CvAR ED primer.ipynb

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    "# CVaR Primer and Application to Economic Dispatch\n",
    "\n",
    "### Objective\n",
    "This notebook contains test code for the calculation of the Conditional Value at Risk (CVaR) and its application to the Economic Dispatch (ED) problem. \n",
    "\n",
    "The objective is to implement the method presented in [Bertsimas and Brown \\[1\\]](http://pubsonline.informs.org/doi/abs/10.1287/opre.1080.0646?journalCode=opre) for the formulation of uncertainty sets in linear programming settings. \n",
    "\n",
    "In this case, the data source for the uncertainty set are probabilistic forecasts of renewable energy and the linear programming setting is the Robust Counterpart of the ED problem.  \n",
    "\n",
    "More definitions and details are provided below. "
   ]
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   "source": [
    "using Distributions\n",
    "using JuMP\n",
    "using Plots\n",
    "using Gurobi\n",
    "using LaTeXStrings \n",
    "gr();"
   ]
  },
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   "source": [
    "## Risk measures\n",
    "\n",
    "This section of the notebook explores the concepts of risk measures, in particular [Value at Risk](https://en.wikipedia.org/wiki/Value_at_risk) (VaR) and [Conditional Value at Risk](https://en.wikipedia.org/wiki/Expected_shortfall) (CVaR), also known as Expected Shortfall. \n",
    "\n",
    "Consider a random variable $\\widetilde{z}$ in the probability space $\\mathcal{Z}$ and a function $\\mu: \\mathcal{Z} \\rightarrow \\mathbb{R}$, the function $\\mu$ is called the risk measure. In general terms, it translates the possible lossess/gains defined by the random variable $\\widetilde{z}$ into a real value. A more mathematically precise definition is provided in [1], this notebook has the objective of presenting a first approach to the problem. \n",
    "\n",
    "VaR short definition: Represents the losses at the $\\varepsilon$-percentile. This can be interpreted as having a $\\varepsilon\\%$ chance of incurring the VaR losses. Mathematically, the VaR can defined as follows: \n",
    "\n",
    "$$\n",
    "VaR_{\\varepsilon}(\\widetilde{z}) \\triangleq \\min \\left\\{ z \\, |  \\, \\mathbb{P}(\\widetilde{z} \\le z) \\ge \\varepsilon \\right\\}\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "CVaR short definition: The expected shortfall at the $\\varepsilon$-level is the expected value of all the worse than $\\varepsilon\\%$ cases. Mathematically, the CVaR can be defined as follows: \n",
    "\n",
    "$$\n",
    "CVaR_{\\varepsilon}(\\widetilde{z}) \\triangleq \\min_{a} \\left(a+\\frac{1}{\\varepsilon}\\mathbb{E}\\left[\\left(-\\widetilde{z} -a\\right)^{+}\\right]\\right)\n",
    "$$\n",
    "\n",
    "It can be shown that $CVaR_{\\varepsilon}(\\widetilde{z}) \\le VaR_{\\varepsilon}(\\widetilde{z})$. Hence, the CVaR is often used as a conservative approximation of VaR \n",
    "\n",
    "There are important disctinctions between the measures that make CVaR prefereable for optimization applications, this characteristics are discussed at lenght by [Rockafellar and Uryasev \\[2\\]](http://www.ise.ufl.edu/uryasev/files/2011/11/CVaR1_JOR.pdf) and be summarized as follows: \n",
    "\n",
    "- Is has a linear representation, thus, it doesn't sacrifice the linearity of the LP. \n",
    "- Provides information of the loss accross the distribution's tail. \n",
    "- CVar is a coherent risk metric, which means that satisfies al desirable properties for a risk metric: \n",
    "    - Translation invariance: $\\mu\\left(\\widetilde{z} + c\\right) = \\mu\\left(\\widetilde{z}\\right) - c \\quad \\forall c \\in \\mathbb{R}$   \n",
    "    - Positive homogeneity: For $\\lambda \\ge 0$, then $\\mu\\left(\\lambda\\widetilde{z}\\right) = \\lambda \\mu\\left(\\widetilde{z}\\right)$\n",
    "    - Monotonicity: given 2 random variables in the same space $\\mathcal{Z}$, $\\widetilde{z}$ and $\\widetilde{y}$, if $\\widetilde{z} \\ge \\widetilde{y}$, then $\\mu\\left(\\widetilde{z}\\right) \\ge \\mu\\left(\\widetilde{y}\\right)$\n",
    "    - Convexity: given 2 random variables in the same space $\\mathcal{Z}$, $\\widetilde{z}$ and $\\widetilde{y}$ and $\\lambda \\in [0,1]$, then $\\mu\\left(\\lambda\\widetilde{z}+(1-\\lambda)\\widetilde{y}\\right) \\le \\lambda\\mu\\left(\\widetilde{z}\\right) + \\left(1-\\lambda\\right)\\mu\\left(\\widetilde{y}\\right) $ \n",
    "    \n",
    "The interest of describing the mathematical properties of these two risk measures lies on the fact that they will be used as part of the process of decision making under uncertainty, in particular the coherence property of the CVaR measure allows the formulation of robust counterparts to linear programming problems with uncertain data"
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    "### Calculation of VaR and CVaR\n",
    "\n",
    "For the purpose of describing and explaning the aforementioned measures and their characteristics some dummy data is generated. In this case, a data set similar to what is expected from a renewable energy probablistic forecast is used.  \n",
    "\n",
    "The data for this test is as follows: "
   ]
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    "ϵ = 0.01 #desired confidence level\n",
    "d = Normal(1,0.5)\n",
    "x = -0.8:0.01:1.3\n",
    "p = pdf.(d,x)\n",
    "p_n = p/sum(p) #normalization of the probabilities such that \\sum(p) = 1\n",
    "M = length(x)\n",
    "expected_value = mean(Truncated(d, -0.8, 1.3))\n",
    "#data similar to what would be expected from a PV probabilistic forecast.\n",
    "println(\"The expected value of the data is $expected_value\")"
   ]
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   "cell_type": "markdown",
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    "The VaR of for a given confidence interval $\\varepsilon \\in (0,1]$, is the $\\varepsilon$ percentile of the data provided. \n",
    "\n",
    "The CVaR for a given confidence interval $\\varepsilon \\in (0,1]$, is defined as the expected value of the losses higher than the $\\varepsilon$-quantile of the distribution. \n",
    "\n",
    "Both risk measures can be calculated for the data as follows:  "
   ]
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    {
     "name": "stdout",
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     "text": [
      "The expected value of the data is 0.7707977497442995\n"
     ]
    }
   ],
   "source": [
    "VaR = quantile(Truncated(d, -0.8, 1.3),ϵ)\n",
    "idx_cvar = (x .<= VaR);\n",
    "println(\"The Value at Risk of the data is $VaR\")\n",
    "p_cvar = p_n[idx_cvar];\n",
    "c_var = sum(x[idx_cvar].*p_n[idx_cvar])/ϵ;\n",
    "println(\"The Conditional Value at Risk of the data is $c_var\")"
   ]
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   "cell_type": "markdown",
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    "As predicted by the theory, the CVaR is larger (in the sense of the lossess) than the VaR. This reflects the fact that CVaR provides information about the expected value in the tail of the distribution and not just the value at the percentile of interest. In classical financial applications of CVaR the decicion maker is interested only in one of the tails, in this case the negative valued one.\n",
    "\n",
    "The following plot depicts the results of the analysis for the provided data. "
   ]
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     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Value at Risk of the data is -0.2182523997475212\n",
      "The Conditional Value at Risk of the data is -0.373607638124112\n"
     ]
    }
   ],
   "source": [
    "plot(x,p,lab = \"Random Variable\", ylabel = \"Probability [0-1]\")\n",
    "plot!([VaR], linetype=:vline, lab = \"VaR\")\n",
    "plot!([c_var], linetype=:vline, lab = \"CVaR\")\n",
    "plot!(x[idx_cvar],p[idx_cvar], fill=(0, :red), lab = \"CVaR_area\", xlabel = \"Gains\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bertsimas and Brown model \n",
    "\n",
    "[Bertsimas and Brown \\[1\\]](http://pubsonline.informs.org/doi/abs/10.1287/opre.1080.0646?journalCode=opre) presents a methodology for constructing uncertainty sets within a robust optimization framework for linear optimization problems with uncertain data. The motivation for the formulation is twofold; first, generally there are only a finite number of observations of the unknown data. Second, even if equiped with a perfect description of the uncertain data it is not clear how to construct an uncertainty set. \n",
    "\n",
    "In this notebook, we will attempt to reproduce the formulation from Bertsimas for a data set with distinct probabilities as expected from a probabilistic forecast. The objective is to obtain the same results for the risk measures using an statistical framework as well as an optimization setting. For the sake of a clear explanation, recall the definition of CVaR: \n",
    "\n",
    "$$\n",
