RT Energy and AS Co-Optimization – Original and Modified Approach
Proposed Resource Specific AS Offer Submission Rule Modification:
The proposed modifications to the Resource Specific AS Offer Submission rules are:
a) Cascade of AS prices (MCPC) from higher quality AS to lower quality AS
If a higher quality AS is offered then, the submitted prices for lower quality AS offer for the same offered MW
quantity must be equal to or less than the offered higher quality AS.
For example, if 10 MW of RegUp is offered at 15 $/MW/h, then the submitted offer price for lower quality AS
(RRS, Non-Spin, …) must be equal to or less than 15 $/MW/h
b) Increased liquidity in the AS market
If higher qualities AS offers are submitted with no prices for lower quality AS, then instead of the current
practice of not using the offered AS MW for lower quality AS, ERCOT systems WILL consider these AS offer
MW for lower quality AS at the same price as submitted for the higher quality AS.
For example, if 10 MW of RegUp offered at 15 $/MW/h but no prices submitted for lower quality AS (RRS,
Non-Spin…), then, ERCOT systems will consider this offer for the qualified lower quality AS (RRS, Non-Spin,…)
at 15 $/MW/h.
Energy and AS co-optimization in the Day-Ahead and Real-Time Market
In this section, modifications to the original approach in the concept paper are presented. The table below
summarizes the key points of the original and modifications to the original for comparison.
AS
DayProducts Ahead
RealTime
AS
Demand
Curves
AS
MCPC
Original
Only awarded if
 RegUp,
offers submitted.
 RegDn,
i.e. willing seller
 RRS,
 Non-Spin
 RegUp,
 RegDn,
 RRS,
 Non-Spin
a) For On-Line
Non-Spin, if
qualified,
then
automatically
considered to
be offered at
zero $/MW
b) For other
types of AS,
will be
awarded only
if AS offer
submitted.
c) Same MCPC
for both OnLine and OffLine Non-Spin
 The value of the maximum price on
the ORDC needs to be coordinated
with VOLL, SWOC.
 Demand Curves for Regulation
(Up/Down), and RRS are carved out
of the ORDC for spinning (On-Line)
reserves
 Demand Curve for Non-Spin (includes
On-Line and Off-Line) is carved out of
the ORDC that represents both
spinning and non-spinning (On-Line +
Off-Line) reserves
Modification
DayAhead
RealTime
 RegUp,
 RegDn,
 RRS,
 Spinning Operating
Reserve (SOR),
 Non-Spinning
Operating Reserve
(NSOR)
 RegUp,
 RegDn,
 RRS,
 Spinning Operating
Reserve (SOR),
 Non-Spinning
Operating Reserve
(NSOR)
Only awarded if
offers submitted. i.e.
willing seller
a) For SOR,
automatically
considered to be
offered at zero
$/MW
b) For other types
of AS, will be
awarded only if
AS offer
submitted.
c) Separate prices
(MCPC) for SOR
and NSOR
 The value of the maximum price on each AS demand
curve needs to be coordinated with VOLL,SWOC
 Demand Curves for Regulation (Up/Down), and RRS
do NOT have to be based on ORDC
 Demand curve for SOR can be either the current
ORDC for spinning (On-Line) reserves or based on
EUE
 Demand curve for NSOR can be either the current
ORDC for spinning and non-spinning reserves (OnLine + Off-Line) or based on EUE
 The SOR MCPC is the sum of the shadow prices of
the constraints to procure SOR and NSOR
AS Demand Curves - Original
$/MW
Minimum Contingency
X=2000 MW
ORDC – Spinning Reserves
RegUp
Demand Curve
RRS Demand
Curve
MW
$/MW
Minimum Contingency
X=2000 MW
ORDC – Combined Spinning & Non-Spinning Reserves
NSpin Demand Curve
MW
AS Demand Curves – Modified (SOR,NSOR demand curves based on ORDC)
$/MW
$/MW
RegUp
Demand Curve
RRS Demand Curve
MW
MW
$/MW
SOR Demand Curve from ORDC Spinning
Reserves with Minimum Contingency (X)
removed
MW
$/MW
NSOR Demand Curve from ORDC
Combined Spinning & Non-Spinning
Reserves with Minimum Contingency (X)
removed
MW
Formulation of Equations
The simplified formulation of the optimization problem (objective, constraints, pricing analysis) is presented below
where energy and the various AS products are co-optimized in both Day-Ahead and Real-Time Markets.
