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scratch.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "dc7753d5-1d20-47e3-8ab2-774766013f5f",
"metadata": {},
"source": [
"# `growth_new_capacity_up`, `historical_new_capacity`, and `CAP_NEW`\n",
"\n",
"Example notebook for https://github.com/iiasa/message_ix/issues/932\n",
"\n",
"## Deriving the relationship\n",
"\n",
"Consider:\n",
"- Continuous time $t$ in units of years.\n",
"- A continuous function $n(t)$ which is *additions to capacity per unit time*.\n",
"- Two consecutive periods:\n",
" - Period $a$ spanning time $t_0$ to $t_a$ with duration $d_a$ (in years).\n",
" - Period $b$ spanning time $t_a$ to $t_b$ with duration $d_b$.\n",
"- Growth rates $r_a$ and $r_b$ (equivalent to ``growth_new_capacity_up``), each of which may be 0 or positive.\n",
"- An initial value $n(t_0) = N_0$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "887523fe-6a36-4d1b-a49a-515e08fa702b",
"metadata": {},
"outputs": [],
"source": [
"from sympy import Symbol, integrate, log, simplify, symbols\n",
"\n",
"t = Symbol(\"t\")\n",
"d_a, d_b = symbols(\"d_a d_b\", positive=True)\n",
"t_0 = Symbol(\"t_0\", nonnegative=True)\n",
"t_a = t_0 + d_a\n",
"t_b = t_a + d_b\n",
"\n",
"r_a, r_b = symbols(\"r_a r_b\", nonnegative=True)\n",
"\n",
"N_0 = Symbol(\"N_0\", nonnegative=True)"
]
},
{
"cell_type": "markdown",
"id": "33f743ba-7e97-4eff-9355-cc4ad6840c35",
"metadata": {},
"source": [
"Express $n_a(t)$, the upper bound on $n(t)$ in the period $a$.\n",
"This is the exponential growth function with initial value $N_0$, rate $r_a$, and duration measured from the start of period $a$, i.e. $t_0$:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "45dd5781-216c-4a93-9356-841116a46546",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle N_{0} \\left(r_{a} + 1\\right)^{t - t_{0}}$"
],
"text/plain": [
"N_0*(r_a + 1)**(t - t_0)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n_a = N_0 * (1 + r_a) ** (t - t_0)\n",
"n_a"
]
},
{
"cell_type": "markdown",
"id": "dd1ca0fc-00f7-46d3-a331-b41729355ec8",
"metadata": {},
"source": [
"Likewise, $n_b(t)$ is the upper bound on $n(t)$ in the period $b$.\n",
"It has a similar form, with its maximum *initial* value being $n_a(t_a)$,\n",
"its rate $r_b$,\n",
"and the duration measured from the start of period $b$, i.e. $t_a$:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "68dc3b4c-7e9a-4c11-9618-d6f43bc6979d",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle N_{0} \\left(r_{a} + 1\\right)^{d_{a}} \\left(r_{b} + 1\\right)^{- d_{a} + t - t_{0}}$"
],
"text/plain": [
"N_0*(r_a + 1)**d_a*(r_b + 1)**(-d_a + t - t_0)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n_b = n_a.subs(t, t_a) * (1 + r_b) ** (t - t_a)\n",
"n_b"
]
},
{
"cell_type": "markdown",
"id": "c9ae905a-cfac-4adb-a7d9-30d38fc6779f",
"metadata": {},
"source": [
"The maximum *total* addition to capacity over period $a$ is the integral of $n_a(t)$ on the interval $(t_0, t_a)$."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "cd64afa1-1a8d-4366-9773-da3a26a54394",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\begin{cases} \\frac{N_{0} \\left(r_{a} + 1\\right)^{d_{a}}}{\\log{\\left(r_{a} + 1 \\right)}} - \\frac{N_{0}}{\\log{\\left(r_{a} + 1 \\right)}} & \\text{for}\\: r_{a} \\neq 0 \\\\- N_{0} t_{0} + N_{0} \\left(d_{a} + t_{0}\\right) & \\text{otherwise} \\end{cases}$"
