Add Wind+PEM tracking and stochastic bidding documentation

docs/reference_guides/apps/grid_integration/Bidder.rst

@@ -13,6 +13,155 @@
 each of these uncertain price and power output pairs formulates a segment in the
 bidding curves.
 
+Here we present a stochastic bidding model for a renewable integated energy system (Wind generator + PEM).
+
+
+Day-Ahead Bidding Problem for Wind + PEM IES
+---------------------------------------------
+
+The objective function is the expected profit, which equals the revenue substracts the cost. 
+We want to consider the revenue from the electricity market and the hydrogen market.
+
+.. math:: \max \quad \sum_{s \in S, t \in T^{DA}}\omega_{s}[{(\pi_{t,s}^{DA}P_{t,s}^{DA} + \pi_{t,s}^{RT}(P_{t,s}^{RT}-P_{t,s}^{DA}))\Delta t  + Pr^{H}m_{t,s}^{H}- c_{t,s}}] - C_{fix}
+
+s.t.
+
+.. math:: P_{t,s}^{DA} \leq P_{t,s}^{RT} \quad \forall t, s \quad \quad (1)
+.. math:: P_{t,s} = P^{RT}_{t,s} \quad \forall t, s \quad \quad (2)
+.. math:: (\pi_{t,s'}^{DA} - \pi_{t,s}^{DA})(P_{t,s'}^{DA} - P_{t,s}^{DA}) \geq 0 \quad \forall s \in S, \forall s' \in S \backslash s, \forall t \in T^{DA} \quad \quad (3)
+.. math:: P_{t,s}^{DA} \leq P_{t,s}^{wind} \quad \forall t, s \quad \quad (4)
+.. math:: P_{t,s}^{RT} \leq P_{t,s}^{wind} \quad \forall t, s \quad \quad (5)
+.. math:: P_{t,s}^{wind} \leq f_{t} P_{max}^{wind} \quad \forall t, s \quad \quad (6)
+.. math:: P_{t,s}^{DA} + P_{t,s}^{PEM} \leq P_{max}^{wind} \quad \forall t, s \quad \quad (7)
+.. math:: P_{t,s}^{PEM} \leq P_{max}^{PEM} \quad \forall t, s \quad \quad (8)
+.. math:: m_{t,s}^{H} = P_{t,s}^{PEM}C_{H}\Delta t \quad \forall t, s \quad \quad (9)
+.. math:: c_{t,s} = C^{op} P_{t,s}^{PEM}\quad \forall t, s \quad \quad (10)
+.. math:: C_{fix} = C_{fix}^{wind}P_{max}^{wind} + C_{fix}^{PEM}P_{max}^{PEM} \quad \quad (11)
+
+Equation (1) requires the day-ahead offering power is less or equal to the real-time offering power 
+in order to avoid underbidding. Equation (2) states that the RT offering power is the same as the 
+IES power output to the grid. In the bidding mode, the market rules require the offering power is 
+non-decreasing (convex) with its marginal cost in an energy bid. This rule is represented by equation (3).
+Equation (4) to equation (9) are process model constraints. Equation (10) calculates the operation costs for IES
+and equation (11) calculate the fixed cost for IES.
+
+**Parameters**
+
+:math:`\omega_{s}`: Frequency of each scenario.
+
+:math:`\pi^{DA}_{t,s}`: Day-ahead LMP forecasting from forecaster at hour t for scenario s, \$/MWh.
+
+:math:`\pi^{RT}_{t,s}`: Real-time LMP forecasting from forecaster at hour t for scenario s, \$/MWh.
+
+:math:`Pr^{H}`: Market price for hydrogen， \$/kg.
+
+:math:`P_{max}^{PEM}`: PEM max capacity, MW.
+
+:math:`f_{t}`: Wind power generation capacity factor at hour t, MW/MW.
+
+:math:`P_{max}^{wind}`: Wind generator max capacity, MW.
+
+:math:`C^{op}`: PEM operation cost coefficient, \$/MW.
+
+:math:`C_{fix}^{wind}`: Wind generator fixed cost  coefficient, \$/MW.
+
+:math:`C_{fix}^{PEM}`:  PEM fixed cost coefficient, \$/MW.
+
+:math:`C_{H}`: Electricity to hydrogen conversion rate, kg/MWh.
+
+
+**Variables**
+
+:math:`P_{t,s}`: IES power output to the grid at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{DA}`: Day-ahead offering power at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{RT}`: Real-time offering power at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{wind}`: Wind power generation at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{PEM}`:  Power delivered to PEM at hour t in scenario s, MW.
+
+:math:`m_{t,s}^{H}`: Hydrogen production mass at hour t in scenario s, kg.
+
+:math:`c_{t,s}`: IES operational cost at hour t in scenario s, \$.
+
+
+Real-time Bidding Problem for Wind+PEM IES
+------------------------------------------
+