    "CVaR_{\\varepsilon}(\\widetilde{z}) \\triangleq \\min_{a} \\left(a+\\frac{1}{\\varepsilon}\\mathbb{E}\\left[\\left(-\\widetilde{z} -a\\right)^{+}\\right]\\right)\n",
    "$$  \n",
    "\n",
    "Assuming that the random data $\\widetilde{z}$ is provided corresponds is discretized in $M$ samples with assigned probabilities $p^k$. The $CVaR$ for a confidence level $\\varepsilon$ can be calculated as follows: \n",
    "\n",
    "$$\n",
    "\\mathcal{U} = \\text{conv}\\left( \\left\\{\\frac{1}{\\varepsilon} \\sum_{i \\in I} p^i z^i  + \\left( 1-\\frac{1}{\\varepsilon}\\sum_{i \\in I}^M p^i\\right) z^j: I \\subset \\{1,...,M\\}, \\ j \\subset \\{1,...,M\\} \\setminus I,\\sum_{i \\in I} p^i \\le \\varepsilon \\right\\}\\right)\n",
    "$$\n",
    "\n",
    "This set formulation defines a polytope, which has the interpretation of the convex hull of all $j$-point averages of \n",
    "$\\widetilde{z}$. The intuition behind this model is that it is possible to define a convex hull from the discrete samples of the random variables, and the size is determined by a generating family $\\mathcal{Q} = \\left\\{conv\\left(\\left\\{q^1,...q^k\\right\\}\\right): q_k \\le \\frac{p_k}{\\varepsilon}\\right\\}$.\n",
    "\n",
    "This set can be implemented as an optimization problem as follows:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{a,y} \\ &a + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y^k\\\\\n",
    "       s.t.\\ &a + y^k \\ge z^k \\ &\\forall k = 1,...,M\\\\\n",
    "             & y^k \\ge 0 \\ &\\forall k = 1,...,M\\\\           \n",
    "\\end{aligned}\n",
    "$$         \n",
    "\n",
    "Where the dummy variables $y^k$ are introduced to handle the non-linear $(\\cdot)^{+}$ operator in the definition of the CVaR. This formulation also provides the value of the VaR for the same $\\varepsilon$, calculated with the variable $a$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "cvar_opt = Model(solver=GurobiSolver(Presolve=1));\n",
    "@variable(cvar_opt, a)\n",
    "@variable(cvar_opt, y[k = 1:M] >= 0);\n",
    "@constraint(cvar_opt, a + y[k = 1:M] .>= -1* x[k = 1:M]);\n",
    "@objective(cvar_opt, Min, a + 1/(ϵ)*sum((p_n[k]*y[k]) for k = 1:M))  ;\n",
    "solve(cvar_opt);\n",
    "println(\"\")            \n",
    "println(\"CVaR using primal optimization model $(-1*getobjectivevalue(cvar_opt))\")\n",
    "println(\"VaR using primal optimization model $(-1*getvalue(a))\")            "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The previous problem still requires the enumeration of all possible scenarios $k$, generating $M$ inequality constraints, which doesn't represent any improvement over classical stochastic optimization from the computational perspective. To overcome this limitation, the dual of the optimization problem can be formulated as follows and presented in [Natarajan, et. al. \\[3\\]](http://pubsonline.informs.org/doi/abs/10.1287/opre.1080.0683?journalCode=opre). \n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\max_{\\bf{u}} \\ &\\sum_{k=1}^M u^k z^k\\\\\n",
    "       s.t.\\ & \\sum_{k=1}^M u^k  = 1\\\\\n",
    "             & 0 \\le u^k \\le \\frac{1}{\\varepsilon}p^k  &\\forall k = 1,...,M\\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "According to [Natarajan, et. al. \\[3\\]](http://pubsonline.informs.org/doi/abs/10.1287/opre.1080.0683?journalCode=opre), if the loss function is affine, the dual is equivalent to: \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "-\\min_{\\bf{u},\\bar{z}} \\ &\\bar{z}\\\\\n",
    "       s.t.\\ & \\bar{z} = \\sum_{k=1}^M u^k z^k\\\\\n",
    "             & \\sum_{k=1}^M u^k  = 1\\\\\n",
    "             & 0 \\le u^k \\le \\frac{1}{\\varepsilon}p^k  &\\forall k = 1,...,M\\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The calibration parameter $\\varepsilon$ describes the confidence level. When $\\varepsilon \\rightarrow 1$ the value of CVaR tends to the $\\mathbb{E}[\\widetilde{z}]$ and when $\\varepsilon \\rightarrow 0$ the value of CVaR corresponds to the $\\inf$ of the values in the sample. In both optimization models this property holds, but it is more intuitive in the dual formulation. \n",
    "\n",
    "For $\\varepsilon = 1$, variables $u^k$ can take at maximum the value of $p^k$ and given the constraint that limits the values of $u^k$ to be in a convex hull forces $u^k = p^k$. Hence, $\\bar{z} = \\mathbb{E}[\\widetilde{z}]$. On the other hand when $\\varepsilon \\rightarrow 0$ the upper boundary of $u^k \\rightarrow \\infty$; however, the convex hull constraints $\\bf{u}$ to be $j$-th canonical vector with a 1 in the entry $j$ corresponding to the minimum value of $\\widetilde{z}$.  "
   ]
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  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "cvar_opt2 = Model(solver=GurobiSolver(Presolve=1))\n",
    "@variable(cvar_opt2, z)\n",
    "@variable(cvar_opt2, 0 <= u[k = 1:M] <= p_n[k]/(ϵ))\n",
    "@constraint(cvar_opt2, z == sum(u[k]*x[k] for k = 1:M))\n",
    "@constraint(cvar_opt2, sum(u[k] for k = 1:M)  == 1)\n",
    "@objective(cvar_opt2, Min, z)  \n",
    "solve(cvar_opt2)\n",
    "println(\"\")   \n",
    "println(\"CVaR using primal optimization model $(getobjectivevalue(cvar_opt2))\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results of the computational experiment show that the dual formulation is more efficient and describes the uncertainty set in a more compact way. The primal formulation for the example case has 211 rows and 212 columns, while the dual contains 2 rows and 212 columns. As a result, the dual converges within 1 iteration, while the primal formulation takes 59. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basic one unit dispatch with CVaR constraints\n",
    "\n",
    "Having tested two possible formulations for the calculation of the CVaR of a random variable, this section of the notebook showcases the application of these models to the ED problem. In this particular case, the test system has only one conventional generator, one PV power plant and one load. The problem considers that the operator obtained a probabilistic forecast for the PV power plant from a provider that consists of the following data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only\n",
      "Optimize a model with 211 rows, 212 columns and 422 nonzeros\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e+00, 1e+00]\n",
      "  Objective range  [2e-03, 1e+00]\n",
      "  Bounds range     [0e+00, 0e+00]\n",
      "  RHS range        [1e-02, 1e+00]\n",
      "Presolve removed 52 rows and 52 columns\n",
      "Presolve time: 0.00s\n",
      "Presolved: 159 rows, 160 columns, 318 nonzeros\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0    3.7039900e-01   0.000000e+00   0.000000e+00      0s\n",
      "       0    3.7039900e-01   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 0 iterations and 0.00 seconds\n",
      "Optimal objective  3.703989998e-01\n",
      "\n",
      "CVaR using primal optimization model -0.3703989997685551\n",
      "VaR using primal optimization model -0.22\n",
      "Academic license - for non-commercial use only\n",
      "Optimize a model with 2 rows, 212 columns and 422 nonzeros\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e-02, 1e+00]\n",
      "  Objective range  [1e+00, 1e+00]\n",
      "  Bounds range     [2e-03, 1e+00]\n",
      "  RHS range        [1e+00, 1e+00]\n",
      "Presolve removed 1 rows and 52 columns\n",
      "Presolve time: 0.00s\n",
      "Presolved: 1 rows, 160 columns, 160 nonzeros\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0   -8.0000000e-01   9.983211e-01   0.000000e+00      0s\n",
      "       1   -3.7039900e-01   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 1 iterations and 0.00 seconds\n",
      "Optimal objective -3.703989998e-01\n",
      "\n",
      "CVaR using primal optimization model -0.3703989997685551\n"
     ]
    }
   ],
   "source": [
    "d = Normal(58,10)\n",
    "solar_output = 17:0.1:95 #In MW\n",
    "p = pdf.(d,solar_output)\n",
    "p_n_pv = p/sum(p) #Normalized probability\n",
    "M = length(solar_output);\n",
    "#The system data for the rest of the components is as follows\n",
    "load = 125;\n",
    "generator_max = 85;\n",
    "generator_min = 50;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "<g clip-path=\"url(#clip3100)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:16; text-anchor:middle;\" transform=\"rotate(-90, 14.4, 185.966)\" x=\"14.4\" y=\"185.966\">Probabily [0-1]</text>\n",
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       "  77.4341,350.263 78.0601,350.26 78.686,350.257 79.312,350.253 79.9379,350.25 80.5638,350.246 81.1898,350.242 81.8157,350.239 82.4417,350.235 83.0676,350.23 \n",
       "  83.6935,350.226 84.3195,350.222 84.9454,350.217 85.5714,350.212 86.1973,350.207 86.8233,350.202 87.4492,350.196 88.0751,350.191 88.7011,350.185 89.327,350.179 \n",
       "  89.953,350.173 90.5789,350.166 91.2048,350.159 91.8308,350.152 92.4567,350.145 93.0827,350.138 93.7086,350.13 94.3345,350.122 94.9605,350.113 95.5864,350.105 \n",