],
"text/plain": [
"Piecewise((N_0*(r_a + 1)**d_a/log(r_a + 1) - N_0/log(r_a + 1), Ne(r_a, 0)), (-N_0*t_0 + N_0*(d_a + t_0), True))"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"integrate(n_a, (t, t_0, t_a))"
]
},
{
"cell_type": "markdown",
"id": "ab16ed1f-63f5-47cb-9ca5-6b506e63fcd7",
"metadata": {},
"source": [
"The MESSAGE variable ``CAP_NEW`` measures the *average or mean* additions to capacity within a period;\n",
"in other words, the value of a constant $n(t)$ that has the same integral.\n",
"We call this $N_a$:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "d300521a-fe59-47e4-9c61-ec5b3491639b",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{\\begin{cases} \\frac{N_{0} \\left(r_{a} + 1\\right)^{d_{a}}}{\\log{\\left(r_{a} + 1 \\right)}} - \\frac{N_{0}}{\\log{\\left(r_{a} + 1 \\right)}} & \\text{for}\\: r_{a} \\neq 0 \\\\- N_{0} t_{0} + N_{0} \\left(d_{a} + t_{0}\\right) & \\text{otherwise} \\end{cases}}{d_{a}}$"
],
"text/plain": [
"Piecewise((N_0*(r_a + 1)**d_a/log(r_a + 1) - N_0/log(r_a + 1), Ne(r_a, 0)), (-N_0*t_0 + N_0*(d_a + t_0), True))/d_a"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"N_a = integrate(n_a, (t, t_0, t_a)) / d_a\n",
"N_a"
]
},
{
"cell_type": "markdown",
"id": "028f7beb-90d0-4914-a327-5bc6703f7d78",
"metadata": {},
"source": [
"Likewise for period $b$:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "80148475-1e51-4dad-943b-3cfc37e21f04",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{\\begin{cases} \\frac{N_{0} \\left(r_{a} + 1\\right)^{d_{a}} \\left(r_{b} + 1\\right)^{d_{b}}}{\\log{\\left(r_{b} + 1 \\right)}} - \\frac{N_{0} \\left(r_{a} + 1\\right)^{d_{a}}}{\\log{\\left(r_{b} + 1 \\right)}} & \\text{for}\\: r_{b} \\neq 0 \\\\- N_{0} \\left(d_{a} + t_{0}\\right) \\left(r_{a} + 1\\right)^{d_{a}} + N_{0} \\left(r_{a} + 1\\right)^{d_{a}} \\left(d_{a} + d_{b} + t_{0}\\right) & \\text{otherwise} \\end{cases}}{d_{b}}$"
],
"text/plain": [
"Piecewise((N_0*(r_a + 1)**d_a*(r_b + 1)**d_b/log(r_b + 1) - N_0*(r_a + 1)**d_a/log(r_b + 1), Ne(r_b, 0)), (-N_0*(d_a + t_0)*(r_a + 1)**d_a + N_0*(r_a + 1)**d_a*(d_a + d_b + t_0), True))/d_b"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"N_b = integrate(n_b, (t, t_a, t_b)) / d_b\n",
"N_b"
]
},
{
"cell_type": "markdown",
"id": "f54854fb-d757-4844-8ee5-4630e97846d1",
"metadata": {},
"source": [
"The MESSAGE GAMS implementation constrains $N_b$ (``CAP_NEW(b)``) as a function of $N_a$ (``CAP_NEW(a)``).\n",