+.. math:: \max \quad \sum_{t \in T_{DA}}\hat{\pi}_{t}^{DA}\hat{P}_{t}^{DA}\Delta t + \sum_{t\in T_{RT}, s\in S}\omega_{s}[\pi_{t,s}^{RT}(P_{t,s}^{RT} -\hat{P}_{t}^{DA})\Delta t + Pr^{H}m_{t,s}^{H} - c_{t,s} - \omega_{t}^{RT} P_{t,s}^{underbid}] - C_{fix}
+
+s.t.
+
+.. math:: \hat{P}^{DA}_{t} \leq P_{t,s}^{RT} + P_{t,s}^{underbid} \quad \forall t, s \quad \quad (12)
+.. math:: P_{t,s}^{RT} = P_{t,s} \quad \forall t, s \quad \quad (13)
+.. math:: (\pi_{t,s'}^{RT} - \pi_{t,s}^{RT})(P_{t,s'}^{RT} - P_{t,s}^{RT}) \geq 0 \quad \forall s \in S, \forall s' \in S \backslash s, \forall t \in T^{RT} \quad \quad (14)
+.. math:: P_{t,s}^{RT} \leq P_{t,s}^{wind} \quad \forall t, s \quad \quad (15)
+.. math:: P_{t,s}^{wind} \leq f_{t}P_{wind}^{max} \quad \forall t, s \quad \quad (16)
+.. math:: P_{t,s}^{RT} + P_{t,s}^{PEM} \leq P_{t,s}^{wind} \quad \forall t, s \quad \quad (17)
+.. math:: P_{t,s}^{PEM} \leq P_{max}^{PEM} \quad \forall t, s \quad \quad (18)
+.. math:: m_{t,s}^{H} = P_{t,s}^{PEM}C_{H}\Delta t \quad \forall t, s \quad \quad (19)
+.. math:: c_{t,s} = C^{op} P_{t,s}^{PEM}\quad \forall t, s \quad \quad (20)
+.. math:: C_{fix} = C_{fix}^{wind}P_{max}^{wind} + C_{fix}^{PEM}P_{max}^{PEM} \quad \quad (21)
+
+Before the actual operations, electricity markets allow the resources to submit real-time energy bids to 
+correct deviations from the day-ahead market. At this time, both day-ahead LMP :math:`\hat{\pi}_{t}^{DA}`
+and day-ahead dispatch level :math:`\hat{P}_{t}^{DA}` have been realized as a result of the
+day-ahead market clearing. In real-time market, due to the forecaster error and some other reasons, the 
+real-time offering power may not realize promises that generator owner makes in the day-ahead market. We 
+call this 'underbidding' and underbiding energy will be penaltized by the ISO. To prevent the underbidding 
+and loss of revenue, we add a relaxed lower bound for the real-time offering power with a slack
+variable :math:`P_{t,s}^{underbid}` for underbidding in equation (12) and penalized in the objective function.
+
+**Parameters**
+
+:math:`\omega_{s}`: Frequency of each scenario.
+
+:math:`\omega_{t}^{RT}`: Penalty for underbidding at real-time at hour t, \$/MWh.
+
+:math:`\hat{\pi}_{t}^{DA}`: Realized day-ahead energy LMP signals at hour t, \$/MWh.
+
+:math:`\hat{P}_{t}^{DA}`: Realized day-ahead dispatch level at hour t, \$/MWH.
+
+:math:`\pi^{RT}_{t,s}`: Real-time LMP forecasting from forecaster at hour t for scenario s, \$/MWh.
+
+:math:`Pr^{H}`: Market price for hydrogen, \$/kg.
+
+:math:`P^{PEM}_{max}`: PEM max capacity, MW.
+
+:math:`f_{t}`: Wind power generation capacity factor at hour t, MW/MW.
+
+:math:`P_{max}^{wind}`: Wind generator max capacity, MW.
+
+:math:`C^{op}`: PEM operation cost coefficient, \$/MW.
+
+:math:`C_{fix}^{wind}`: Wind generator fixed cost  coefficient, \$/MW.
+
+:math:`C_{fix}^{PEM}`:  PEM fixed cost coefficient, \$/MW.
+
+:math:`C_{H}`: Electricity to hydrogen conversion rate, kg/MWh.
+
+**Variables**
+
+:math:`P_{t,s}`: IES power output to the grid at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{underbid}`: The amount of underbidding power in real-time at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{RT}`: Real-time offering power at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{wind}`: Wind power generation at hour t in scenario s, MW.
+
+:math:`P_{t,s}^{PEM}`: Power delivered to PEM at hour t in scenario s, MW.
+
+:math:`m_{t,s}^{H}`: Hydrogen production mass at hour t in scenario s, kg.
+
+:math:`c_{t,s}`: IES operational cost at hour t in scenario s, \$.
+
+Some wind, battery, PEM models and the double-loop simulation example can be found in Dispatches GitHub repository.
+
+https://dispatches.readthedocs.io/en/main/models/renewables/index.html
+
 .. |example_bid| image:: images/example_bid.png
   :width: 800
   :alt: Alternative text