       "  96.2124,350.096 96.8383,350.087 97.4642,350.077 98.0902,350.067 98.7161,350.057 99.3421,350.046 99.968,350.035 100.594,350.023 101.22,350.011 101.846,349.999 \n",
       "  102.472,349.986 103.098,349.973 103.724,349.96 104.35,349.946 104.976,349.931 105.601,349.916 106.227,349.9 106.853,349.884 107.479,349.867 108.105,349.85 \n",
       "  108.731,349.832 109.357,349.814 109.983,349.795 110.609,349.775 111.235,349.755 111.861,349.734 112.487,349.712 113.113,349.689 113.739,349.666 114.365,349.642 \n",
       "  114.991,349.617 115.617,349.592 116.242,349.565 116.868,349.538 117.494,349.51 118.12,349.481 118.746,349.451 119.372,349.42 119.998,349.388 120.624,349.355 \n",
       "  121.25,349.321 121.876,349.286 122.502,349.25 123.128,349.213 123.754,349.174 124.38,349.135 125.006,349.094 125.632,349.052 126.258,349.008 126.883,348.963 \n",
       "  127.509,348.917 128.135,348.87 128.761,348.821 129.387,348.77 130.013,348.718 130.639,348.665 131.265,348.61 131.891,348.553 132.517,348.494 133.143,348.434 \n",
       "  133.769,348.372 134.395,348.308 135.021,348.243 135.647,348.175 136.273,348.106 136.899,348.034 137.524,347.961 138.15,347.885 138.776,347.807 139.402,347.727 \n",
       "  140.028,347.645 140.654,347.56 141.28,347.473 141.906,347.384 142.532,347.292 143.158,347.198 143.784,347.101 144.41,347.001 145.036,346.899 145.662,346.793 \n",
       "  146.288,346.685 146.914,346.574 147.54,346.46 148.165,346.343 148.791,346.223 149.417,346.1 150.043,345.973 150.669,345.843 151.295,345.71 151.921,345.573 \n",
       "  152.547,345.432 153.173,345.288 153.799,345.141 154.425,344.989 155.051,344.834 155.677,344.674 156.303,344.511 156.929,344.343 157.555,344.172 158.181,343.996 \n",
       "  158.806,343.815 159.432,343.63 160.058,343.441 160.684,343.247 161.31,343.048 161.936,342.845 162.562,342.636 163.188,342.423 163.814,342.204 164.44,341.98 \n",
       "  165.066,341.751 165.692,341.517 166.318,341.277 166.944,341.031 167.57,340.78 168.196,340.522 168.821,340.259 169.447,339.99 170.073,339.715 170.699,339.434 \n",
       "  171.325,339.146 171.951,338.852 172.577,338.551 173.203,338.244 173.829,337.93 174.455,337.608 175.081,337.28 175.707,336.945 176.333,336.603 176.959,336.253 \n",
       "  177.585,335.896 178.211,335.531 178.837,335.158 179.462,334.778 180.088,334.39 180.714,333.993 181.34,333.589 181.966,333.176 182.592,332.755 183.218,332.325 \n",
       "  183.844,331.887 184.47,331.439 185.096,330.983 185.722,330.518 186.348,330.044 186.974,329.56 187.6,329.067 188.226,328.565 188.852,328.053 189.478,327.531 \n",
       "  190.103,326.999 190.729,326.457 191.355,325.906 191.981,325.343 192.607,324.771 193.233,324.188 193.859,323.594 194.485,322.99 195.111,322.374 195.737,321.748 \n",
       "  196.363,321.111 196.989,320.462 197.615,319.802 198.241,319.131 198.867,318.448 199.493,317.753 200.119,317.047 200.744,316.328 201.37,315.598 201.996,314.855 \n",
       "  202.622,314.101 203.248,313.333 203.874,312.554 204.5,311.762 205.126,310.957 205.752,310.139 206.378,309.309 207.004,308.465 207.63,307.609 208.256,306.739 \n",
       "  208.882,305.856 209.508,304.96 210.134,304.05 210.76,303.127 211.385,302.19 212.011,301.239 212.637,300.275 213.263,299.297 213.889,298.305 214.515,297.299 \n",
       "  215.141,296.279 215.767,295.245 216.393,294.196 217.019,293.134 217.645,292.057 218.271,290.966 218.897,289.86 219.523,288.74 220.149,287.606 220.775,286.457 \n",
       "  221.401,285.293 222.026,284.115 222.652,282.923 223.278,281.716 223.904,280.494 224.53,279.257 225.156,278.006 225.782,276.74 226.408,275.459 227.034,274.164 \n",
       "  227.66,272.854 228.286,271.53 228.912,270.191 229.538,268.837 230.164,267.469 230.79,266.086 231.416,264.688 232.042,263.276 232.667,261.85 233.293,260.41 \n",
       "  233.919,258.955 234.545,257.485 235.171,256.002 235.797,254.504 236.423,252.993 237.049,251.467 237.675,249.928 238.301,248.375 238.927,246.808 239.553,245.227 \n",
       "  240.179,243.634 240.805,242.026 241.431,240.406 242.057,238.773 242.683,237.126 243.308,235.467 243.934,233.795 244.56,232.111 245.186,230.414 245.812,228.705 \n",
       "  246.438,226.984 247.064,225.251 247.69,223.506 248.316,221.751 248.942,219.983 249.568,218.205 250.194,216.416 250.82,214.616 251.446,212.806 252.072,210.985 \n",
       "  252.698,209.155 253.323,207.315 253.949,205.465 254.575,203.606 255.201,201.738 255.827,199.861 256.453,197.976 257.079,196.083 257.705,194.182 258.331,192.273 \n",
       "  258.957,190.357 259.583,188.433 260.209,186.504 260.835,184.567 261.461,182.625 262.087,180.676 262.713,178.723 263.339,176.764 263.964,174.8 264.59,172.832 \n",
       "  265.216,170.859 265.842,168.883 266.468,166.904 267.094,164.921 267.72,162.936 268.346,160.949 268.972,158.959 269.598,156.968 270.224,154.976 270.85,152.983 \n",
       "  271.476,150.989 272.102,148.996 272.728,147.003 273.354,145.011 273.98,143.02 274.605,141.03 275.231,139.043 275.857,137.058 276.483,135.076 277.109,133.098 \n",
       "  277.735,131.123 278.361,129.152 278.987,127.186 279.613,125.225 280.239,123.269 280.865,121.319 281.491,119.376 282.117,117.439 282.743,115.51 283.369,113.588 \n",
       "  283.995,111.675 284.621,109.77 285.246,107.874 285.872,105.988 286.498,104.111 287.124,102.245 287.75,100.39 288.376,98.546 289.002,96.7139 289.628,94.894 \n",
       "  290.254,93.0868 290.88,91.2927 291.506,89.5122 292.132,87.7456 292.758,85.9936 293.384,84.2565 294.01,82.5347 294.636,80.8287 295.262,79.139 295.887,77.466 \n",
       "  296.513,75.8102 297.139,74.1719 297.765,72.5516 298.391,70.9497 299.017,69.3667 299.643,67.803 300.269,66.259 300.895,64.7351 301.521,63.2317 302.147,61.7493 \n",
       "  302.773,60.2883 303.399,58.849 304.025,57.4319 304.651,56.0372 305.277,54.6656 305.903,53.3172 306.528,51.9925 307.154,50.6919 307.78,49.4157 308.406,48.1643 \n",
       "  309.032,46.938 309.658,45.7372 310.284,44.5622 310.91,43.4134 311.536,42.2911 312.162,41.1956 312.788,40.1272 313.414,39.0863 314.04,38.0731 314.666,37.0879 \n",
       "  315.292,36.131 315.918,35.2028 316.544,34.3034 317.169,33.4331 317.795,32.5922 318.421,31.7809 319.047,30.9995 319.673,30.2481 320.299,29.5271 320.925,28.8366 \n",
       "  321.551,28.1769 322.177,27.5481 322.803,26.9504 323.429,26.3839 324.055,25.849 324.681,25.3456 325.307,24.8741 325.933,24.4344 326.559,24.0268 327.185,23.6513 \n",
       "  327.81,23.3081 328.436,22.9973 329.062,22.7189 329.688,22.4731 330.314,22.2599 330.94,22.0794 331.566,21.9317 332.192,21.8167 332.818,21.7346 333.444,21.6853 \n",
       "  334.07,21.6688 334.696,21.6853 335.322,21.7346 335.948,21.8167 336.574,21.9317 337.2,22.0794 337.826,22.2599 338.451,22.4731 339.077,22.7189 339.703,22.9973 \n",
       "  340.329,23.3081 340.955,23.6513 341.581,24.0268 342.207,24.4344 342.833,24.8741 343.459,25.3456 344.085,25.849 344.711,26.3839 345.337,26.9504 345.963,27.5481 \n",
       "  346.589,28.1769 347.215,28.8366 347.841,29.5271 348.466,30.2481 349.092,30.9995 349.718,31.7809 350.344,32.5922 350.97,33.4331 351.596,34.3034 352.222,35.2028 \n",
       "  352.848,36.131 353.474,37.0879 354.1,38.0731 354.726,39.0863 355.352,40.1272 355.978,41.1956 356.604,42.2911 357.23,43.4134 357.856,44.5622 358.482,45.7372 \n",
       "  359.107,46.938 359.733,48.1643 360.359,49.4157 360.985,50.6919 361.611,51.9925 362.237,53.3172 362.863,54.6656 363.489,56.0372 364.115,57.4319 364.741,58.849 \n",
       "  365.367,60.2883 365.993,61.7493 366.619,63.2317 367.245,64.7351 367.871,66.259 368.497,67.803 369.123,69.3667 369.748,70.9497 370.374,72.5516 371,74.1719 \n",
       "  371.626,75.8102 372.252,77.466 372.878,79.139 373.504,80.8287 374.13,82.5347 374.756,84.2565 375.382,85.9936 376.008,87.7456 376.634,89.5122 377.26,91.2927 \n",
       "  377.886,93.0868 378.512,94.894 379.138,96.7139 379.764,98.546 380.389,100.39 381.015,102.245 381.641,104.111 382.267,105.988 382.893,107.874 383.519,109.77 \n",
       "  384.145,111.675 384.771,113.588 385.397,115.51 386.023,117.439 386.649,119.376 387.275,121.319 387.901,123.269 388.527,125.225 389.153,127.186 389.779,129.152 \n",
       "  390.405,131.123 391.03,133.098 391.656,135.076 392.282,137.058 392.908,139.043 393.534,141.03 394.16,143.02 394.786,145.011 395.412,147.003 396.038,148.996 \n",
       "  396.664,150.989 397.29,152.983 397.916,154.976 398.542,156.968 399.168,158.959 399.794,160.949 400.42,162.936 401.046,164.921 401.671,166.904 402.297,168.883 \n",
       "  402.923,170.859 403.549,172.832 404.175,174.8 404.801,176.764 405.427,178.723 406.053,180.676 406.679,182.625 407.305,184.567 407.931,186.504 408.557,188.433 \n",
       "  409.183,190.357 409.809,192.273 410.435,194.182 411.061,196.083 411.687,197.976 412.312,199.861 412.938,201.738 413.564,203.606 414.19,205.465 414.816,207.315 \n",
       "  415.442,209.155 416.068,210.985 416.694,212.806 417.32,214.616 417.946,216.416 418.572,218.205 419.198,219.983 419.824,221.751 420.45,223.506 421.076,225.251 \n",