"In particular we compute a ratio $k = N_b / N_a$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "7b9b0b26-3e43-48cd-9512-d5d217e54e3d",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\begin{cases} \\frac{d_{a} \\left(r_{a} + 1\\right)^{d_{a}} \\left(\\left(r_{b} + 1\\right)^{d_{b}} - 1\\right) \\log{\\left(r_{a} + 1 \\right)}}{d_{b} \\left(\\left(r_{a} + 1\\right)^{d_{a}} - 1\\right) \\log{\\left(r_{b} + 1 \\right)}} & \\text{for}\\: r_{a} \\neq 0 \\wedge r_{b} \\neq 0 \\\\\\frac{d_{a} \\left(r_{a} + 1\\right)^{d_{a}} \\log{\\left(r_{a} + 1 \\right)}}{\\left(r_{a} + 1\\right)^{d_{a}} - 1} & \\text{for}\\: r_{a} \\neq 0 \\\\\\frac{\\left(r_{a} + 1\\right)^{d_{a}} \\left(\\left(r_{b} + 1\\right)^{d_{b}} - 1\\right)}{d_{b} \\log{\\left(r_{b} + 1 \\right)}} & \\text{for}\\: r_{b} \\neq 0 \\\\\\left(r_{a} + 1\\right)^{d_{a}} & \\text{otherwise} \\end{cases}$"
],
"text/plain": [
"Piecewise((d_a*(r_a + 1)**d_a*((r_b + 1)**d_b - 1)*log(r_a + 1)/(d_b*((r_a + 1)**d_a - 1)*log(r_b + 1)), Ne(r_a, 0) & Ne(r_b, 0)), (d_a*(r_a + 1)**d_a*log(r_a + 1)/((r_a + 1)**d_a - 1), Ne(r_a, 0)), ((r_a + 1)**d_a*((r_b + 1)**d_b - 1)/(d_b*log(r_b + 1)), Ne(r_b, 0)), ((r_a + 1)**d_a, True))"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k = simplify(N_b / N_a)\n",
"k"
]
},
{
"cell_type": "markdown",
"id": "84300794-35ea-4d1e-aab8-3af6c39595d6",
"metadata": {},
"source": [
"We can further simplify by a series of substitutions:\n",
"- $x_a = (1 + r_a) ^ {d_a}$\n",
"- $y_a = \\frac{x_a - 1}{d_a \\log{(1 + r_a)}}$"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "a19170c3-d693-4ab7-94d5-6750b899f590",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\begin{cases} \\frac{d_{a} x_{a} \\left(x_{b} - 1\\right) \\log{\\left(r_{a} + 1 \\right)}}{d_{b} \\left(x_{a} - 1\\right) \\log{\\left(r_{b} + 1 \\right)}} & \\text{for}\\: r_{a} \\neq 0 \\wedge r_{b} \\neq 0 \\\\\\frac{d_{a} x_{a} \\log{\\left(r_{a} + 1 \\right)}}{x_{a} - 1} & \\text{for}\\: r_{a} \\neq 0 \\\\\\frac{x_{a} \\left(x_{b} - 1\\right)}{d_{b} \\log{\\left(r_{b} + 1 \\right)}} & \\text{for}\\: r_{b} \\neq 0 \\\\x_{a} & \\text{otherwise} \\end{cases}$"
],
"text/plain": [
"Piecewise((d_a*x_a*(x_b - 1)*log(r_a + 1)/(d_b*(x_a - 1)*log(r_b + 1)), Ne(r_a, 0) & Ne(r_b, 0)), (d_a*x_a*log(r_a + 1)/(x_a - 1), Ne(r_a, 0)), (x_a*(x_b - 1)/(d_b*log(r_b + 1)), Ne(r_b, 0)), (x_a, True))"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x_a, x_b = symbols(\"x_a x_b\")\n",
"k1 = k.subs((1 + r_a) ** d_a, x_a).subs((1 + r_b) ** d_b, x_b)\n",
"k1"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "558a24a2-0203-4e73-9474-438d72ce91ce",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\begin{cases} \\frac{x_{a} y_{b}}{y_{a}} & \\text{for}\\: r_{a} \\neq 0 \\wedge r_{b} \\neq 0 \\\\\\frac{x_{a}}{y_{a}} & \\text{for}\\: r_{a} \\neq 0 \\\\x_{a} y_{b} & \\text{for}\\: r_{b} \\neq 0 \\\\x_{a} & \\text{otherwise} \\end{cases}$"
],
"text/plain": [
"Piecewise((x_a*y_b/y_a, Ne(r_a, 0) & Ne(r_b, 0)), (x_a/y_a, Ne(r_a, 0)), (x_a*y_b, Ne(r_b, 0)), (x_a, True))"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y_a, y_b = symbols(\"y_a y_b\")\n",
"k2 = k1.subs((d_a * log(1 + r_a)) / (x_a - 1), 1 / y_a).subs((x_b - 1) / (d_b * log(1 + r_b)), y_b)\n",
"k2"
]
},
{
"cell_type": "markdown",
"id": "b0f53e73-10e7-4074-b538-c8512ed8f15d",
"metadata": {},
"source": [
"If we make $y_a$ a conditional expression such that $y_a = 1$ when $r_a = 0$\n",
"(this is simple to construct using the GAMS language)\n",
"we can always use the first expression in this 4-way conditional."
