95813c08e930cf008edad8e","href":"https://static.observableusercontent.com/files/1fce78c96c0d15a6630cb5b14d13c3c53ba00913fe30732bfd7a28bf12281150a8cf180821fa154e4ea9939de73f88db9bb8c6d1f95813c08e930cf008edad8e","name":"Sources_Variability_Uncertainty.png","type":"image/png","lastModified":1738152866271.225,"size":56396}] 2f:{"title":"Energy & Revenue Variability","cells":[{"id":758,"value":"html`\n

Energy & Revenue Variability

Ngonye Falls Hydroelectric Project

\n
\n${book_index('energy-variability')}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"As a run-of-river hydro plant, with no storage and on an unregulated river, the 180MW Ngonye Falls project will generate a varying amount of electricity from one year to the next. \nDespite unusal features of the river and the plant design which tend to smooth out the differences, years with higher than average rainfall and therefore higher flow of water in the river will produce more electricity than years with lower flow.\n

\nWith a fixed-tariff, energy-only PPA, variations in energy yield from one year to the next will also lead to variations in project revenue which, in turn, will affect the ability of the project to service its debt obligations and provide a return to equity investors.\n\n","pinned":false,"hidden":false,"mode":"html"},{"id":64,"value":"html`

900?900:width}px\" src=\"${await FileAttachment(\"ngonye_models.png\").url()}\">

\n Block diagram showing the relationship between the Power & Energy Model, Financial Model and various parameters for the Ngonye Falls hydroelectric project.\n

\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":114,"value":"Using 100 years of daily hydrology data for the Zambezi river and a Power and Energy (P&E) model that simulates operation of the project over that 100 year period, a detailed statistical model of the expected future plant output can be constructed. ","pinned":false,"hidden":false,"mode":"html"},{"id":93,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":103,"value":"$32","pinned":false,"hidden":false,"mode":"html"},{"id":530,"value":"

Revenue variability and uncertainty

","pinned":false,"hidden":false,"mode":"html"},{"id":535,"value":"The variability and uncertainty in power & energy and therefore project revenue comes from two sources, Model Error and Resource Variability, that have very different drivers and characteristics.\n

\nFor the Ngonye Falls run-of-river project, resource variability is the critical factor in determining whether the project's financing will be robust.","pinned":false,"hidden":false,"mode":"html"},{"id":395,"value":"html`

900?900:width}px\" src=\"${await FileAttachment(\"Sources_Variability_Uncertainty.png\").url()}\">

\n \n

\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":558,"value":"$33","pinned":false,"hidden":false,"mode":"html"},{"id":69,"value":"

A] Base-case

","pinned":false,"hidden":false,"mode":"html"},{"id":237,"value":"The best estimate of the base-case, long-term performance expectation for the power station is based on the past, historic average performance if the plant had operated over the historic hydrology.\n

\nThe average performance of the plant over the historic period can be estimated by the mean or the median (P50) of the annual energy generation. The median (P50) value is defined as the annual energy value that is exceeded for 50% of years and can be visualised on a exceedance probability plot.","pinned":false,"hidden":false,"mode":"html"},{"id":130,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":313,"value":"The mean of the same data is found in the usual way by divding the sum of all the annual energy values by the number of years in the historic data.\n

\nWhen the probability distribution (histogram) for a set of samples is symetrical - without long tails on the left or the right - then the mean and the median will have the same value. In the case of the annual energy generation over the historic data for the Ngonye Falls project the mean and median are very similar which confirms there is no systematic skew in the probability distribution and either value gives a good measure for the 'average' annual energy generation.","pinned":false,"hidden":false,"mode":"html"},{"id":243,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":327,"value":"

B] Downside scenarios and debt service

","pinned":false,"hidden":false,"mode":"html"},{"id":330,"value":"$36","pinned":false,"hidden":false,"mode":"html"},{"id":318,"value":"