Simplifications:
(i)
(ii)
(iii)
(iv)
(v)
Transmission constraints in both Day-Ahead and Real-Time, PTPs, block offers and bids in Day-Ahead are
not considered
Constraints on how much AS can be awarded to a single resource based on a % of HSL or ramp capability
are not considered
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟
𝐴𝑆𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
, 𝐶𝑖
, 𝐶𝑖
are the submitted bids ($/MWh) , energy offers ($/MWh), AS offers
($/MW) respectively. For simplicity, these bids and offers are considered to be constant for the entire
MW bid or offered.
In the Day-Ahead Market, Resources must submit offers to sell energy and AS (all the types) in order to
be awarded.
The equations below are describing the modified approach. If the reader is interested in the equations
for the original approach in the concept paper, then in the equations below:
a. Consider SOR offers (spinning operating reserve) as Non-Spin (both On-Lin and Off-Line)
b. Replace the demand curve for SOR with Non-Spin demand curve.
c. Remove terms with NSOR (non-spinning operating reserve) – off-line non-spin is already included in
(v) a. above.
Day-Ahead Market Objective Function:
𝑁𝑒𝑏
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑
𝑂𝑏𝑗 ≡ 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 {− ∑ 𝐶𝑖
𝑖=1
𝑁𝑟𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝑖=1
𝑁𝑟𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝑖=1
𝑁𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
𝑅𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑅𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝐶𝑖
𝐶𝑖
𝐶𝑖𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑 × 𝑀𝑊𝑖𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑖=1
𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑
−∑
× 𝑀𝑊𝑖
𝐶𝑖𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑 × 𝑀𝑊𝑖𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑖=1
𝑁𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝑖=1
𝐶𝑖𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑 × 𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑁𝑔
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟
+ ∑ 𝐶𝑖
× 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
𝑁𝑔+𝑁𝑐𝑙
+∑
𝑖=1
𝑖=1
𝑁𝑓𝑔+𝑁𝑓𝑐𝑙
+∑
𝑖=1
𝑁𝑓𝑐𝑙
+∑
𝑖=1
𝐹𝑅𝑅𝑆𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝐹𝑅𝑅𝑆𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
∑
𝑖=1
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
𝑁𝑔+𝑁𝑐𝑙
+∑
× 𝑀𝑊𝑖𝐹𝑅𝑅𝑆𝐷𝑛𝐴𝑤𝑎𝑟𝑑 +
∑
𝑅𝑒𝑔𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑖=1
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
𝑅𝑅𝑆𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
× 𝑀𝑊𝑖
𝑁𝑔
𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑆𝑁𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
× 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 + ∑ 𝐶𝑖
𝑖=1
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
× 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑
𝑖=1
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
+
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
× 𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 }
Real-Time Market Objective Function:
𝑁𝑒𝑏
𝑁𝑃𝐵
𝑂𝑏𝑗 ≡ 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 { ∑
𝐶𝑖𝑃𝑜𝑤𝑒𝑟𝐵𝑎𝑙𝑎𝑛𝑐𝑒
𝑖=1
𝑁𝑟𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝑖=1
𝑁𝑟𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝑖=1
𝑁𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑
− ∑ 𝐶𝑖
𝑅𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑅𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝐶𝑖
𝐶𝑖𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑 × 𝑀𝑊𝑖𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
−∑
𝑖=1
𝐶𝑖𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑 × 𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑁𝑔
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟
+ ∑ 𝐶𝑖
× 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
𝑁𝑔+𝑁𝑐𝑙
+∑
𝑖=1
𝑖=1
+∑
𝑁𝑓𝑐𝑙
+∑
𝑖=1
𝐹𝑅𝑅𝑆𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑖=1