       "  421.702,226.984 422.328,228.705 422.953,230.414 423.579,232.111 424.205,233.795 424.831,235.467 425.457,237.126 426.083,238.773 426.709,240.406 427.335,242.026 \n",
       "  427.961,243.634 428.587,245.227 429.213,246.808 429.839,248.375 430.465,249.928 431.091,251.467 431.717,252.993 432.343,254.504 432.969,256.002 433.594,257.485 \n",
       "  434.22,258.955 434.846,260.41 435.472,261.85 436.098,263.276 436.724,264.688 437.35,266.086 437.976,267.469 438.602,268.837 439.228,270.191 439.854,271.53 \n",
       "  440.48,272.854 441.106,274.164 441.732,275.459 442.358,276.74 442.984,278.006 443.609,279.257 444.235,280.494 444.861,281.716 445.487,282.923 446.113,284.115 \n",
       "  446.739,285.293 447.365,286.457 447.991,287.606 448.617,288.74 449.243,289.86 449.869,290.966 450.495,292.057 451.121,293.134 451.747,294.196 452.373,295.245 \n",
       "  452.999,296.279 453.625,297.299 454.25,298.305 454.876,299.297 455.502,300.275 456.128,301.239 456.754,302.19 457.38,303.127 458.006,304.05 458.632,304.96 \n",
       "  459.258,305.856 459.884,306.739 460.51,307.609 461.136,308.465 461.762,309.309 462.388,310.139 463.014,310.957 463.64,311.762 464.266,312.554 464.891,313.333 \n",
       "  465.517,314.101 466.143,314.855 466.769,315.598 467.395,316.328 468.021,317.047 468.647,317.753 469.273,318.448 469.899,319.131 470.525,319.802 471.151,320.462 \n",
       "  471.777,321.111 472.403,321.748 473.029,322.374 473.655,322.99 474.281,323.594 474.907,324.188 475.532,324.771 476.158,325.343 476.784,325.906 477.41,326.457 \n",
       "  478.036,326.999 478.662,327.531 479.288,328.053 479.914,328.565 480.54,329.067 481.166,329.56 481.792,330.044 482.418,330.518 483.044,330.983 483.67,331.439 \n",
       "  484.296,331.887 484.922,332.325 485.548,332.755 486.173,333.176 486.799,333.589 487.425,333.993 488.051,334.39 488.677,334.778 489.303,335.158 489.929,335.531 \n",
       "  490.555,335.896 491.181,336.253 491.807,336.603 492.433,336.945 493.059,337.28 493.685,337.608 494.311,337.93 494.937,338.244 495.563,338.551 496.189,338.852 \n",
       "  496.814,339.146 497.44,339.434 498.066,339.715 498.692,339.99 499.318,340.259 499.944,340.522 500.57,340.78 501.196,341.031 501.822,341.277 502.448,341.517 \n",
       "  503.074,341.751 503.7,341.98 504.326,342.204 504.952,342.423 505.578,342.636 506.204,342.845 506.83,343.048 507.455,343.247 508.081,343.441 508.707,343.63 \n",
       "  509.333,343.815 509.959,343.996 510.585,344.172 511.211,344.343 511.837,344.511 512.463,344.674 513.089,344.834 513.715,344.989 514.341,345.141 514.967,345.288 \n",
       "  515.593,345.432 516.219,345.573 516.845,345.71 517.471,345.843 518.096,345.973 518.722,346.1 519.348,346.223 519.974,346.343 520.6,346.46 521.226,346.574 \n",
       "  521.852,346.685 522.478,346.793 523.104,346.899 523.73,347.001 524.356,347.101 524.982,347.198 525.608,347.292 526.234,347.384 526.86,347.473 527.486,347.56 \n",
       "  528.111,347.645 528.737,347.727 529.363,347.807 529.989,347.885 530.615,347.961 531.241,348.034 531.867,348.106 532.493,348.175 533.119,348.243 533.745,348.308 \n",
       "  534.371,348.372 534.997,348.434 535.623,348.494 536.249,348.553 536.875,348.61 537.501,348.665 538.127,348.718 538.752,348.77 539.378,348.821 540.004,348.87 \n",
       "  540.63,348.917 541.256,348.963 541.882,349.008 542.508,349.052 543.134,349.094 543.76,349.135 544.386,349.174 545.012,349.213 545.638,349.25 546.264,349.286 \n",
       "  546.89,349.321 547.516,349.355 548.142,349.388 548.768,349.42 549.393,349.451 550.019,349.481 550.645,349.51 551.271,349.538 551.897,349.565 552.523,349.592 \n",
       "  553.149,349.617 553.775,349.642 554.401,349.666 555.027,349.689 555.653,349.712 556.279,349.734 556.905,349.755 557.531,349.775 558.157,349.795 558.783,349.814 \n",
       "  559.409,349.832 560.034,349.85 560.66,349.867 561.286,349.884 561.912,349.9 562.538,349.916 563.164,349.931 563.79,349.946 564.416,349.96 565.042,349.973 \n",
       "  565.668,349.986 \n",
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "println(\"Mean forecast value $(mean(Truncated(d, 10, 80))) MW\")\n",
    "plot(solar_output,p, lab = \"Solar Power Output\", xaxis = \"Power Output [MW]\", yaxis = \"Probabily [0-1]\")\n",
    "plot!([(mean(Truncated(d, 17, 95)))], linetype=:vline, lab = \"Expected Value of the PV output\")"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analysis of risk for ED \n",
    "\n",
    "Following the same method as before, this section performs the statistic analysis of the risk measure for the net load values in the ED case under study. First, we will explore the risk of not being able to supply energy for all the load, corresponding to the Left-hand-side tail of the forecast where the solar PV output is below the expected value.\n",
    "\n",
    "In order to perform this analysis, we formulate the risk in terms of the netload, which corresponds to the difference between the load and the solar output $P_{load} - P_{pv}$. Consider that $P_{pv}$ is a random variable, thus, making the net load a random variable as well.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean forecast value 57.640292220935876 MW\n",
      "VaR of Load Shedding is 83.4471562393571\n"
     ]
    }
   ],
   "source": [
    "ϵ = 0.05\n",
    "VaR_pv = (load - quantile(Truncated(d, 17, 95),ϵ))\n",
    "println(\"VaR of Load Shedding is $VaR_pv\")\n",
    "idx_cvar_pv = ((load-solar_output) .> VaR_pv);\n",
    "p_cvar_pv = p_n_pv[idx_cvar_pv];\n",
    "c_var_pv = sum((load -solar_output[idx_cvar_pv]).*p_n_pv[idx_cvar_pv])/ϵ;\n",
    "println(\"CVaR of Load Shedding is $c_var_pv\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following plot explains better the application of the concept to the dispatch problem . In this case, the value of the load to be supplied for $1-\\varepsilon$ confidence interval is 83.44 MW which is very safe considering that the generator capacity is equal to 85 MW. If the generator is dispatched at a level of 70 MW, the VaR of the load will be 13.44 MW; however, if the generator is dispatched at its maximum output, according to the metric, there is no VaR of load shedding.   \n",
    "\n",
    "However, if we use the CVaR measure is possible to determine that the expected value in these case is equal to 87.56 MW which is above the capacity of the generator, thus, in dispatched at maximum capacity in the tail we can expect 2.56 MW of load shedding. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CVaR of Load Shedding is 87.56729121415364\n"
     ]
    }
   ],
   "source": [
    "net_load = load - solar_output;\n",
    "plot(net_load,p, lab = \"Total Net Load\", ylabel = \"Probability\", xticks = 20:10:125)\n",
    "plot!([generator_max], linetype=:vline, lab = \"Generator Upper Limit\")\n",
    "plot!([VaR_pv], linetype=:vline, lab = \"Load VaR\")\n",
    "plot!([c_var_pv], linetype=:vline, lab = \"Load CVaR\")\n",
    "plot!(net_load[idx_cvar_pv],p[idx_cvar_pv], fill=(0, :red), lab = \"CVaR Load\", xlabel = \"netload\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "e_var (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function e_var()\n",
    "    var_mat = []\n",
    "    c_var_mat = []\n",
    "    solar_output = 17:0.1:95 #In MW\n",
    "    for ϵ = 0.01:0.01:0.999\n",
    "        VaR_pv = (quantile(Truncated(d, 17, 95),ϵ) - load)\n",
    "        idx_cvar_pv = ((solar_output-load) .<= VaR_pv);\n",
    "        p_cvar_pv = p_n_pv[idx_cvar_pv];\n",
    "        c_var_pv = sum((solar_output[idx_cvar_pv]-load).*p_n_pv[idx_cvar_pv])/ϵ;\n",
    "        push!(c_var_mat, c_var_pv)\n",
    "        push!(var_mat, VaR_pv)\n",
    "    end\n",
    "    return c_var_mat, var_mat\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A different way of showcasing these results is by plotting the supply requirement for different values of $\\varepsilon$ in the case of load shedding. The plot shows that CVaR as a risk measure dominates over VaR, at the largest possible value of $\\varepsilon$, CVaR converges to the expected value of the solar power. \n",
    "\n",
    "This is a more practial explanation of what the risk measure means, if we assume that the probabilities provided by the probabilistic forecast are a perfect representation of the uncertain variables there is no risk of losing load because the generator maximum capacity is above the supply required. However, for small confidence intervals it is possible to assess the risks of not having enough installed capacity to supply the load. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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    }
   ],
   "source": [
    "t1, t2 = e_var();\n",
    "plot(0.01:0.01:0.999,-1*t2, yticks = 40:10:100, xticks = 0:0.1:1,\n",
    "    lab = \"VaR of Net Load\", yaxis= \"Supply required\")\n",
    "plot!(0.01:0.01:0.999,-1*t1, lab = \"CVaR of Net Load\", xaxis = L\"$\\epsilon$\")\n",
    "plot!([generator_max], linetype=:hline, lab = \"Generator Upper Limit\")\n",