Drift

\nThe average - mean or median (P50) - of the annual energy generation of the historic hydrology data will vary depending on how far back in the data these values are calculated. When considering the expected average energy over the period of an offtake agreement (25 years) or financing period (18 years) it might be more appropriate to use the more recent past rather than the whole series as this will make the measure more robust against very long term shifts in climate or the characteristics of the river basin.","pinned":false,"hidden":false,"mode":"html"},{"id":178,"value":"$37","pinned":false,"hidden":false,"mode":"ojs"},{"id":578,"value":"For the annual energy production of the Ngonye Falls project an expanding mean from the most recent year back in time shows very little movement indicating that an average over the full timeseries is the moost appropriate base case value for financial modelling.","pinned":false,"hidden":false,"mode":"html"},{"id":580,"value":"

Timeframe

","pinned":false,"hidden":false,"mode":"html"},{"id":600,"value":"$38","pinned":false,"hidden":false,"mode":"html"},{"id":473,"value":"$39","pinned":false,"hidden":false,"mode":"ojs"},{"id":593,"value":"

Autocorrelation

","pinned":false,"hidden":false,"mode":"html"},{"id":610,"value":"A timeseries exhibits autocorrelation when the value in one period is, to some extent, dependent on values in previous periods. Where a timeseries has autocorrelation then the assumption that the value in any period is completely independent of all other periods no longer holds.\n

\nThe autocorrelation plot of mean annual energy for the Ngonye Falls generation shows that there is a strong correlation with an offset of one year and a weaker correlation over two years.","pinned":false,"hidden":false,"mode":"html"},{"id":351,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs"},{"id":655,"value":"A method to take account of autocorrelation in analysing the variability in the timeseries over a fixed timeframe is to use a moving average with a window fixed to the timeframe of interest. This method gives the average annual energy production for a moving window of years and maintains the actual order of years observed in the historic data including the impact of any autocorrelation.","pinned":false,"hidden":false,"mode":"html"},{"id":618,"value":"$3b","pinned":false,"hidden":false,"mode":"ojs"},{"id":659,"value":"Once the moving average values have been calculated then standard statistical measures can be taken of these averages.","pinned":false,"hidden":false,"mode":"html"},{"id":641,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs"},{"id":677,"value":"

Time value of money

","pinned":false,"hidden":false,"mode":"html"},{"id":679,"value":"Analysis based only on energy - whether whole series measures, timeframe scaled statistic or statistics based on moving averages - and that only take simple samples of these distributions (eg 15-year P75 shown above) into the financial model cannot properly take account of the time value of money where the timing of particular values are important. For example, a series of low energy yields and revenue early in the financing period will have a far greater impact on project economics than the same series of values later in the financing period.\n

\nThe most robust way to consider the time value of money as well as autocorrelation and proper timeframes is to run the financial model multiple times using a moving window over the timeseries hydrology (and therefore annual energy) and then undertaking a statistical analysis of the whole basket of these model runs.","pinned":false,"hidden":false,"mode":"html"},{"id":696,"value":"{\n let cap = ``\n\n var data = [{\n y: movingWindowIRRs,\n text: movingWindowIRRs.map(v=>v),\n //x: rollingMean[yearsNo].means.map((v,i)=>i+1924),\n mode: 'lines+markers', showlegend:false,\n hoverinfo:'x+text', name: '',\n marker:{size: 4},\n line: { shape: 'spline', color: colr_b(1), width: 2.5 },\n type: 'scatter'\n }]\n \n var layout = {\n width: width>800?770:width-30,\n height:300,\n margin: {t:10, b:60, l:50, r:10},\n font: font, showlegend: true,\n legend: {orientation:'h', x: 1, xanchor: 'right', y: 1 }, \n xaxis: {\n showgrid: true, zeroline: false, \n ticks: 'outside', tickformat:'.0f'\n },\n yaxis: {\n title: 'IRR %',\n zeroline: false,showline: false,separatethousands: true,\n }\n }\n const div=container(cap, width>800?790:width-10)\n plt.newPlot(div, data, layout,{displayModeBar: false});\n return div \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":704,"value":"TODO: Convert these charts to minimum CFADS or some other measure of debt sustainability","pinned":false,"hidden":false,"mode":"html"},{"id":707,"value":"{\n let cap = ``\n\n var data = [{\n y: movingWindowIRRsExceedance.map(v=>v.irr),\n text: movingWindowIRRsExceedance.map(v=>v.irr),\n x: movingWindowIRRsExceedance.map(v=>v.exceedance),\n mode: 'lines+markers',\n hoverinfo:'x+text', name: '',\n marker:{size: 4},\n line: { shape: 'spline', color: colr_b(1), width: 2.5 },\n type: 'scatter'\n }]\n var layout = {\n width: width>800?770:width-30,\n height:300,\n margin: {t:10, b:60, l:50, r:10},\n font: font, showlegend: true,\n legend: {orientation:'h', x: 1, xanchor: 'right', y: 1 }, \n xaxis: {\n showgrid: true, zeroline: false, dtick:0.1,\n ticks: 'outside', tickformat:',.0%',\n title: 'Exceedance Probability'\n },\n yaxis: {\n title: 'IRR%',\n zeroline: false,showline: false,separatethousands: true,\n }\n }\n const div=container(cap, width>800?790:width-10)\n plt.newPlot(div, data, layout,{displayModeBar: false});\n return div \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":75,"value":"