𝐹𝑅𝑅𝑆𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
× 𝑀𝑊𝑖
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
× 𝑀𝑊𝑖𝐹𝑅𝑅𝑆𝐷𝑛𝐴𝑤𝑎𝑟𝑑 +
𝑅𝑒𝑔𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑁𝑔+𝑁𝑐𝑙
+∑
∑
∑
𝑅𝑅𝑆𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑖=1
𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
(𝐶𝑖
= 0 𝑓𝑜𝑟 𝑅𝑒𝑎𝑙 − 𝑇𝑖𝑚𝑒) × 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑
𝑖=1
𝑁𝑔
𝑆𝑁𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
+ ∑ 𝐶𝑖
𝑖=1
× 𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 }
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑖=1
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
+
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
𝐶𝑖𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑 × 𝑀𝑊𝑖𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝑖=1
𝑁𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑
𝑁𝑓𝑔+𝑁𝑓𝑐𝑙
× 𝑀𝑊𝑖
𝑖=1
𝐶𝑖
𝑖=1
𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑
−∑
×
𝑀𝑊𝑖𝑃𝑜𝑤𝑒𝑟𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝑀𝑊
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
× 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
× 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑
Subject to:
Ignoring transmission constraints and focusing on power balance, AS procurement and the main Resource limit
constraints, the set of constraints are given below:
System wide constraints:
a) Power Balance: (Shadow price =
𝜆)
Day-Ahead:
𝑁𝑔
𝑁𝑒𝑏
∑ 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
− ∑ 𝑀𝑊𝑖
𝑖=1
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
=0
𝑖=1
Real-Time:
𝑁𝑔
𝑁𝑃𝐵
∑ 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
+ ∑ 𝑀𝑊𝑖𝑃𝑜𝑤𝑒𝑟𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝑀𝑊 − 𝐺𝑇𝐵𝐷 = 0
𝑖=1
𝑖=1
a) RegUp Procurement (including FRRS-Up): (Shadow price =
𝑁𝑔+𝑁𝑐𝑙
∑
𝑖=1
𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
𝑁𝑓𝑔+𝑁𝑓𝑐𝑙
+∑
𝑀𝑊𝑖
𝑖=1
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
𝛼)
𝑁𝑟𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑
−∑
𝑖=1
b) FRRS-Up maximum procurement limit: (Shadow price =
𝑖=1
𝑀𝑊𝑖
c) RegDn Procurement (including FRRS-Dn): (Shadow price =
𝑁𝑔+𝑁𝑐𝑙
𝑖=1
𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
𝑁𝑓𝑐𝑙
+∑
𝑖=1
𝑁𝑓𝑐𝑙
𝑀𝑎𝑥𝐹𝑅𝑅𝑆𝐷𝑛 − ∑
𝑖=1
e) RRS Procurement: (Shadow price =
𝑖=1
f)
𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 − ∑
𝛽)
𝑁𝑟𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑
𝑖=1
𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝜔𝐹𝑅𝑅𝑆𝐷𝑛 )
𝑀𝑊𝑖𝐹𝑅𝑅𝑆𝐷𝑛𝐴𝑤𝑎𝑟𝑑 ≥ 0
𝑁𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑
𝑖=1
≥0
≥0
𝛾)
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
∑
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
𝑀𝑊𝑖𝐹𝑅𝑅𝑆𝐷𝑛𝐴𝑤𝑎𝑟𝑑 − ∑
d) FRRS-Dn maximum procurement limit: (Shadow price =
𝑅𝑒𝑔𝑈𝑝𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑
𝜔𝐹𝑅𝑅𝑆𝑈𝑝 )
𝑁𝑓𝑔+𝑁𝑓𝑐𝑙
𝑀𝑎𝑥𝐹𝑅𝑅𝑆𝑈𝑝 − ∑
∑
𝑀𝑊𝑖
𝑀𝑊𝑖𝑅𝑅𝑆𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑 ≥ 0
RRS maximum procurement from “blocky” Load Resource: (Shadow price =
𝜔𝑅𝑅𝑆𝑏𝑙 )
≥0
𝑁𝑏𝑙
∑ 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 − 𝑀𝑎𝑥𝑅𝑅𝑆𝑏𝑙 ≥ 0
𝑖=1
g) SOR Procurement: (Shadow price =
𝛿)
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 − ∑
∑
𝑖=1
h) SNSOR Procurement: (Shadow price =
𝑁𝑔+𝑁𝑐𝑙+𝑁𝑏𝑙
𝑖=1
+ ∑ 𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 − ∑
𝑖=1
𝑀𝑊𝑖𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑 ≥ 0
𝜙)
𝑁𝑔
𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑
∑
𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑
𝑖=1
𝑁𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑
𝑖=1
𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐷𝑒𝑚𝑎𝑛𝑑𝐴𝑤𝑎𝑟𝑑 ≥ 0
Individual Energy Bid constraints:
i)
Energy Bid MW constraint for every energy bid 𝑖 = 1,2,3 … 𝑁𝑒𝑏 : (Shadow price = 𝜃𝑖𝑒𝑏 respectively)
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝑀𝑊𝑖 − 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
≥0
Individual Resource constraints:
Each Resource will have its own set of constraints to ensure awards are within bounds of its own upper (HSL/MPC)
and low (LSL/LPC) limits.
