Properly account for Delta_T in objective function terms (#41)

* Fix #39. Properly account for Delta_T in objective function terms.

Ref: #39

* Correct mistake in handling of ramp wear & tear for Detla_T ~= 1

CHANGES.md

@@ -5,13 +5,20 @@ Change history for MOST
 since 1.2
 ---------
 
+#### 10/25/23
+ - Fix [issue #39][10] in which the value of `mdi.Delta_T`, the number of
+ hours represented by each period, was not being accounted for in most
+ of the terms in the objective function.
+ *Thanks to Stefano Nicolin.*
+
 #### 10/4/23
  - Fix [issue #37][9] which caused a fatal error in storage input checks
  with multiple storage units under some circumstances.
  *Thanks to Keir Steegstra.*
 
 #### 2/3/23
- - Remove extra column in ExpectedRampCost and ignore for single period.
+ - Remove extra column in mdo.results.ExpectedRampCost and ignore for
+ single period.
 
 
 Version 1.2 - *Dec 13, 2022*
@@ -310,3 +317,4 @@ Version 1.0 - *Jun 1, 2016*
 [7]: https://arxiv.org/abs/2204.08140
 [8]: https://github.com/MATPOWER/most/issues/29
 [9]: https://github.com/MATPOWER/most/issues/37
+[10]: https://github.com/MATPOWER/most/issues/39

docs/src/MOST-manual/MOST-manual.tex

@@ -858,30 +858,30 @@ \subsubsection{Objective Function}
 \item[--] expected cost of active power dispatch and redispatch
 \end{itemize}
 \begin{align}
-f_p(p,p_+,p_-) &= \sum_{t\in T} \sum_{j\in J^t} \sum_{k\in K^{tj}} \psi_\alpha^{tjk} \sum_{i\in I^{tjk}}
+f_p(p,p_+,p_-) &= \Delta \cdot \sum_{t\in T} \sum_{j\in J^t} \sum_{k\in K^{tj}} \psi_\alpha^{tjk} \sum_{i\in I^{tjk}}
  \Bigl[\widetilde{C}_P^{ti}(p^{tijk}) + C_{P+}^{ti}(p_+^{tijk}) + C_{P-}^{ti}(p_-^{tijk}) \Bigr]
 \label{eq:most_energy_cost}
 \end{align}
 \begin{itemize}
 \item[--] cost of zonal reserves\footnotemark[\value{footnote}]
 \begin{equation}
-f_z(r_z) = \sum_{t\in T} \sum_{j\in J^t} \sum_{k\in K^{tj}} \psi_\alpha^{tjk} \sum_{i\in I^{tjk}} C_z^{ti}(r_z^{tijk})
+f_z(r_z) = \Delta \cdot \sum_{t\in T} \sum_{j\in J^t} \sum_{k\in K^{tj}} \psi_\alpha^{tjk} \sum_{i\in I^{tjk}} C_z^{ti}(r_z^{tijk})
 \label{eq:most_zres_cost}
 \end{equation}
 \item[--] cost of endogenous contingency reserves\footnotemark[\value{footnote}]
 \begin{equation}
-f_r(r_+, r_-) = \sum_{t\in T} \gamma^t \sum_{i\in I^t} \left[C_{R+}^{ti}(r_+^{ti}) + C_{R-}^{ti}(r_-^{ti}) \right]
+f_r(r_+, r_-) = \Delta \cdot \sum_{t\in T} \gamma^t \sum_{i\in I^t} \left[C_{R+}^{ti}(r_+^{ti}) + C_{R-}^{ti}(r_-^{ti}) \right]
 \label{eq:most_cres_cost}
 \end{equation}
 \item[--] expected cost of load-following ramping (wear and tear)
 \begin{equation}
-f_\delta(p) = \sum_{t\in T} \! \gamma^t \!\!\!\!\! \sum_{\begin{aligned}\scriptstyle j_1 &\scriptstyle \in J^{t-1}\\[-6pt] \scriptstyle j_2 &\scriptstyle \in J^t\end{aligned}} \!\!\!\!\!
+f_\delta(p) = \Delta \cdot \sum_{t\in T} \! \gamma^t \!\!\!\!\! \sum_{\begin{aligned}\scriptstyle j_1 &\scriptstyle \in J^{t-1}\\[-6pt] \scriptstyle j_2 &\scriptstyle \in J^t\end{aligned}} \!\!\!\!\!
  \phi^{t j_2 j_1} \!\!\!\! \sum_{i\in I^{tj_2 0}} \!\!\! C_\delta^i(p^{tij_20} - p^{(t-1)ij_10})
 \label{eq:rampcost}
 \end{equation}
 \item[--] cost of load-following ramp reserves
 \begin{equation}
-f_{\rm lf}(\delta_+, \delta_-) = \sum_{t\in T} \gamma^t \sum_{i\in I^t} \left[C_{\delta+}^{ti}(\delta_+^{ti}) + C_{\delta-}^{ti}(\delta_-^{ti}) \right]
+f_{\rm lf}(\delta_+, \delta_-) = \Delta \cdot \sum_{t\in T} \gamma^t \sum_{i\in I^t} \left[C_{\delta+}^{ti}(\delta_+^{ti}) + C_{\delta-}^{ti}(\delta_-^{ti}) \right]
 \label{eq:most_rampres_cost}
 \end{equation}
 \item[--] cost of initial stored energy and value (since it is negative) of expected leftover stored energy in terminal states
@@ -890,7 +890,7 @@ \subsubsection{Objective Function}
 \end{equation}
 \item[--] no load, startup and shutdown costs
 \begin{equation}
-f_{\rm uc}(u,v,w) = \sum_{t\in T} \gamma^t \sum_{i\in I^t} \Bigl( C_P^{ti}(0) u^{ti} + C_v^{ti} v^{ti} + C_w^{ti} w^{ti} \Bigr)
+f_{\rm uc}(u,v,w) = \sum_{t\in T} \gamma^t \sum_{i\in I^t} \Bigl( \Delta \cdot C_P^{ti}(0) u^{ti} + C_v^{ti} v^{ti} + C_w^{ti} w^{ti} \Bigr)
 \end{equation}
 \end{itemize}
 