    "scatter!([1],[load-mean(Truncated(d, 17, 95))],lab = \"Maximum confidence interval CVaR\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to test the optimization formulation, the next cells will show the models for CVaR adapted and calculated to replicate the results from the statistical analysis in terms of calculation of the supply needs. For this application, the loss function is defined as the difference between load and solar generation in the following fashion. \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{a,y} \\ &a + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y^k\\\\\n",
    "       s.t.\\ &a + y^k \\ge -(P_{pv}^k - P_{load}) \\ &\\forall k = 1,...,M\\\\\n",
    "             & y^k \\ge 0 \\ &\\forall k = 1,...,M\\\\           \n",
    "\\end{aligned}\n",
    "$$  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "cvar_opt_pv = Model(solver=GurobiSolver(Presolve=1));\n",
    "@variable(cvar_opt_pv, a)\n",
    "@variable(cvar_opt_pv, y[k = 1:M] >= 0);\n",
    "@constraint(cvar_opt_pv, a + y[k = 1:M] .>= -1* (solar_output[k = 1:M] - load));\n",
    "@objective(cvar_opt_pv, Min, a + 1/(ϵ)*sum((p_n_pv[k]*y[k]) for k = 1:M))  ;\n",
    "solve(cvar_opt_pv);\n",
    "println(\"\")            \n",
    "println(\"CVaR using primal optimization model $(getobjectivevalue(cvar_opt_pv))\")\n",
    "println(\"VaR using primal optimization $(getvalue(a))\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The challenge in the context of ED is that it is possible to suffer curtailment and not only loss of load, and the convention for risk measures is that a negative value implies that the risk associated with the violation of the uncertainty constraint is acceptable. \n",
    "\n",
    "For this reason, a formulation that considers CVaR as a risk measure has to be adapted to take into account under and over supply. Thus, it is possible to formulate the problem of oversupply as a negative netload as follows: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "ϵ = 0.05\n",
    "VaR_pv = (load - quantile(Truncated(d, 17, 95),1-ϵ)) #For this calculation is easier to use the complement \\epsilon\n",
    "println(\"VaR of curtailment is $VaR_pv\")\n",
    "idx_cvar_pv = ((load - solar_output) .< VaR_pv);\n",
    "p_cvar_pv = p_n_pv[idx_cvar_pv];\n",
    "c_var_pv = sum((load-solar_output[idx_cvar_pv]).*p_n_pv[idx_cvar_pv])/(ϵ);\n",
    "println(\"CVaR of the net load is $c_var_pv\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following plot explains better the application of the concept to the dispatch problem in the case of power curtailment. In this case, the value of power that can be absorbed for $\\varepsilon$ confidence interval is 50.56 MW which is acceptable considering that the generator minimum capacity is equal to 50 MW. Likewise in the load shedding case, if the generator is dispatched at 70 MW, the renewable energy at risk of curtailment is 19.44 MW.   \n",
    "\n",
    "However, if we use the CVaR measure is possible to determine that the expected value in these case is equal to 46.7 MW which is below the capacity of the generator, thus, in the worst case scenario in the tail we can expect 3.3 MW of power curtailment.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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       "  496.814,346.612 496.189,346.501 495.563,346.388 494.937,346.271 494.311,346.15 493.685,346.027 493.059,345.9 492.433,345.77 491.807,345.637 491.181,345.5 \n",
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       "  385.273,62.931 421.273,62.931 \n",
       "  \"/>\n",
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       "  385.273,93.171 421.273,93.171 \n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 427.273, 97.671)\" x=\"427.273\" y=\"97.671\">Minimal Loading CVaR</text>\n",
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       "<g clip-path=\"url(#clip3700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 427.273, 112.791)\" x=\"427.273\" y=\"112.791\">CVaR Net Load</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only\n",
      "Optimize a model with 781 rows, 782 columns and 1562 nonzeros\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e+00, 1e+00]\n",
      "  Objective range  [2e-05, 1e+00]\n",
      "  Bounds range     [0e+00, 0e+00]\n",
      "  RHS range        [3e+01, 1e+02]\n",
      "Presolve time: 0.00s\n",
      "Presolved: 781 rows, 782 columns, 1562 nonzeros\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0      handle free variables                          0s\n",
      "     247    8.7616871e+01   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 247 iterations and 0.00 seconds\n",
      "Optimal objective  8.761687090e+01\n",
      "\n",
      "CVaR using primal optimization model 87.6168708994765\n",
      "VaR using primal optimization 83.4\n",
      "VaR of curtailment is 50.56128526476101\n",
      "CVaR of the net load is 46.30441082451583\n"
     ]
    }
   ],
   "source": [
    "net_load = load - solar_output;\n",
    "plot(net_load,p, lab = \"Total Net Load\", ylabel = \"Probability\", xticks = 20:10:125)\n",
    "plot!([generator_min], linetype=:vline, lab = \"Generator Lower Limit\")\n",
    "plot!([VaR_pv], linetype=:vline, lab = \"Minimal Loading VaR\")\n",
    "plot!([c_var_pv], linetype=:vline, lab = \"Minimal Loading CVaR\")\n",
    "plot!(net_load[idx_cvar_pv],p[idx_cvar_pv], fill=(0, :red), lab = \"CVaR Net Load\", xlabel = \"Net load\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "e_var (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function e_var()\n",
    "    var_mat = []\n",
    "    c_var_mat = []\n",
    "    solar_output = 17:0.1:95 #In MW\n",
    "    for ϵ = 0.01:0.01:0.999\n",
    "        VaR_pv = (load - quantile(Truncated(d, 17, 95),1-ϵ)) \n",
    "        #For this calculation is easier to use the complement \\epsilon, makes the implementation easier\n",
    "        idx_cvar_pv = ((load - solar_output) .< VaR_pv);\n",
    "        p_cvar_pv = p_n_pv[idx_cvar_pv];\n",
    "        c_var_pv = sum((load-solar_output[idx_cvar_pv]).*p_n_pv[idx_cvar_pv])/(ϵ);\n",
    "        push!(c_var_mat, c_var_pv)\n",
    "        push!(var_mat, VaR_pv)\n",
    "    end\n",
    "    return c_var_mat, var_mat\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These results can also be plotted for different values of $\\varepsilon$. In this case, the curves are inverted because the interested is related to the risk of hitting the lower limit of the generator. The relevant metric is no longer the maximum load to be supplied but the minimum loading.  \n",
    "\n",
    "For small confidence intervals it is possible to assess the risks of having to curtail power because the minimum generation is reached. Again, the CVaR dominated the VaR measure, but it does it by expecting that the minimum loading is reached for a large value of $\\varepsilon$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "ename": "BoundsError",
     "evalue": "\u001b[91mBoundsError: attempt to access 90-element StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}} at index [1:99]\u001b[39m",
     "output_type": "error",
     "traceback": [
      "\u001b[91mBoundsError: attempt to access 90-element StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}} at index [1:99]\u001b[39m",
      "",
      "Stacktrace:",
      " [1] \u001b[1mthrow_boundserror\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}}, ::Tuple{UnitRange{Int64}}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./abstractarray.jl:434\u001b[22m\u001b[22m",
      " [2] \u001b[1mcheckbounds\u001b[22m\u001b[22m at \u001b[1m./abstractarray.jl:362\u001b[22m\u001b[22m [inlined]",
      " [3] \u001b[1mgetindex\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}}, ::UnitRange{Int64}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./twiceprecision.jl:210\u001b[22m\u001b[22m",
      " [4] \u001b[1mgr_display\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Plots.Subplot{Plots.GRBackend}, ::Measures.Length{:mm,Float64}, ::Measures.Length{:mm,Float64}, ::Array{Float64,1}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Plots/src/backends/gr.jl:1039\u001b[22m\u001b[22m",
      " [5] \u001b[1mgr_display\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Plots.Plot{Plots.GRBackend}, ::String\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Plots/src/backends/gr.jl:583\u001b[22m\u001b[22m",
      " [6] \u001b[1m_show\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Base.AbstractIOBuffer{Array{UInt8,1}}, ::MIME{Symbol(\"image/svg+xml\")}, ::Plots.Plot{Plots.GRBackend}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Plots/src/backends/gr.jl:1380\u001b[22m\u001b[22m",
      " [7] \u001b[1mshow\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Base.AbstractIOBuffer{Array{UInt8,1}}, ::MIME{Symbol(\"image/svg+xml\")}, ::Plots.Plot{Plots.GRBackend}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Plots/src/output.jl:202\u001b[22m\u001b[22m",
      " [8] \u001b[1m#sprint#230\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Void, ::Function, ::Int64, ::Function, ::MIME{Symbol(\"image/svg+xml\")}, ::Vararg{Any,N} where N\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./strings/io.jl:66\u001b[22m\u001b[22m",