docs/reference_guides/apps/grid_integration/Tracker.rst

@@ -9,6 +9,59 @@ integrated energy system which consists of a thermal generator and an energy sto
 The figure shows that to track the dispatch (load) the energy system can optimally
 use power output from charging and discharging cycle.
 
+
+Tracking Problem for Wind+PEM IES
+----------------------------------
+
+Here we present a tracking problem example for wind + PEM IES.
+
+
+.. math:: \min \quad \sum_{t \in T^{RT}} (c_{t,0} + \omega_{t}(P^{+}_{t,0} + P^{-}_{t,0})) - C_{fix}
+
+s.t.
+
+.. math:: P_{t,0} + P_{t,0}^{-} = \hat{P}^{RT}_{t} + P^{+}_{t,0} \quad \forall t
+
+.. math:: P_{t,0} + \hat{P}_{t,0}^{PEM} \leq \hat{P}_{wind}^{RT} \quad \forall t
+
+.. math:: P_{t,0}^{PEM} \leq P_{max}^{PEM} \quad \forall t
+
+.. math:: m_{t,0}^{H} = P_{t,0}^{PEM}C_{H}\Delta t \quad \forall t
+
+.. math:: c_{t,0} = C^{op} P_{t,0}^{PEM}\quad \forall t
+
+.. math:: C_{fix} = C_{fix}^{wind}P_{max}^{wind} + C_{fix}^{PEM}P_{max}^{PEM}
+
+**Parameters**
+
+:math:`\hat{P}^{RT}_{t}`: Realized real-time dispatch level at hour t, MW.
+
+:math:`\hat{P}_{wind}^{RT}`: Realized wind generation power at hour t, MW.
+
+:math:`P_{max}^{PEM}`: PEM max capacity, MW.
+
+:math:`C^{op}`: PEM operation cost coefficient, \$/MW.
+
+:math:`C_{fix}^{wind}`: Wind generator fixed cost  coefficient, \$/MW.
+
+:math:`C_{fix}^{PEM}`:  PEM fixed cost coefficient, \$/MW.
+
+:math:`C_{H}`: Electricity to hydrogen conversion rate, kg/MWh.
+
+**Variables**
+
+:math:`P_{t,0}`: IES power output to the grid at hour t in scenario 0, MW.
+
+:math:`P_{t,0}^{PEM}`: Power delivered to PEM at hour t in scenario 0, MW.
+
+:math:`P_{t,0}^{+}`: Power over-delivered by the IES to the grid at hour t in scenario 0, MW.
+
+:math:`P_{t,0}^{-}`: Power under-delivered by the IES to the grid at hour t in scenario 0, MW.
+
+:math:`m_{t,0}^{H}`: Hydrogen production mass at hour t in scenario 0, kg.
+
+:math:`c_{t,0}`: IES operational cost at hour t in scenario 0, \$.
+
 .. |tracking_example| image:: images/tracking_example.png
   :width: 1200
   :alt: Alternative text

docs/reference_guides/apps/grid_integration/index.rst

@@ -13,7 +13,7 @@ the operational (hours to year timescale) interactions between energy systems an
 wholesale electricity markets. For more information, please look at the introduction section.
 
 .. toctree::
-   :maxdepth: 2
+   :maxdepth: 1
 
    Introduction
    Implementation