@@ -1469,7 +1469,7 @@ \subsubsection{{\tt xgd} -- Extra Generator Data ({\tt xGenData})}
  \item [*] {All fields are $n_g \times 1$ vectors of per-generator values.}
  \item [\dag] {These are defaults provided by \code{loadxgendata}. If \code{gen} is provided, either directly or as the \code{gen} field of \mpc{}, then \code{P = gen(:, PG)}, \code{C = gen(:, GEN\_STATUS)} and \code{R = 2*(gen(:, PMAX) - MIN(0, gen(:, PMIN)))}, otherwise $\code{C}~=~1$, $\code{R}~=~0$ and no default is provided for \code{P} (corresponding field is not optional).}
  \item [\dag\dag] {Ramping costs/restrictions from the initial dispatch at $t=0$ are ignored for single-period problems.}
- \item [\ddag] {Each of these costs $C(\cdot)$ is presented in the formulation as a general function, but is implmented as a simple linear function of the form $C(x) = a x$, where the linear coefficient being supplied is $a$. The only exception is the ramping cost, which has the quadratic form $C_\delta^i(x) = a x^2$.}
+ \item [\ddag] {Each of these costs $C(\cdot)$ is presented in the formulation as a general function, but is implmented as a simple linear function of the form $C(x) = a x$, where the linear coefficient being supplied is $a$. The only exception is the ramping cost, which has the quadratic form $C_\delta^i(x) = \frac{1}{\Delta^2} a x^2$.}
  \item [\S] {Requires that \code{CommitKey} be present and non-empty.}
  \item [\P] {Sign is based on \code{C}\tnote{\dag}, i.e. $+\infty$ for \code{C} = 1, $-\infty$ for \code{C} = 0.}
 \end{tablenotes}
@@ -1722,7 +1722,7 @@ \subsubsection{Input Data}
 \code{mpc} & I & & base system data, standard \matpower{} case struct\tnote{\ddag}, with \baseMVA{}, \bus{}, \gen{}, \branch{} and \gencost{} fields \\
 \code{offer(t)} & I & & struct with offer data for period~$t$, see Table~\ref{tab:md_inputoffer} for details of sub-fields \\
 \code{OpenEnded} & I & 1 & ignore terminal dispatch ramp constraints, \emph{deprecated} \\
-\code{RampWearCostCoeff(i,t)} & I & 0 & $n_g \times n_t$, cost of ramping of generator~$i$ from period~$t-1$ to $t$, coefficient $C_\delta^i$ for square of dispatch difference in \eqref{eq:rampcost} \\
+\code{RampWearCostCoeff(i,t)} & I & 0 & $n_g \times n_t$, cost of ramping of generator~$i$ from period~$t-1$ to $t$, coefficient $C_\delta^i$ for square of dispatch difference in \eqref{eq:rampcost}\tnote{\S} \\
 \code{Storage} & B & & struct with parameters for storage units, see Table~\ref{tab:md_inputstorage} for the input fields \\
 \code{TerminalPg(i)} & I & & $n_g \times 1$, injection of generator~$i$ at $t = n_t$, \emph{deprecated, untested} \\
 \code{tstep(t)} & B & & $n_t \times 1$ struct of parameters related to period~$t$ \\
@@ -1736,6 +1736,7 @@ \subsubsection{Input Data}
  \item [*] {I = input, O = output, B = both, opt = taken from \matpower{} options.}
  \item [\dag] {See Section~\ref{sec:contab} for details. Note that, while \code{loadmd} assigns the same \code{contab} to all $t$ and $j$, it is possible to set different \code{contab} values manually and they will be respected by \code{most}.}
  \item [\ddag] {See Appendix~\ref{MUM-app:caseformat} in the \mum{} for details.}
+ \item [\S] {More precisely, $C_\delta^i(x) = \frac{1}{\Delta^2} a x^2$}, where $a$ is the corresponding value of \code{RampWearCostCoeff(i,t)}.
 \end{tablenotes}
 \end{threeparttable}
 \end{table}
@@ -3441,6 +3442,7 @@ \subsubsection*{Changes}
 