      " [9] \u001b[1mdisplay_dict\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Plots.Plot{Plots.GRBackend}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Plots/src/output.jl:295\u001b[22m\u001b[22m",
      " [10] \u001b[1m(::Compat.#inner#14{Array{Any,1},IJulia.#display_dict,Tuple{Plots.Plot{Plots.GRBackend}}})\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Compat/src/Compat.jl:332\u001b[22m\u001b[22m",
      " [11] \u001b[1mexecute_request\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::ZMQ.Socket, ::IJulia.Msg\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/IJulia/src/execute_request.jl:209\u001b[22m\u001b[22m",
      " [12] \u001b[1m(::Compat.#inner#14{Array{Any,1},IJulia.#execute_request,Tuple{ZMQ.Socket,IJulia.Msg}})\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/Compat/src/Compat.jl:332\u001b[22m\u001b[22m",
      " [13] \u001b[1meventloop\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::ZMQ.Socket\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/jdlara/.julia/v0.6/IJulia/src/eventloop.jl:8\u001b[22m\u001b[22m",
      " [14] \u001b[1m(::IJulia.##15#18)\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./task.jl:335\u001b[22m\u001b[22m"
     ]
    }
   ],
   "source": [
    "t1, t2 = e_var();\n",
    "plot(0.1:0.01:0.999, t2, xticks = 0:0.1:1, \n",
    "    lab = \"VaR of Net Load\", yaxis= \"Min Loading\")\n",
    "plot!(0.1:0.01:0.999, t1, yaxis = 20:10:120, lab = \"CVaR of Net Load\", xaxis = L\"$\\epsilon$\")\n",
    "plot!([generator_min], linetype=:hline, lab = \"Generator lower Limit\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to test the optimization formulation to capture the Left-hand-side tail of the distribution, the optimization formulation has to be modified slightly. Namely, the sign of the loss function has to be reversed as follows: \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{a,y} \\ &a + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y^k\\\\\n",
    "       s.t.\\ &a + y^k \\ge (P_{pv}^k - P_{load}) \\ &\\forall k = 1,...,M\\\\\n",
    "             & y^k \\ge 0 \\ &\\forall k = 1,...,M\\\\           \n",
    "\\end{aligned}\n",
    "$$  \n",
    "\n",
    "And the result of the optimization problem will be the negative of the CVaR and VaR. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "cvar_opt_pv = Model(solver=GurobiSolver(Presolve=1));\n",
    "@variable(cvar_opt_pv, a)\n",
    "@variable(cvar_opt_pv, y[k = 1:M] >= 0);\n",
    "@constraint(cvar_opt_pv, a + y[k = 1:M] .>= (solar_output[k = 1:M] - load));\n",
    "@objective(cvar_opt_pv, Min, a + 1/(ϵ)*sum((p_n_pv[k]*y[k]) for k = 1:M))  ;\n",
    "solve(cvar_opt_pv);\n",
    "println(\"\")\n",
    "println(\"CVaR using primal optimizatio model $(getobjectivevalue(cvar_opt_pv))\")\n",
    "println(\"VaR using primal optimization $(getvalue(a))\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two sided CVaR ED formulation v1 (using the primal of CVaR)\n",
    "\n",
    "The classic and simplest version of the ED problem for one generator $P_g$, one load $P_L$ and a PV power plant $\\bar{P_{pv}}$ can be formulated as follows: \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{P_g} \\ &P_g\\\\\n",
    "       s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "             & P_g + \\bar{P_{pv}} - P_L = 0\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Where the objective is to find the least cost dispatch that meets the net load calculated using the expected value of the solar power output $\\bar{P_{pv}}$. However, in order to formulate the problem using risk measures it is necessary to consider all the information about the random variable.  \n",
    "\n",
    "First, the loss function $\\mathcal{L}()$ for the CVaR can be defined as the violation of the power balance, in this setting the power balance is modeled using two independent loss functions corresponding to the CVaR of the upper tail and the lower tail of the distribution of the random variable $\\widetilde{P_{pv}}$. It is known that one tail of the distribution represents the risk of under supply (load shedding) and the other tail corresponds to the risk of over supply (power curtailment). In order to introduce the two tailed risks, measured with the CVaR the power balance constraints are reformulated as follows: \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{P_g} \\ &\\delta_1 + \\delta_2 + P_g\\\\\n",
    "       s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "             & CVaR_{\\varepsilon}\\left(P_g + \\widetilde{P_{pv}} - P_L\\right) \\le \\delta_1 & \\\\ \n",
    "             & CVaR_{\\varepsilon}\\left(-P_g - \\widetilde{P_{pv}} + P_L \\right) \\le \\delta_2 &\\\\\n",
    "\\end{aligned}             \n",
    "$$\n",
    "\n",
    "This mathematical program calculate the optimal value $P_g^*$ that also reduces the power balance constraint violation risk. Furthermore, the model can be reformulated as the robust counter part of the ED problem where uncertain variable is the expected value of the PV power plants $\\bar{P_{pv}}$, calculated within the uncertainty set defined by the risk measure. \n",
    "\n",
    "The aforementioned reformulation implies that two values $\\bar{P_{pv}}$ need be calculated, one for the curtailment risk and one for the load shedding risk. Moreover, in this setting $\\mathcal{U}_{pv}$ is actually two independent uncertainty sets corresponding to the CVaR of the upper tail and the lower tail of the distribution. It is possible to formulate a robust counterpart of the optimization model using the uncertainty set structure studied in the previous sections as follows: \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\begin{aligned}\n",
    "\\min_{P_g, \\delta_1, \\delta_2} \\ &\\delta_1 + \\delta_2 + P_g\\\\\n",
    "s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "&\\left\\{\\begin{array}{l@{\\;}ll}\n",
    "\\min_{a_1, \\bf{y_1}} & a_1 + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y_1^k &\\\\\n",
    " s.t. & a_1 + y_1^k \\ge -\\left(P_g + P_{pv}^k - P_L \\right)  &\\forall k = 1,...,M\\\\\n",
    " & y_1^k \\ge 0 \\quad &\\forall k = 1,...,M\n",
    "\\end{array}\\right\\} \\le \\delta_1\\\\\n",
    "&\\left\\{\\begin{array}{l@{\\;}ll}\n",
    "\\min_{a_2, \\bf{y_2}} & a_2 + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y_2^k &\\\\\n",
    " s.t. & a_2 + y_2^k \\ge \\quad \\left(P_g + P_{pv}^k - P_L \\right)  &\\forall k = 1,...,M\\\\\n",
    " & y_2^k \\ge 0 \\quad &\\forall k = 1,...,M\n",
    "\\end{array}\\right\\} \\le \\delta_2\\\\\n",
    "&\\delta_1 \\ge 0, \\quad \\delta_2 \\ge 0\n",
    "\\end{aligned}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "The model can be rewritten as follows: \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{P_g, \\delta_1, \\delta_2} \\ &\\delta_1 + \\delta_2 + P_g\\\\\n",
    "s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "& a_1 + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y_1^k \\le \\delta_1\\\\\n",
    "& a_1 + y_1^k \\ge -\\left(P_g + P_{pv}^k - P_L \\right) &\\forall k = 1,...,M\\\\\n",
    "& y_1^k \\ge 0 \\ &\\forall k = 1,...,M\\\\\n",
    "& a_2 + \\frac{1}{\\varepsilon}\\sum_{k=1}^M p^k y_2^k \\le \\delta_2\\\\\n",
    "& a_2 + y_2^k \\ge \\left(P_g + P_{pv}^k - P_L \\right) &\\forall k = 1,...,M\\\\\n",
    "& y_2^k \\ge 0 \\ &\\forall k = 1,...,M\\\\\n",
    "&\\delta_1 \\ge 0, \\quad \\delta_2 \\ge 0\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The results of implementing this model are as follows: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "code_folding": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only\n",
      "Optimize a model with 781 rows, 782 columns and 1562 nonzeros\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e+00, 1e+00]\n",
      "  Objective range  [2e-05, 1e+00]\n",
      "  Bounds range     [0e+00, 0e+00]\n",
      "  RHS range        [3e+01, 1e+02]\n",
      "Presolve time: 0.00s\n",
      "Presolved: 781 rows, 782 columns, 1562 nonzeros\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0      handle free variables                          0s\n",
      "     700   -4.6421003e+01   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 700 iterations and 0.01 seconds\n",
      "Optimal objective -4.642100281e+01\n",
      "\n",
      "CVaR using primal optimizatio model -46.42100281403589\n",
      "VaR using primal optimization -50.599999999999994\n"
     ]
    }
   ],
   "source": [
    "ϵ = 0.05\n",
    "cvar_edv1 = Model(solver=GurobiSolver(Presolve=1))\n",
    "\n",
    "#Generator feasible space\n",
    "@variable(cvar_edv1, generator_min <= p_g <= generator_max);\n",
    "#Slack variables for the power balance\n",
    "@variable(cvar_edv1, delta_1 >= 0)\n",
    "@variable(cvar_edv1, delta_2 >= 0)\n",
    "\n",
    "#Variables to estimate the VaR \n",
    "@variable(cvar_edv1, a1)\n",
    "@variable(cvar_edv1, a2)\n",
    "\n",
    "#Slack Variables CVaR\n",