j)
LSL Constraint for every modeled Generation Resource 𝑖 = 1,2,3 … 𝑁𝑔 : (Shadow price = 𝜃𝑖𝐿𝑆𝐿 )
𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
− 𝐿𝑆𝐿𝑖 ≥ 0
k) HSL Constraint for every modeled Generation Resource 𝑖 = 1,2,3 … 𝑁𝑔 : (Shadow price = 𝜃𝑖𝐻𝑆𝐿 )
Online:
𝐻𝑆𝐿𝑖 − 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 − 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
Offline:
𝐻𝑆𝐿𝑖 − 𝑀𝑊𝑖𝑆𝑁𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
l)
AS Offer MW constraint for every modeled Generation Resource 𝑖 = 1,2,3 … 𝑁𝑔 : (Shadow price =
𝑔−𝐴𝑆𝑢𝑝
𝜃𝑖
Online:
𝑔−𝐴𝑆𝑑𝑛
, 𝜃𝑖
respectively)
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 − 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝐷𝑛−𝑖 − 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
≥0
Offline:
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖𝑅𝑆𝑁𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
m) MPC & LPC Constraint for every modeled “Blocky” Load Resource 𝑖 = 1,2,3 … 𝑁𝑏𝑙 : (Shadow price = 𝜃𝑖𝑏𝑙 )
Note that a “blocky” Load Resource is awarded only one AS product.
𝑀𝑃𝐶𝑖 − 𝐿𝑃𝐶𝑖 − 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 ≥ 0
or
𝑀𝑃𝐶𝑖 − 𝐿𝑃𝐶𝑖 − 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
n) AS Offer MW constraint for every modeled “Blocky” Load Resource 𝑖 = 1,2,3 … 𝑁𝑏𝑙 : (Shadow price = 𝜃𝑖𝑏𝑙−𝐴𝑆 )
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 ≥ 0
or
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
o) MPC & LPC Constraint for every modeled Controllable Load Resource 𝑖 = 1,2,3 … 𝑁𝑐𝑙 : (Shadow price = 𝜃𝑖𝑐𝑙 )
𝑀𝑃𝐶𝑖 − 𝐿𝑃𝐶𝑖 − 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 − 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
p) AS Offer MW constraint for every Controllable Load Resource 𝑖 = 1,2,3 … 𝑁𝑐𝑙 : (Shadow price =
𝑐𝑙−𝐴𝑆𝑢𝑝
𝜃𝑖
, 𝜃𝑖𝑐𝑙−𝐴𝑆𝑑𝑛 respectively)
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 − 𝑀𝑊𝑖𝑆𝑂𝑅𝐴𝑤𝑎𝑟𝑑 ≥ 0
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝐷𝑛−𝑖 − 𝑀𝑊𝑖
𝑅𝑒𝑔𝐷𝑛𝐴𝑤𝑎𝑟𝑑
≥0
q) HSL & LSL Constraint for every “Quick/Fast” Resource qualified for FRRS-Up and FFR1 and partly modeled as
𝑓𝑔
Generation Resource 𝑖 = 1,2,3 … 𝑁𝑓𝑔 : (Shadow price = 𝜃𝑖 )
𝐻𝑆𝐿𝑖 − 𝐿𝑆𝐿𝑖 − 𝑀𝑊𝑖
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
≥0
r) AS Offer MW constraint for every “Quick/Fast” Resource qualified for FRRS-Up and FFR1 and partly modeled
𝑓𝑔−𝐴𝑆