 \subsubsection*{Bugs Fixed}
 \begin{itemize}
+\item Fix issue \#39 in which the value of \code{mdi.Delta\_T}, the number of hours represented by each period, was not being accounted for in most of the terms in the objective function. \emph{Thanks to Stefano Nicolin.}
 \item Fix issue \#37 which caused a fatal error in storage input checks with multiple storage units under some circumstances. \emph{Thanks to Keir Steegstra.}
 \end{itemize}
 

lib/most.m

@@ -56,9 +56,8 @@
 % MDO MOST data structure, output
 % (see MOST User's Manual for details)
 
-
 % MOST
-% Copyright (c) 2010-2022, Power Systems Engineering Research Center (PSERC)
+% Copyright (c) 2010-2023, Power Systems Engineering Research Center (PSERC)
 % by Carlos E. Murillo-Sanchez, PSERC Cornell & Universidad Nacional de Colombia
 % and Ray Zimmerman, PSERC Cornell
 %
@@ -82,7 +81,7 @@
  fprintf( ' ----- Built on MATPOWER -----\n');
  fprintf( ' by Carlos E. Murillo-Sanchez, Universidad Nacional de Colombia--Manizales\n');
  fprintf( ' and Ray D. Zimmerman, Cornell University\n');
- fprintf( ' (c) 2012-2022 Power Systems Engineering Research Center (PSERC) \n');
+ fprintf( ' (c) 2012-2023 Power Systems Engineering Research Center (PSERC) \n');
  fprintf( '=============================================================================\n');
 end
 