\n\n

Data

","pinned":false,"hidden":false,"mode":"html"},{"id":691,"value":"movingWindowIRRsExceedance = [...Array(21).keys()].map(v=> {\n return {\n exceedance:v/20,\n irr: d3.quantile(movingWindowIRRs.toSorted(), 1-(v/20))\n }\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":688,"value":"movingWindowIRRs=[13.62,\n13.62,\n13.60,\n13.62,\n13.47,\n13.48,\n13.66,\n13.61,\n13.63,\n13.59,\n13.55,\n13.53,\n13.40,\n13.21,\n13.26,\n13.30,\n13.26,\n13.30,\n13.22,\n13.15,\n13.25,\n13.68,\n14.05,\n14.08,\n14.18,\n14.32,\n14.37,\n14.23,\n14.14,\n14.32,\n14.74,\n14.72,\n14.74,\n14.97,\n15.11,\n15.35,\n15.20,\n15.02,\n15.04,\n15.04,\n15.13,\n15.34,\n15.30,\n15.04,\n14.75,\n14.78,\n14.82,\n14.70,\n14.77,\n14.61,\n14.45,\n14.02,\n13.43,\n12.96,\n12.82,\n12.74,\n12.70,\n12.51,\n12.44,\n12.17,\n11.92,\n11.73,\n11.72,\n11.67,\n11.82,\n11.87,\n11.98,\n12.31,\n13.03,\n13.51,\n13.70,\n13.79,\n13.96,\n14.11]","pinned":false,"hidden":false,"mode":"ojs"},{"id":662,"value":"fifteenyearp75RollingMean=rollingMeanExceedance.filter(v=>v.duration == 15)[0].exceedance[15]","pinned":false,"hidden":false,"mode":"ojs"},{"id":631,"value":"rollingMeanExceedance={\n return rollingMean.map(d=>{\n return {\n duration: d.duration,\n exceedance: [...Array(21).keys()].map(v=> {\n return {\n exceedance:v/20,\n meanEnergy: d3.quantile(d.means.toSorted(), 1-(v/20))\n }\n })\n }\n })\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":612,"value":"rollingMean={\n let durations = [5, 10, 15, 18, 20, 25, 50, 100]\n return durations.map(d=>{\n return {\n duration: d,\n means: yearly_energy.slice(0,yearly_energy.length-d).map((v,i)=>{\n return d3.mean(yearly_energy.slice(i,i+d+1).map(y=>y.energy))\n })\n }\n })\n \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":436,"value":"durationVariability=(await d3.csv(peFolder + peModelName + '_pe_durationvariability.csv?' + Math.floor(Math.random() * 100), d3.autoType))","pinned":false,"hidden":false,"mode":"ojs"},{"id":342,"value":"acf = (await d3.csv(server + `/other/base_pe_autocorrel.csv?` + Math.floor(Math.random() * 1000),d3.autoType)).slice(0,8)","pinned":false,"hidden":false,"mode":"ojs"},{"id":801,"value":"annualExceedance=(await d3.csv(peFolder + peModelName + '_pe_annualexceedances.csv?' + Math.floor(Math.random() * 100), d3.autoType))","pinned":false,"hidden":false,"mode":"ojs"},{"id":133,"value":"","pinned":false,"hidden":false,"mode":"ojs"},{"id":73,"value":"yearly = (await d3.csv(hydrologyFolder + \"yearly.csv?\" + Math.floor(Math.random() * 1000),d3.autoType))","pinned":false,"hidden":false,"mode":"ojs"},{"id":159,"value":"yearlyExpandingMean = yearly_energy.map((y,i)=>{\n return {\n energyMean: d3.mean(yearly_energy.slice(i).map(v=>v.energy)),\n years: yearly_energy.length-i,\n from: y.waterYear\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":97,"value":"yearly_energy = await d3.csv(peFolder + peModelName + \"_pe_yearly.csv?\" + Math.floor(Math.random() * 1000),d3.autoType)","pinned":false,"hidden":false,"mode":"ojs"},{"id":345,"value":"model=models.find(md=>md.modelRef==peModelName)","pinned":false,"hidden":false,"mode":"ojs"},{"id":791,"value":"models = (await d3.csv(server + '/ngonyePlantModels/models/modelStatistics.csv?' + Math.floor(Math.random() * 900),d3.autoType))","pinned":false,"hidden":false,"mode":"ojs"},{"id":789,"value":"hydrologyFolder = `${server}/syntheticFlowSeries/${hydrologySet}/processed/`","pinned":false,"hidden":false,"mode":"ojs"},{"id":787,"value":"peFolder = `${server}/ngonyePlantModels/models/${peModelName}/`","pinned":false,"hidden":false,"mode":"ojs"},{"id":785,"value":"hydrologySet='2024'","pinned":false,"hidden":false,"mode":"ojs"},{"id":783,"value":"peModelName='sh_2024_7.8mhead'","pinned":false,"hidden":false,"mode":"ojs"},{"id":781,"value":"server='https://data.westernpower.org'","pinned":false,"hidden":false,"mode":"ojs"},{"id":78,"value":"