as Generation Resource 𝑖 = 1,2,3 … 𝑁𝑓𝑔 : (Shadow price = 𝜃𝑖
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖
)
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
≥0
s) MPC & LPC Constraint for every “Quick/Fast” Resource qualified for FRRS-Up, FRRS-Dn and RRS and partly
𝑓𝑐𝑙
modeled as Controllable Load Resource 𝑖 = 1,2,3 … 𝑁𝑓𝑐𝑙 : (Shadow price = 𝜃𝑖 )
𝑀𝑃𝐶𝑖 − 𝐿𝑃𝐶𝑖 − 𝑀𝑊𝑖𝐹𝑅𝑅𝑆𝐷𝑛𝐴𝑤𝑎𝑟𝑑 − 𝑀𝑊𝑖
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
− 𝑀𝑊𝑖𝑅𝑅𝑆𝐴𝑤𝑎𝑟𝑑 ≥ 0
t) AS Offer MW constraint for every “Quick/Fast” Resource qualified for FRRS-Up and FRRS-Dn and partly
𝑓𝑐𝑙−𝐴𝑆𝑢𝑝
modeled as Controllable Load Resource 𝑖 = 1,2,3 … 𝑁𝑓𝑐𝑙 : (Shadow price = 𝜃𝑖
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝑈𝑝−𝑖 − 𝑀𝑊𝑖
𝐹𝑅𝑅𝑆𝑈𝑝𝐴𝑤𝑎𝑟𝑑
≥0
𝐴𝑆𝑂𝑓𝑓𝑒𝑟𝑀𝑊𝐷𝑛−𝑖 − 𝑀𝑊𝑖𝐹𝑅𝑅𝑆𝐷𝑛𝐴𝑤𝑎𝑟𝑑 ≥ 0
𝑓𝑐𝑙−𝐴𝑆𝑑𝑛
, 𝜃𝑖
respectively)
Lagrangian Function:
The objective and constraints are combined to form the Lagrange function:
ℒ = {𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 − ∑ 𝑆ℎ𝑎𝑑𝑜𝑤𝑝𝑟𝑖𝑐𝑒𝑖 × 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑖 }
𝑖
At optimal solution ∇ℒ = 0 (optimality condition)
i.e. the partial derivative of ℒ with respect to each award
(𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
, 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
(𝜆, 𝛼, 𝜃𝑖𝑒𝑏 , 𝜃𝑖𝐿𝑆𝐿 , 𝜃𝑖𝐻𝑆𝐿 , 𝑒𝑡𝑐. )
, 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
, 𝑒𝑡𝑐) and the shadow prices
will equate to zero at the optimal solution.
Taking the partial derivative of ℒ with respect to each award
(𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
, 𝑀𝑊𝑖
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
and rearranging the terms by ∆𝑀𝑊𝑖
, 𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑𝐴𝑤𝑎𝑟𝑑
, ∆𝑀𝑊𝑖
, 𝑒𝑡𝑐. )
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟𝐴𝑤𝑎𝑟𝑑
, ∆𝑀𝑊𝑖
𝑅𝑒𝑔𝑈𝑝𝐴𝑤𝑎𝑟𝑑
, 𝑒𝑡𝑐. we get:
a) Day Ahead: For each energy bid 𝑖 = 1,2,3, … 𝑁𝑒𝑏, the following equation holds true
𝐸𝑛𝑒𝑟𝑔𝑦𝐵𝑖𝑑
𝐶𝑖
= 𝜆 − 𝜃𝑖𝑒𝑏
If the energy bid i is marginal to the power balance constraint, then 𝜃𝑖𝑒𝑏 = 0 and the energy bid i sets the
shadow price for the power balance constraint (System Lambda (𝜆))
Real-Time: If the Power Balance Penalty Curve is marginal to energy, the following equation holds true and
the Power Balance Penalty Curve sets the shadow price for the power balance constraint (System Lambda
(𝜆)):
𝐶𝑖𝑃𝑜𝑤𝑒𝑟𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = 𝜆