@@ -509,7 +508,7 @@
  for k = 1:mdi.idx.nc(t,j)+1
  mpc = mdi.flow(t,j,k).mpc;
  c00tjk = totcost(mpc.gencost, zeros(ng,1));
- mdi.UC.c00(:, t) = mdi.UC.c00(:, t) + mdi.CostWeightsAdj(k, j, t) * c00tjk;
+ mdi.UC.c00(:, t) = mdi.UC.c00(:, t) + mdi.Delta_T * mdi.CostWeightsAdj(k, j, t) * c00tjk;
  c0col = COST + mpc.gencost(:,NCOST) - 1;
  ipoly = find(mpc.gencost(:, MODEL) == POLYNOMIAL);
  ipwl = find(mpc.gencost(:, MODEL) == PW_LINEAR);
@@ -1723,19 +1722,19 @@
  om.init_indexed_name('qdc', 'RampWear', {nt+1, nj_max, nj_max});
  end
  for j = 1:mdi.idx.nj(1)
- w = mdi.tstep(1).TransMat(j,1); % the probability of going from initial state to jth
- Q = spdiags(w * baseMVA^2 * mdi.RampWearCostCoeff(:,1), 0, ng, ng);
+ w = mdi.Delta_T * mdi.tstep(1).TransMat(j,1); % the probability of going from initial state to jth
+ Q = spdiags(w * baseMVA^2 * mdi.RampWearCostCoeff(:,1) / mdi.Delta_T^2, 0, ng, ng);
  c = -w * baseMVA * mdi.RampWearCostCoeff(:,1) .* mdi.InitialPg;
- vs = struct('name', {'Pg'}, 'idx', {{1,j,1}});
  k0 = w * 0.5 * mdi.RampWearCostCoeff(:,1)' * mdi.InitialPg.^2;
+ vs = struct('name', {'Pg'}, 'idx', {{1,j,1}});
  om.add_quad_cost('RampWear', {1,j,1}, Q, c, k0, vs);
  end
  % Then the remaining periods
  for t = 2:nt
  for j2 = 1:mdi.idx.nj(t)
  for j1 = 1:mdi.idx.nj(t-1)
- w = mdi.tstep(t).TransMat(j2,j1) * mdi.CostWeights(1, j1, t-1);
- h = w * baseMVA^2 * mdi.RampWearCostCoeff(:,t);
+ w = mdi.Delta_T * mdi.tstep(t).TransMat(j2,j1) * mdi.CostWeights(1, j1, t-1);
+ h = w * baseMVA^2 * mdi.RampWearCostCoeff(:,t) / mdi.Delta_T^2;
  i = (1:ng)';
  j = ng+(1:ng)';
  Q = sparse([i;j;i;j], [i;i;j;j], [h;-h;-h;h], 2*ng, 2*ng);
@@ -1750,11 +1749,11 @@
  % that makes sense for nt+1; all other fields in mdi.tstep(nt+1) can be empty.
  if ~mdi.OpenEnded
  for j = 1:mdi.idx.nj(nt)
- w = mdi.tstep(nt+1).TransMat(1, j) * mdi.CostWeights(1, j, nt);
- Q = spdiags(w * baseMVA^2 * mdi.RampWearCostCoeff(:,nt+1), 0, ng, ng);
+ w = mdi.Delta_T * mdi.tstep(nt+1).TransMat(1, j) * mdi.CostWeights(1, j, nt);
+ Q = spdiags(w * baseMVA^2 * mdi.RampWearCostCoeff(:,nt+1) / mdi.Delta_T^2, 0, ng, ng);
  c = -w * baseMVA * mdi.RampWearCostCoeff(:,nt+1) .* mdi.TerminalPg;
- vs = struct('name', {'Pg'}, 'idx', {{nt,j,1}});
  k0 = w * 0.5 * mdi.RampWearCostCoeff(:,nt+1)' * mdi.TerminalPg.^2;
+ vs = struct('name', {'Pg'}, 'idx', {{nt,j,1}});
  om.add_quad_cost('RampWear', {nt+1,j,1}, Q, c, k0, vs);
  end
  end
@@ -1773,7 +1772,7 @@
  for t = 1:nt
  for j = 1:mdi.idx.nj(t)
  for k = 1:mdi.idx.nc(t,j)+1
- w = mdi.CostWeightsAdj(k,j,t);  %% NOTE (k,j,t) order !!!
+ w = mdi.Delta_T * mdi.CostWeightsAdj(k,j,t); %% NOTE (k,j,t) order !!!
 