Libraries

","pinned":false,"hidden":false,"mode":"html"},{"id":769,"value":"import {book_index} from \"@andyflem/commonlibrary\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":447,"value":"function formatEnergy(energy) {\n return d3.format(',.0f')(energy) + 'GWh';\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":440,"value":"$3d","pinned":false,"hidden":false,"mode":"ojs"},{"id":95,"value":"function colr(i = 1, a = 1) {\n return d3hsv.hsv(26, 0.85, ((6 - i + 1.5) * 15)/100, a).toString();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":86,"value":"function colr_b(i = 1, a = 1) {\n return d3hsv.hsv(215, 1, ((6 - i + 1.5) * 15)/120, a).toString();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":82,"value":"function movingAverage(values, N) {\n let a = new Array(Math.floor((N-1)/2)).fill(NaN)\n let b = new Array(N-Math.round((N-1)/2)).fill(NaN)\n return a.concat(windows(values,N).map(v=>d3.mean(v))).concat(b);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":84,"value":"function windows(inputArray, size) {\n return Array.from(\n {length: inputArray.length - (size - 1)}, //get the appropriate length\n (_, index) => inputArray.slice(index, index+size) //create the windows\n )\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"import {font} from \"@andyflem/commonlibrary\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"cumec=`m\\u00b3/s`","pinned":false,"hidden":false,"mode":"ojs"},{"id":21,"value":"style=html`css `","pinned":false,"hidden":false,"mode":"ojs"},{"id":15,"value":"html`\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"makeTrans=function(col,opacity) {\n let c=d3.color(col)\n c.opacity=opacity\n return c.formatRgb()\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"container = function(captionText, width_over) {\n const div=DOM.element('div')\n div.setAttribute(\"class\",\"border m-2 p-2\")\n div.setAttribute(\"style\",`width: ${width_over?width_over:(width>800?800:width)}px`)\n if (captionText) {\n const caption=DOM.element('figcaption')\n caption.setAttribute(\"style\",\"max-width:100%\")\n caption.append(captionText)\n div.append(caption)\n }\n return div\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":9,"value":"d3hsv = require(\"d3-hsv@0.1\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"d3 = require(\"d3@5\",\"d3-hsv@0.1\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"plt = require(\"https://cdn.plot.ly/plotly-latest.min.js\")","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W3e","stdlib":"1","titling":"auto"}