b) For each energy offer from modeled Generation Resource 𝑖 = 1,2,3, … 𝑁𝑔 , the following equation holds true
𝐸𝑛𝑒𝑟𝑔𝑦𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
= 𝜆 − 𝜃𝑖𝐻𝑆𝐿 + 𝜃𝑖𝐿𝑆𝐿
If the energy offer i is marginal to the power balance constraint, then, 𝜃𝑖𝐻𝑆𝐿 = 0, 𝜃𝑖𝐿𝑆𝐿 = 0 and the energy
offer i sets the shadow price for the power balance constraint (System Lambda (𝜆))
c) For each RegUp offer from modeled Generation Resource 𝑖 = 1,2,3, … 𝑁𝑔 , the following equation holds true
𝑅𝑒𝑔𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑔−𝐴𝑆𝑢𝑝
= 𝛼 − 𝜃𝑖𝐻𝑆𝐿 − 𝜃𝑖
If the RegUp offer i is marginal to the RegUp Procurement constraint, then in most cases, 𝜃𝑖𝐻𝑆𝐿 =
𝑔−𝐴𝑆𝑢𝑝
0, 𝜃𝑖
= 0 and the RegUp Offer i sets the shadow price for the RegUp Procurement constraint ( this is
the RegUp MCPC (𝛼) )
d) For each RegDn offer from modeled Generation Resource 𝑖 = 1,2,3, … 𝑁𝑔 , the following equation holds true
𝑅𝑒𝑔𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑔−𝐴𝑆𝑑𝑛
= 𝛽 − 𝜃𝑖𝐿𝑆𝐿 − 𝜃𝑖
If the RegDn offer i is marginal to the RegDn Procurement constraint, then, in most cases, 𝜃𝑖𝐻𝑆𝐿 =
𝑔−𝐴𝑆𝑢𝑝
0, 𝜃𝑖
= 0 and the RegDn Offer i sets the shadow price for the RegDn Procurement constraint ( this is
the RegDn MCPC (𝛽) )
e) For each RRS offer from modeled Generation Resource 𝑖 = 1,2,3, … 𝑁𝑔 , the following equation holds true
𝑅𝑅𝑆𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑔−𝐴𝑆𝑢𝑝
= 𝛾 − 𝜃𝑖𝐻𝑆𝐿 − 𝜃𝑖
𝑔−𝐴𝑆𝑢𝑝
If the RRS offer i is marginal to the RRS Procurement constraint, then, 𝜃𝑖𝐻𝑆𝐿 = 0, 𝜃𝑖
= 0 and the RRS
Offer i sets the shadow price for the RRS Procurement constraint (this is the RRS MCPC (𝛾) )
f)
For each SOR offer from modeled Generation Resource 𝑖 = 1,2,3, … 𝑁𝑔 , the following equation holds true
𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑔−𝐴𝑆𝑢𝑝
= 𝛿 + 𝜙 − 𝜃𝑖𝐻𝑆𝐿 − 𝜃𝑖
𝑔−𝐴𝑆𝑢𝑝
If the SOR offer i is marginal to the SOR Procurement constraint, then, 𝜃𝑖𝐻𝑆𝐿 = 0, 𝜃𝑖
=0 .
Please note that the MCPC for SOR is the sum of the shadow price for the SOR and SNSOR (𝛿 + 𝜙)
procurement constraints.
g) For each SNSOR offer from modeled offline Generation Resource 𝑖 = 1,2,3, … 𝑁𝑔 , the following equation
holds true
𝑆𝑁𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑔−𝐴𝑆𝑢𝑝
= 𝜙 − 𝜃𝑖𝐻𝑆𝐿 − 𝜃𝑖
𝑔−𝐴𝑆𝑢𝑝
If the SNSOR offer i is marginal to the SNSOR Procurement constraint, then, 𝜃𝑖𝐻𝑆𝐿 = 0, 𝜃𝑖
= 0.