]
},
{
"cell_type": "markdown",
"id": "463c9d16-86f5-4884-878b-9cbabc1d8d38",
"metadata": {},
"source": [
"### Examples\n",
"\n",
"We consider some concrete / quantified examples based on this formulation.\n",
"\n",
"**Example 1:** Two periods of equal duration:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "28d2e1f0-395a-4296-8e47-1f5c2edeea9e",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.0510100501$"
],
"text/plain": [
"1.05101005010000"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"subs_1 = [(d_a, 5), (d_b, 5), (r_a, 0.01), (r_b, 0.01)]\n",
"k.subs(subs_1)"
]
},
{
"cell_type": "markdown",
"id": "4db03302-84f1-4b83-a26f-68e88340c6a7",
"metadata": {},
"source": [
"Note that this *happens* to be the same as:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "16196e00-7a0c-453e-b3ac-352d19c5b7c7",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0510100501000001"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1 + 0.01) ** 5"
]
},
{
"cell_type": "markdown",
"id": "cf478b40-6739-4e0a-9380-0b878a674d84",
"metadata": {},
"source": [
"This is because $n_a(t) = N_a$ occurs at the same point within interval $a$ as $n_b(t) = N_b$ within interval $b$,\n",
"and both intervals are of equal size, so the distance between these two points is exactly $d_a = d_b = 5$."
]
},
{
"cell_type": "markdown",
"id": "740f03b1-3653-44f4-986e-ae53303776ed",
"metadata": {},
"source": [
"**Example 2:** Two periods of equal duration, but double the duration in Example 1: "
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "d3e03870-28f3-4ed3-b05a-76c628f32a9d",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.1046221254112$"
],
"text/plain": [
"1.10462212541120"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k.subs([(d_a, 10), (d_b, 10), (r_a, 0.01), (r_b, 0.01)])"
]
},
{
"cell_type": "markdown",
"id": "aee6f742-ae02-4e51-b47f-83452624db45",
"metadata": {},
"source": [
"Similar to Example (1), this is $(1 + 0.01) ^ {10}$.\n",
"\n",
"**Example 3:** Period $b$ twice as long as period $a$:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "dd41cd17-39fa-4b73-be18-5e7a614cd222",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.0778160877556$"
],
"text/plain": [
"1.07781608775560"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k.subs([(d_a, 5), (d_b, 10), (r_a, 0.01), (r_b, 0.01)])"
]
},
{
"cell_type": "markdown",
"id": "c1bcb0ac-f8c6-4d20-a6da-32dfe6d6f1f9",
"metadata": {},
"source": [
"**Example 4:** Period $b$ half the length of period $a$:"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "3d40951d-e7a4-4256-a32c-c4d488504ec5",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.07714940290746$"
],
"text/plain": [
"1.07714940290746"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k.subs([(d_a, 10), (d_b, 5), (r_a, 0.01), (r_b, 0.01)])"
]
},
{
"cell_type": "markdown",
"id": "dd6caf6c-6de0-4dcf-b58c-072fc4f99022",
"metadata": {},
"source": [
"**Example 5:** Same total duration as Examples (3) and (4), with periods of equal duration:"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "2ba4a693-3d6d-4a71-b9a3-5e32260ba883",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.07748269376821$"
],
"text/plain": [
"1.07748269376821"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k.subs([(d_a, 7.5), (d_b, 7.5), (r_a, 0.01), (r_b, 0.01)])"
]
},
{
"cell_type": "markdown",
"id": "3b5c9166-9597-48e5-bda7-a91d2c60f1de",
"metadata": {},
"source": [
"Two points to observe:\n",
"1. The values in Example (3), (4), and (5) all differ.\n",
"2. https://github.com/iiasa/message_ix/pull/654 introduced an adjustment factor like $d_a / d_b$ to the existing ratio $N_b / N_a = (1 + r_b) ^ {d_b}$, but the values in Examples (3), (4), and (5) cannot be computed in this way."