    "@variable(cvar_edv1, y1[k = 1:M] >= 0);\n",
    "@variable(cvar_edv1, y2[k = 1:M] >= 0);\n",
    "\n",
    "#Robust Power balance constraints \n",
    "#RHS value of the CVaR constraints (+)\n",
    "@constraint(cvar_edv1, a1 + 1/(ϵ)*sum((p_n_pv[k]*y1[k]) for k = 1:M) <= delta_1);\n",
    "@constraint(cvar_edv1, a1 + y1[k = 1:M] .>= -1*(p_g + solar_output[k = 1:M] - load));           \n",
    "\n",
    "#LHS of the CVaR constraints (-)            \n",
    "@constraint(cvar_edv1, a2 + 1/(ϵ)*sum((p_n_pv[k]*y2[k]) for k = 1:M) <= delta_2);\n",
    "@constraint(cvar_edv1, a2 + y2[k = 1:M] .>= (p_g + solar_output[k = 1:M] - load));                                   \n",
    "\n",
    "@objective(cvar_edv1, Min, delta_1 + delta_2 + p_g);     "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve(cvar_edv1)\n",
    "println(\"\")\n",
    "println(\"The optimal Dispatch is $(getvalue(p_g))\")\n",
    "println(\"CVaR of Load Shedding is $(getvalue(delta_1))\")\n",
    "println(\"CVaR of Curtailment is $(getvalue(delta_2))\")"
   ]
  },
  {
   "cell_type": "markdown",
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    "The results of the model show the lowest cost dispatch and the CVaR values for each one of the possible cost drivers, load shedding and power curtailment. However, since the objective function is linear and both terms $\\delta_1$ and $\\delta_2$ are equally weighted, the optimizer dispatches the generator to the minimum value. This creates a comparatively high risk of Load Shedding. On the other hand, if the $\\delta$ terms a unevenvely weighted it will dispatch the generator to its maximum in order to reduce the risk of load shedding. \n",
    "\n",
    "This is known as a bang-bang solution, where the optimizer only chooses one of the extremes. The performance of the model can be improved by using a convex cost function that better balances both risks. "
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two sided CVaR ED formulation v2 (using the dual of CVaR)\n",
    "\n",
    "The previous formulation has the disadvantage that forces to declare a constraint for each scenario $z^k$; hence, the large size of rows described by the solver. However, it is possible to use the alternative formulation of the uncertainty set and arrange the problem in the following fashion:\n",
    "\n",
    "Where: \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "CVaR_{\\varepsilon}\\left(\\mathcal{L}\\left(\\widetilde{P_{pv}}\\right)\\right) = &\\left\\{\\begin{array}{l@{\\;}ll}\n",
    "-\\min_{\\bf{u},\\bar{z}} & \\mathcal{L}(\\bar{z}) \\\\\n",
    "& \\bar{z} = \\sum_{k=1}^M u^k P_{pv}^k \\\\\n",
    "&\\sum_{k=1}^M u_2^k = 1\\\\\n",
    "&0 \\le u_1^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\n",
    "\\end{array}\\right\\}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Exploiting the affine property of the loss function, the ED problem can be reformulated in an equivalent form as follows:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{P_g, \\delta_1, \\delta_2} \\ &\\delta_1 + \\delta_2 + P_g\\\\\n",
    "s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "&\\left\\{\\begin{array}{l@{\\;}ll}\n",
    "-\\min_{\\bf{u_1}} & - \\left(P_g +\\sum_{k=1}^M u_1^k P_{pv}^k - P_L \\right) &\\\\\n",
    "s.t.&\\sum_{k=1}^M u_1^k = 1\\\\\n",
    " & 0 \\le u_1^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\\\\\n",
    "\\end{array}\\right\\} \\le \\delta_1\\\\\n",
    "&\\left\\{\\begin{array}{l@{\\;}ll}\n",
    "-\\min_{\\bf{u_2}} & (P_g +\\sum_{k=1}^M u_2^k P_{pv}^k - P_L) \\\\\n",
    "s.t.&\\sum_{k=1}^M u_2^k = 1\\\\\n",
    " & 0 \\le u_2^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\n",
    "\\end{array}\\right\\} \\le \\delta_2\\\\\n",
    "&\\delta_1 \\ge 0, \\quad \\delta_2 \\ge 0\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Which leads to the following relaxed formulation:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{P_g, \\delta_1, \\delta_2} \\ &(\\delta_1 + \\delta_2)^2 + P_g\\\\\n",
    "       s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "             & - \\left(P_g +\\sum_{k=1}^M u_1^k P_{pv}^k - P_L \\right) \\ge \\delta_1\\\\\n",
    "             & \\sum_{k=1}^M u_1^k = 1\\\\\n",
    "             & 0 \\le u_1^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\\\\\n",
    "             & P_g +\\sum_{k=1}^M u_2^k P_{pv}^k - P_L  \\ge \\delta_2\\\\\n",
    "             & \\sum_{k=1}^M u_2^k = 1\\\\\n",
    "             & 0 \\le u_2^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\\\\ \n",
    "             & \\delta_1 \\ge 0, \\ \\delta_2 \\ge 0\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The results of implementing this model are as follows: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       ":Optimal"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only\n",
      "Optimize a model with 1564 rows, 1567 columns and 6252 nonzeros\n",
      "Coefficient statistics:\n",
      "  Matrix range     [2e-05, 1e+00]\n",
      "  Objective range  [1e+00, 1e+00]\n",
      "  Bounds range     [5e+01, 8e+01]\n",
      "  RHS range        [3e+01, 1e+02]\n",
      "Presolve time: 0.00s\n",
      "Presolved: 1564 rows, 1567 columns, 6252 nonzeros\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0      handle free variables                          0s\n",
      "     456    9.1195868e+01   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 456 iterations and 0.02 seconds\n",
      "Optimal objective  9.119586809e+01\n",
      "\n",
      "The optimal Dispatch is 50.0\n",
      "CVaR of Load Shedding is 37.616870899476545\n",
      "CVaR of Curtailment is 3.5789971859640817\n"
     ]
    }
   ],
   "source": [
    "ϵ = 0.05 \n",
    "cvar_edv2 = Model(solver=GurobiSolver(Presolve=1))\n",
    "\n",
    "#Generator feasible space\n",
    "@variable(cvar_edv2, generator_min <= p_g <= generator_max)\n",
    "\n",
    "#Slack variables for the power balance\n",
    "@variable(cvar_edv2, delta_1 >= 0)\n",
    "@variable(cvar_edv2, delta_2 >= 0)\n",
    "\n",
    "#Slack Variables CVaR\n",
    "@variable(cvar_edv2, 0<= u1[k = 1:M] <= p_n_pv[k]/(ϵ));\n",
    "@variable(cvar_edv2, 0<= u2[k = 1:M] <= p_n_pv[k]/(ϵ));\n",
    "\n",
    "#Robust Power balance constraints \n",
    "@constraint(cvar_edv2,  -(p_g + sum((u1[k]*(solar_output[k] - load) for k = 1:M))) >= delta_1);\n",
    "@constraint(cvar_edv2,   (p_g + sum((u2[k]*(solar_output[k] - load) for k = 1:M))) >= delta_2);\n",
    "\n",
    "@constraint(cvar_edv2, sum(u1[k] for k = 1:M)  == 1);\n",
    "@constraint(cvar_edv2, sum(u2[k] for k = 1:M)  == 1);\n",
    "\n",
    "@objective(cvar_edv2, Min, (-delta_1-delta_2) + 1*p_g )\n",
    "solve(cvar_edv2)                                                            "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "println(\"The optimal Dispatch is $(getvalue(p_g))\")\n",
    "println(\"CVaR of load shedding is $(getvalue(delta_1))\")\n",
    "println(\"CVaR of Curtailment is $(getvalue(delta_2))\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results obtained from using the dual of the uncertainty set are the same as in the previous case, with the advantage of reducing the number of rows to 4. As a result, it is possible to model $\\mathcal{U}_{pv}$ using only 4 equations instead of $M$. Moreover, if we compare the iteration number in each model, there is a performance gain from 456 to 15. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two sided CVaR ED with 2 PV panel power plants\n",
    "\n",
    "This example illustrates the application and interpretation of the 2-sided CVaR in the context of multiple renewable energy power plants. The system has the same configuration as in the previous cases with an increased load and the installation of two PV power plants. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only\n",
      "Optimize a model with 4 rows, 1565 columns and 3128 nonzeros\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e+00, 1e+02]\n",
      "  Objective range  [1e+00, 1e+00]\n",
      "  Bounds range     [2e-05, 8e+01]\n",
      "  RHS range        [1e+00, 1e+00]\n",
      "Presolve time: 0.00s\n",
      "Presolved: 4 rows, 1565 columns, 3128 nonzeros\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0   -1.3264430e+03   2.040548e+02   0.000000e+00      0s\n",
      "       5    8.8041319e+00   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 5 iterations and 0.00 seconds\n",
      "Optimal objective  8.804131915e+00\n",
      "The optimal Dispatch is 50.0\n",
      "CVaR of load shedding is 37.6168708994765\n",
      "CVaR of Curtailment is 3.578997185964082\n"
     ]
    }
   ],
   "source": [
    "d1 = Normal(58,15)\n",
    "d2 = Normal(40,12)\n",
    "solar_output1 = 17:5:95 #In MW\n",
    "solar_output2 = 7:5:85 #In MW\n",
    "p1 = pdf.(d1,solar_output1)\n",
    "p2 = pdf.(d2,solar_output2) \n",
    "#Julia doesn't have a meshgrid command, so all the combinations of the solar output have to \n",
    "# arranged manually. \n",
    "solar_output = [0;0];\n",
    "p = []\n",