Note that the MCPC for SNSOR is the shadow price for the SNSOR procurement constraint
h) For each RRS offer from modeled “blocky” Load Resource 𝑖 = 1,2,3, … 𝑁𝑏𝑙 , the following equation holds true
𝑅𝑅𝑆𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
= 𝛾 − 𝜔𝑅𝑅𝑆𝑏𝑙 − 𝜃𝑖𝑏𝑙 − 𝜃𝑖𝑏𝑙−𝐴𝑆
If the RRS offer i is marginal to the RRS Procurement constraint, then, 𝜔𝑅𝑅𝑆𝑏𝑙 = 0, 𝜃𝑖𝑏𝑙 = 0, 𝜃𝑖𝑏𝑙−𝐴𝑆 = 0 and
the RRS Offer i sets the shadow price for the RRS Procurement constraint (this is the RRS MCPC (𝛾) )
Note that if the RRS offer MW is submitted as a block, then, this offer cannot set the MCPC
If the RRS maximum procurement constraint from “blocky” Load Resource is binding then 𝜔𝑅𝑅𝑆𝑏𝑙 ≠ 0. In this
case the MCPC for RRS is still the shadow price of the RRS procurement constraint (𝛾) and the RRS awards to
“blocky” Load Resources will be paid this price 𝛾
i)
For each SOR offer from modeled “blocky” Load Resource 𝑖 = 1,2,3, … 𝑁𝑏𝑙 , the following equation holds true
𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
= 𝛿 + 𝜙 − 𝜃𝑖𝑏𝑙 − 𝜃𝑖𝑏𝑙−𝐴𝑆
If the SOR offer i is marginal to the SOR Procurement constraint, then, 𝜃𝑖𝑏𝑙 = 0, 𝜃𝑖𝑏𝑙−𝐴𝑆 = 0 and the SOR
Offer i sets the shadow price for the SOR Procurement constraint (this is the SOR MCPC (𝛿) )
Note that if the SOR offer MW is submitted as a block, then, this offer cannot set the MCPC
j)
For each RegUp offer from modeled Controllable Load Resource 𝑖 = 1,2,3, … 𝑁𝑐𝑙 , the following equation
holds true
𝑅𝑒𝑔𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑐𝑙−𝐴𝑆𝑢𝑝
= 𝛼 − 𝜃𝑖𝑐𝑙 − 𝜃𝑖
k) For each RegDn offer from modeled Controllable Load Resource 𝑖 = 1,2,3, … 𝑁𝑐𝑙 , the following equation
holds true
𝑅𝑒𝑔𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
l)
= 𝛽 − 𝜃𝑖𝑐𝑙 − 𝜃𝑖𝑐𝑙−𝐴𝑆𝑑𝑛
For each RRS offer from modeled Controllable Load Resource 𝑖 = 1,2,3, … 𝑁𝑐𝑙 , the following equation holds
true
𝑅𝑅𝑆𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑐𝑙−𝐴𝑆𝑢𝑝
= 𝛾 − 𝜃𝑖𝑐𝑙 − 𝜃𝑖
m) For each SOR offer from modeled Controllable Load Resource 𝑖 = 1,2,3, … 𝑁𝑐𝑙 , the following equation holds
true
𝑆𝑂𝑅𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑐𝑙−𝐴𝑆𝑢𝑝
= 𝛿 + 𝜙 − 𝜃𝑖𝑐𝑙 − 𝜃𝑖
n) For each FRRS-Up offer from “Quick/Fast” Resource partly modeled as Generation Resource 𝑖 = 1,2,3, … 𝑁𝑓𝑔 ,
the following equation holds true
𝐹𝑅𝑅𝑆𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑓𝑔
= 𝛼 − 𝜔𝐹𝑅𝑅𝑆𝑈𝑝 − 𝜃𝑖
𝑓𝑔−𝐴𝑆
− 𝜃𝑖
𝑓𝑔
If the FRRS-Up offer i is marginal to the RegUp Procurement constraint, then, 𝜔𝐹𝑅𝑅𝑆𝑈𝑃 = 0, 𝜃𝑖
=
𝑓𝑔−𝐴𝑆
0, 𝜃𝑖
= 0 and the FRRS-Up Offer i sets the shadow price for the RegUp Procurement constraint (this is
the RegUp MCPC (𝛼) )
If the FRRS-Up maximum procurement constraint from “Quick/Fast” Resource is binding then 𝜔𝐹𝑅𝑅𝑆𝑈𝑃 ≠ 0.