]
},
{
"cell_type": "markdown",
"id": "1d29d59d-2103-4900-8553-94d8c7b42428",
"metadata": {},
"source": [
"**Example 6:** Same total duration as Example (2), but one short and one long period:"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "f782555f-d92a-463a-9dbf-e35124682068",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.10626336814719$"
],
"text/plain": [
"1.10626336814719"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k.subs([(d_a, 1), (d_b, 19), (r_a, 0.01), (r_b, 0.01)])"
]
},
{
"cell_type": "markdown",
"id": "7588648e-e912-4aa0-a750-e46090ff806a",
"metadata": {},
"source": [
"**Example 7:** One long and one short period:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "5debeeaf-bd35-4464-8968-278cc907b579",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.1029833176087$"
],
"text/plain": [
"1.10298331760870"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k.subs([(d_a, 19), (d_b, 1), (r_a, 0.01), (r_b, 0.01)])"
]
},
{
"cell_type": "markdown",
"id": "db7a882e-0765-4d01-b943-01a53da8c461",
"metadata": {},
"source": [
"## Graphical version\n",
"\n",
"Here we graphically examine previous and current values for CAP_NEW in the test suite.\n",
"\n",
"First some common code that computes the total CAP created in each period and cumulatively,\n",
"and draws a plot for comparison."
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "3aa04b74-271b-4e3e-94fd-8e35e59f82d6",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import plotnine as p9\n",
"\n",
"p9.options.set_option(\"figure_format\", \"svg\")\n",
"\n",
"def plot(data, r=0.01, t_max=720):\n",
" tm1, t0, CN0 = 680, 690, 10.0 / 3\n",
" data = (\n",
" pd.concat([pd.DataFrame([[t0, CN0]]), data])\n",
" .set_axis([\"year\", \"CAP_NEW\"], axis=1)\n",
" )\n",
" print(\n",
" data.rename(columns={\"year\": \"period\"})\n",
" .assign(ystart=lambda df: df.period.shift(fill_value=tm1))\n",
" .eval(\"duration_period = period - ystart\")\n",
" .eval(\"CAP = duration_period * CAP_NEW\")\n",
" .assign(CAP_cumulative=lambda df: df.CAP.cumsum() - df.CAP.iloc[0])\n",
" .set_index(\"period\")[[\"duration_period\", \"CAP_NEW\", \"CAP\", \"CAP_cumulative\"]]\n",
" )\n",
"\n",
" data_rect = data.assign(ystart=lambda df: df.year.shift(fill_value=tm1) + 1)\n",
" data_line = (\n",
" pd.Series(np.linspace(t0 + 0.99, t_max + 1), name=\"t\")\n",
" .to_frame()\n",
" .eval(\"CAP_NEW = 3.333333 * (1.0 + @r) ** (t - (@t0 + 1))\")\n",
" ) \n",
" (\n",
" p9.ggplot()\n",
" + p9.geom_rect(\n",
" p9.aes(xmin=\"ystart\", xmax=\"year + 0.99\", ymin=0, ymax=\"CAP_NEW\"),\n",
" data_rect,\n",
" color=\"black\",\n",
" )\n",
" + p9.geom_line(\n",
" p9.aes(x=\"t\", y=\"CAP_NEW\"),\n",
" data_line,\n",
" color=\"red\",\n",
" )\n",
" + p9.lims(y=(0, 10))\n",
" + p9.scale_x_continuous(breaks=range(680, t_max+1, 5), minor_breaks=4)\n",
" + p9.labs(y=\"Additions to capacity\", x=\"Time [years]\")\n",
" + p9.theme_bw()\n",
" ).show()"
]
},
{
"cell_type": "markdown",
"id": "ca165753-4cda-42cd-a198-298c75b18b18",
"metadata": {},
"source": [
"The value $r = 0.01$ used in the test suite is quite small.\n",
"Here's an example plot with a larger $r$ to show that the code correctly computes the exponential,\n",
"and also:\n",
"\n",
"1. Automatically adds the fixed/constant `historical_new_capacity` for the period \"690\".\n",
"2. Draws the periods precisely, e.g. such that \"690\" starts at 0681-01-01 and ends 0690-12-31\n",
" (the calendar year \"690\" is the final year within the multi-year period).\n",
"4. Does compute exponential growth until after the end of the final historical period."
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "0c434c9c-0335-448b-8c95-9714f03de13f",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.0\n"
]
},
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"Consider the values that were previously used as expectations in test_growth_new_capacity_up()."