    "for (i, val_i) in enumerate(solar_output1)\n",
    "    for (j, val_j) in enumerate(solar_output2)\n",
    "        \n",
    "        solar_output = hcat(solar_output,[val_i;val_j])\n",
    "        push!(p,p1[i]*p2[j]) #Joint probabilities, assumming no particular correlation. This has to be captured by\n",
    "                             #the probabilistic forecast.        \n",
    "    end\n",
    "end\n",
    "solar_output = transpose(solar_output[:,2:end]);\n",
    "p_n_pv = p/sum(p) #Normalized probability space\n",
    "M = length(p_n_pv);\n",
    "#The system data for the rest of the components is as follows\n",
    "load = 175;\n",
    "generator_max = 85;\n",
    "generator_min = 50;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The uncertainty space for the solar power production is shown in the figure, where each one of the points represents a possible combination of the power output for the PV systems and the color code represents the probability $p^k$ for the particular combination $P^k_{pv} = [P^k_{pv_1}, P^k_{pv_2}]$. Also, the expected output of both PV systems is shown in the figure.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
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   "source": [
    "scatter(solar_output[:,1],solar_output[:,2],leg=false, zcolor=p, xlabel = \"PV sys 1 [MW]\", ylabel = \"PV sys 2 [MW]\",\n",
    "        xticks = 2:10:95, yticks = 2:10:95, cbar=true, markerstrokealpha = 0.1)\n",
    "scatter!([mean(Truncated(d1, 17, 95))],[mean(Truncated(d2, 7, 85))], annotations=(58,44.5,text(\"Expected Value\",:left)),\n",
    "    c = :red, m = (8, :diamond), xlims = [2,95], ylims = [2,95])"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a previous section it was described that the results from the CVaR optimization model is a generating family $\\mathcal{Q} = \\left\\{conv\\left(\\left\\{q^1,...q^k\\right\\}\\right): q_k \\le \\frac{p_k}{\\varepsilon}\\right\\}$ that represents how much of the uncertainty space is used for the calculation of the expected value. In the optimization model, this generating family is represented by the values of the dual variable $u^k$ for each of the uncertainty sets. \n",
    "\n",
    "The Robust Counterpart for the problem with $N$ PV power plants is as follows:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\min_{P_g, \\delta_1, \\delta_2} \\ &-(\\delta_1 + \\delta_2) + P_g\\\\\n",
    "       s.t.\\ & Pg \\in \\mathcal{X}_g\\\\\n",
    "             & - \\left(P_g +\\sum_{k=1}^M u_1^k \\sum_{n=1}^N P^k_{pv,n} - P_L \\right) \\ge \\delta_1\\\\\n",
    "             & \\sum_{k=1}^M u_1^k = 1\\\\\n",
    "             & 0 \\le u_1^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\\\\\n",
    "             & P_g +\\sum_{k=1}^M u_2^k \\sum_{n=1}^N P^k_{pv,n} - P_L  \\ge \\delta_2\\\\\n",
    "             & \\sum_{k=1}^M u_2^k = 1\\\\\n",
    "             & 0 \\le u_2^k \\le \\frac{1}{\\varepsilon}p^k &\\forall k = 1,...,M\\\\ \n",
    "             & \\delta_1 \\ge 0, \\ \\delta_2 \\ge 0\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "epsilon_testing (generic function with 1 method)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "function epsilon_testing(e)\n",
    "        ϵ = e\n",
    "        cvar_edv3 = Model(solver=GurobiSolver(Presolve=1,OutputFlag=0))\n",
    "\n",
    "        #Generator feasible space\n",
    "        @variable(cvar_edv3, generator_min <= p_g <= generator_max)\n",
    "\n",
    "        #Slack variables for the power balance\n",
    "        @variable(cvar_edv3, delta_1 >= 0)\n",
    "        @variable(cvar_edv3, delta_2 >= 0)\n",
    "\n",
    "        #Slack Variables CVaR\n",
    "        @variable(cvar_edv3, 0<= u1[k = 1:M] <= p_n_pv[k]/(ϵ));\n",
    "        @variable(cvar_edv3, 0<= u2[k = 1:M] <= p_n_pv[k]/(ϵ));\n",
    "\n",
    "        #Robust Power balance constraints \n",
    "        @constraint(cvar_edv3,  -(p_g + sum((u1[k]*(sum(solar_output[k,:]) - load) for k = 1:M))) >= delta_1);\n",
    "        @constraint(cvar_edv3,   (p_g + sum((u2[k]*(sum(solar_output[k,:]) - load) for k = 1:M))) >= delta_2);\n",
    "\n",
    "        @constraint(cvar_edv3, sum(u1[k] for k = 1:M)  == 1);\n",
    "        @constraint(cvar_edv3, sum(u2[k] for k = 1:M)  == 1);\n",
    "\n",
    "        @objective(cvar_edv3, Min, (-delta_1-delta_2) + 0.1*p_g )\n",
    "        solve(cvar_edv3) \n",
    "\n",
    "        uo1 = getvalue(u1);\n",
    "        uo2 = getvalue(u2);  \n",
    "                                                                    \n",
    "        #println(getvalue(delta_1))\n",
    "        #println(getvalue(delta_2))\n",
    "     return uo1, uo2\n",
    "                                                                    \n",
    "end                                                                                                                                   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "uo1_1p, uo2_1p = epsilon_testing(0.01);\n",
    "uo1_5p, uo2_5p = epsilon_testing(0.05);\n",
    "uo1_50p, uo2_50p = epsilon_testing(0.50);\n",
    "uo1_100p, uo2_100p = epsilon_testing(1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the two-system example is possible to visualize the effect of the selection of $\\varepsilon$ in the generating family.\n",
    "\n",
    "These plots show the samples used for the calculation of each one of the CVaR uncertainty set. The figures show the cases for $\\varepsilon = \\{0.01, 0.05, 0.5, 1\\}$. Also, the plots show the expected value in each one of the tails with respect to the rest of the samples. In the case of $\\varepsilon = 1$ it is possible to see that the two uncertainty sets overlap completely and that the expected values match, as predicted by the theory. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
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     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
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     "text": [
      "Academic license - for non-commercial use only\n",
      "Academic license - for non-commercial use only\n",
      "Academic license - for non-commercial use only\n",
      "Academic license - for non-commercial use only\n"
     ]
    }
   ],
   "source": [
    "scatter([solar_output[(uo1_1p .> 0),1],solar_output[(uo1_5p .> 0),1],solar_output[(uo1_50p .> 0),1],solar_output[(uo1_100p .> 0),1],\n",
    "        [sum(solar_output[:,1].*uo1_1p)],[sum(solar_output[:,1].*uo1_5p)],[sum(solar_output[:,1].*uo1_50p)],[sum(solar_output[:,1].*uo1_100p)]],\n",
    "        [solar_output[(uo1_1p .> 0),2],solar_output[(uo1_5p .> 0),2],solar_output[(uo1_50p .> 0),2],solar_output[(uo1_100p .> 0),2],\n",
    "        [sum(solar_output[:,2].*uo1_1p)],[sum(solar_output[:,2].*uo1_5p)],[sum(solar_output[:,2].*uo1_50p)],[sum(solar_output[:,2].*uo1_100p)]],   \n",
    "        leg=false, xlabel = \"PV sys 1 [MW]\", \n",
    "        ylabel = \"PV sys 2 [MW]\",\n",
    "        xticks = 2:10:95, \n",
    "        yticks = 2:10:95,\n",
    "        xlims = [2,95], \n",
    "        ylims = [2,95],\n",
    "        title=[\"1%\" \"5%\" \"50%\" \"100%\"],\n",
    "        title_location = :left,\n",
    "        markeralpha = [1 1 1 0.5],\n",
    "        layout=4)\n",
    "scatter!([solar_output[(uo2_1p .> 0),1],solar_output[(uo2_5p .> 0),1],solar_output[(uo2_50p .> 0),1],solar_output[(uo2_100p .> 0),1],\n",
    "        [sum(solar_output[:,1].*uo2_1p)],[sum(solar_output[:,1].*uo2_5p)],[sum(solar_output[:,1].*uo2_50p)],[sum(solar_output[:,1].*uo2_100p)]],\n",
    "        [solar_output[(uo2_1p .> 0),2],solar_output[(uo2_5p .> 0),2],solar_output[(uo2_50p .> 0),2],solar_output[(uo2_100p .> 0),2],\n",
    "        [sum(solar_output[:,2].*uo2_1p)],[sum(solar_output[:,2].*uo2_5p)],[sum(solar_output[:,2].*uo2_50p)],[sum(solar_output[:,2].*uo2_100p)]],\n",
    "        leg=false,\n",
    "        markeralpha = [1 1 1 0.5],\n",
    "        layout=4)"
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   "source": [
    "## Final Remarks \n",
    "\n",
    "- This notebook showcases the concept of risk measures and the application of the CVaR risk measure to the problem of economic dispatch. \n",
    "\n",
    "- It develops two variants of the same mathematical problem with different computational performance and establishes an equivalency between the two models. \n",
    "\n",
    "- The results also show that the objective function of the form $\\delta_1 + \\delta_2 + P_g$ is unable to capture any risk preferences from the decision maker and it leads to bang-bang solutions that can't be completely implemented in practice. \n",
    "\n",
    "- The integration of more constraints, particularly netword constraints can create conditions for different types of solutions. \n",
    "\n",
    "- The formulations presented here show that is possible to define a robust counterpart of the ED using probabilistic forecasts as inputs while maintaining the feasibility space as a polytope. "
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