In this case the MCPC for RegUp is still the shadow price of the RegUp procurement constraint (𝛼) and the
FRRS-Up awards to “Quick/Fast” Resources will be paid this price 𝛼
o) For each FRRS-Up offer from “Quick/Fast” Resource partly modeled as Load Resource 𝑖 = 1,2,3, … 𝑁𝑓𝑐𝑙 , the
following equation holds true
𝐹𝑅𝑅𝑆𝑈𝑝𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑓𝑐𝑙
= 𝛼 − 𝜔𝐹𝑅𝑅𝑆𝑈𝑝 − 𝜃𝑖
𝑓𝑐𝑙−𝐴𝑆
− 𝜃𝑖
𝑓𝑐𝑙
If the FRRS-Up offer i is marginal to the RegUp Procurement constraint, then, 𝜔𝐹𝑅𝑅𝑆𝑈𝑃 = 0, 𝜃𝑖
=
𝑓𝑐𝑙−𝐴𝑆
0, 𝜃𝑖
= 0 and the FRRS-Up Offer i sets the shadow price for the RegUp Procurement constraint (this is
the RegUp MCPC (𝛼) )
If the FRRS-Up maximum procurement constraint from “Quick/Fast” Resource is binding then 𝜔𝐹𝑅𝑅𝑆𝑈𝑃 ≠ 0.
In this case the MCPC for RegUp is still the shadow price of the RegUp procurement constraint (𝛼) and the
FRRS-Up awards to “Quick/Fast” Resources will be paid this price 𝛼
p) For each FRRS-Dn offer from “Quick/Fast” Resource partly modeled as Load Resource 𝑖 = 1,2,3, … 𝑁𝑓𝑐𝑙 , the
following equation holds true
𝐹𝑅𝑅𝑆𝐷𝑛𝑂𝑓𝑓𝑒𝑟
𝐶𝑖
𝑓𝑐𝑙
= 𝛽 − 𝜔𝐹𝑅𝑅𝑆𝐷𝑛 − 𝜃𝑖
𝑓𝑐𝑙−𝐴𝑆𝑑𝑛
− 𝜃𝑖
𝑓𝑐𝑙
If the FRRS-Dn offer i is marginal to the RegDn Procurement constraint, then, 𝜔𝐹𝑅𝑅𝑆𝐷𝑁 = 0, 𝜃𝑖
=
𝑓𝑐𝑙−𝐴𝑆𝑑𝑛
0, 𝜃𝑖
= 0 and the FRRS-Dn Offer i sets the shadow price for the RegDn Procurement constraint (this
is the RegUp MCPC (𝛽) )
If the FRRS-Dn maximum procurement constraint from “Quick/Fast” Resource is binding then 𝜔𝐹𝑅𝑅𝑆𝐷𝑁 ≠ 0.
In this case the MCPC for RegDn is still the shadow price of the RegDn procurement constraint (𝛽) and the
FRRS-Up awards to “Quick/Fast” Resources will be paid this price 𝛽
Note: If there is scarcity in any of the AS (RegUp, RegDn, RRS, SOR and SNSOR), then the demand curves prices at the
last cleared MW AS demand on the respective demand curves will set the Shadow Prices of the applicable
procurement constraints.
MCPC formula (from shadow prices)
AS Product
RegUp
MCPC
𝛼
FRRS-Up
RegDn
𝛼
𝛽
FRRS-Dn
RRS
SOR
𝛽
𝛾
𝛿+𝜙
SNSOR
𝜙
Comments
Shadow price of the RegUp procurement
(including FRRS-Up) constraint
FRRS-Up is valued the same as RegUp
Shadow price of the RegDn procurement
(including FRRS-Dn) constraint
FRRS-Down is valued the same as RegDn
Shadow Price of the RRS procurement constraint
Sum of the Shadow Prices of the SOR and SNSOR
procurement constraints
Shadow Price of the SNSOR procurement
constraint