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"id": "136afb9b-2b55-4792-9add-f3af47275f39",
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" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.00000\n",
"695 5 3.560014 17.800070 17.80007\n",
"700 5 3.746712 18.733560 36.53363\n",
"710 10 2.074581 20.745810 57.27944\n",
"720 10 2.302091 23.020910 80.30035\n"
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"source": [
"data_prev0 = pd.DataFrame([\n",
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"])\n",
"\n",
"plot(data_prev0)"
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},
{
"cell_type": "code",
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"id": "78f8bee5-a287-47ef-abec-d2a69a458e99",
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{
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"text": [
" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.000000\n",
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"source": [
"data_prev1 = pd.DataFrame([\n",
" [700, 3.746712],\n",
" [710, 4.149163],\n",
" [715, 8.731826],\n",
" [720, 9.182337],\n",
"])\n",
"plot(data_prev1)"
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"Notice that the total `CAP` constructed in `data_prev1` is more than double\n",
"the total `CAP` constructed in `data_prev0`.\n",
"\n",
"Next `CAP_NEW` that is obtained with the revised code and 5 test cases on the #924 branch.\n",
"Observe that:\n",
"\n",
"- Total `CAP` added over the interval 0691-01-01 to 0720-12-31 is the same in all cases.\n",
"- The red line intersects the CAP_NEW value within each period,\n",
" and is higher at the end of each period.\n",
"- Case 4 (`data_case4`) has the same set of periods as `data_prev0` above.\n",
"- Case 5 (`data_case5`) has the same set of periods as `data_prev1` above."
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "c0606931-7d0f-4108-a396-b55a112d9845",
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{
"name": "stdout",
"output_type": "stream",
"text": [
" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.00000\n",
"720 30 3.884281 116.528430 116.52843\n"
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"source": [
"data_case0 = pd.DataFrame([\n",
" [720, 3.884281]\n",
"])\n",
"plot(data_case0)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "a613ef88-2888-4b65-9d0e-3427c08fb63a",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
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" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.000000\n",
"691 1 3.349972 3.349972 3.349972\n",
"720 29 3.902705 113.178445 116.528417\n"
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"source": [
"data_case1 = pd.DataFrame([\n",
" [691, 3.349972],\n",
" [720, 3.902705],\n",
"])\n",
"plot(data_case1)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "5b48ece5-db54-4853-a062-1f156e2cf458",
"metadata": {},
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{
"name": "stdout",
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" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.00000\n",
"700 10 3.504812 35.048120 35.04812\n",
"720 20 4.074015 81.480300 116.52842\n"
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"source": [
"data_case2 = pd.DataFrame([\n",
" [700, 3.504812],\n",
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},
{
"cell_type": "code",
"execution_count": 25,
"id": "4215803e-1545-4b72-9139-08b8c9bd3b01",
"metadata": {},
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{
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" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.000000\n",
"719 29 3.864065 112.057885 112.057885\n",
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],
"source": [
"data_case3 = pd.DataFrame([\n",
" [719, 3.864065],\n",
" [720, 4.470551],\n",
"])\n",
"plot(data_case3)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "be184a0b-14aa-4b24-a337-1e7088ce994b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.000000\n",
"695 5 3.417645 17.088225 17.088225\n",
"700 5 3.591979 17.959895 35.048120\n",
"710 10 3.871493 38.714930 73.763050\n",
"720 10 4.276537 42.765370 116.528420\n"
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],
"source": [
"data_case4 = pd.DataFrame([\n",
" [695, 3.417645],\n",
" [700, 3.591979],\n",
" [710, 3.871493],\n",
" [720, 4.276537],\n",
"])\n",
"plot(data_case4)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "546658b4-4c3c-4204-aa16-6b113c742132",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" duration_period CAP_NEW CAP CAP_cumulative\n",
"period \n",
"690 10 3.333333 33.333333 0.000000\n",
"700 10 3.504812 35.048120 35.048120\n",
"710 10 3.871493 38.714930 73.763050\n",
"715 5 4.170177 20.850885 94.613935\n",
"720 5 4.382897 21.914485 116.528420\n"
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],
"source": [
"data_case5 = pd.DataFrame([\n",
" [700, 3.504812],\n",
" [710, 3.871493],\n",
" [715, 4.170177],\n",
" [720, 4.382897],\n",
"])\n",
"plot(data_case5)"
]
}
],
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