  % weighted polynomial energy costs for committed units
  gc = mdi.flow(t,j,k).mpc.gencost;
@@ -1841,21 +1840,21 @@
  end
 
  % contingency reserve costs
- c = baseMVA * mdi.StepProb(t) * mdi.offer(t).PositiveActiveReservePrice(:);
+ c = mdi.Delta_T * baseMVA * mdi.StepProb(t) * mdi.offer(t).PositiveActiveReservePrice(:);
  vs = struct('name', {'Rpp'}, 'idx', {{t}});
  om.add_quad_cost('Crpp', {t}, [], c, 0, vs);
- c = baseMVA * mdi.StepProb(t) * mdi.offer(t).NegativeActiveReservePrice(:);
+ c = mdi.Delta_T * baseMVA * mdi.StepProb(t) * mdi.offer(t).NegativeActiveReservePrice(:);
  vs = struct('name', {'Rpm'}, 'idx', {{t}});
  om.add_quad_cost('Crpm', {t}, [], c, 0, vs);
  end
  % Assign load following ramp reserve costs. Do first nt periods first
  om.init_indexed_name('qdc', 'Crrp', {mdi.idx.ntramp});
  om.init_indexed_name('qdc', 'Crrm', {mdi.idx.ntramp});
  for t = 1:mdi.idx.ntramp
- c = baseMVA * mdi.StepProb(t) * mdi.offer(t).PositiveLoadFollowReservePrice(:);
+ c = mdi.Delta_T * baseMVA * mdi.StepProb(t) * mdi.offer(t).PositiveLoadFollowReservePrice(:);
  vs = struct('name', {'Rrp'}, 'idx', {{t}});
  om.add_quad_cost('Crrp', {t}, [], c, 0, vs);
- c = baseMVA * mdi.StepProb(t) * mdi.offer(t).NegativeLoadFollowReservePrice(:);
+ c = mdi.Delta_T * baseMVA * mdi.StepProb(t) * mdi.offer(t).NegativeLoadFollowReservePrice(:);
  vs = struct('name', {'Rrm'}, 'idx', {{t}});
  om.add_quad_cost('Crrm', {t}, [], c, 0, vs);
  end

lib/mostver.m

@@ -19,7 +19,7 @@
 v = struct( 'Name', 'MOST', ...
  'Version', '1.2+', ...
  'Release', '', ...
- 'Date', '30-May-2023' );
+ 'Date', '25-Oct-2023' );
 if nargout > 0
  if nargin > 0
  rv = v;

lib/t/t_most_uc.m

@@ -43,7 +43,7 @@ function t_most_uc(quiet, create_plots, create_pdfs, savedir)
 % fcn = {'gurobi'};
 % solvers = {'MOSEK'};
 % fcn = {'mosek'};
-ntests = 68;
+ntests = 69;
 t_begin(ntests*length(solvers), quiet);
 
 if quiet
@@ -235,6 +235,10 @@ function t_most_uc(quiet, create_plots, create_pdfs, savedir)
  plot_case('+ DC Network', mdo, ms, 500, 100, savedir, pp, fname);
  end
  % keyboard;
+ mdi.Delta_T = 2;
+ mdo = most(mdi, mpopt);
+ ms = most_summary(mdo);
+ t_is(ms.f, 2 * ex.f, 8, [t '(Delta_T = 2) : f']);
 
  t = sprintf('%s : + startup/shutdown costs : ', solvers{s});
  if mpopt.out.all