Transcript SCED Linear vs. Quadratic Programming Model

SCED
Security Constrained Economic Dispatch
Linear vs. Quadratic
Programming Model
• When the initial requirements where written, a section was added as
appendix (Section 8) to give a general understanding of the
implementation. At that time, the understanding was that SCED would use
Linear Programming. Later during the design phase it was concluded that
the energy offer price curve would need to be considered as piece wise
linear and also a piece wise linear offer price curve would automatically
resolve breaking a tie in instances of multiple resources being marginal at
the optimum.
• For these reasons SCED Quadratic Programming methodology was used.
This is reflected correctly in the functional specification. While updating
our SCED requirements document to include Baseline 1&2 changes we
unfortunately missed to update the Appendix to reflect this change. We
apologize for any confusion this may have caused and have provided an
updated version of the SCED requirements for your review and comments.
Also during previous TPTF meeting it was mentioned that Quadratic
Programming could lead to duality gap issues. Since SCED problem is
convex, the duality gap is zero and therefore we believe no issue exist.
• The LMP reasonability metric presented in the previous meeting is still
valid and is NOT impacted by the use of Quadratic Programming.
Energy Offer Price Curve
$/MWH
Real Price Curve
Piecewise Linear Approximation
Stepwise Approximation
Pmin
Pmax
MW
Optimization Cost Objective
• Each segment of
Energy Price Curve is represented
separately by segment variable
•The Energy Cost Curve (Objective Function) is equal to
integral of (i.e. area under) Energy Price Curve
Energy Price Curve
Energy Cost Curve (Objective Function)
Piecewise Linear:
Piecewise Quadratic:
Pprice = aslope ∙ Punit+ bconst
Ccost = ½ ∙ aslope ∙ P2unit + Punit∙ bconst + cmincost
Stepwise:
Pprice = bconst
Piecewise Linear:
Ccost = bconst ∙ Punit + cmincost
•The difference is quadratic term marked by red color
•Punit is the delta MW output from the start of the given
price curve segment.
SCED – Mathematical Formulation
Minimize
Sumseg&unit { Ccost = ½ ∙ aslope ∙ P2unit + Punit∙ bconst + cmincost }
Subject to:
sumseg&unit { Punit } = Pload
sumseg&unit { SFunit/line ∙ Punit } ≤ Limitline
Pmin ≤ sumseg{Punit} ≤ Pmax
- Power balance
- Transmission limits
- Unit limits
Note:
•Optimization Objective is bounded, continuous and
convex function (aslope ≥ 0) for each segment. For QP
SCED aslope > 0 and for LP SCED aslope = 0.
•All constraints are linear, i.e. determine a convex set.
SCED – Optimality Conditions
Lagrange Function:
£ = sumseg&unit { ½ ∙ aslope ∙ P2unit + Punit∙ bconst + cmincost } +
λ ∙ (Pload – sumseg&unit { Punit }) +
sumline { ηline ∙ (Limitline – sumseg&unit { SFunit/line ∙ Punit } ) }
Optimality Conditions:
d£/dPunit = aslope ∙ Punit + bconst - λ - sumline { ηline ∙ SFunit/line } = 0
sumseg&unit { Punit } = Pload
sumseg&unit { SFunit/line ∙ Punit } + Fslack = Limitline
ηline ∙ Fslack = 0 - complementary slackness
Pmin ≤ sumseg {Punit }≤ Pmax ; Fslack ≥ 0
Note:
•QP SCED has linear optimality conditions
•Complementary slackness is expressed for QP and LP in the same
way
SCED – Duality Gap
SCED QP Formulation:
•Bounded, continuous and convex optimization
objective
•Linear, i.e. convex constrained set Theory
QP Theory:
•Duality Gap = 0
SCED Implementation:
•Duality Gap ≤ ε (convergence tolerance, default $10-4)
SCED – Tie Breaking Rule
SCED Requirement:
•Dispatch units with the same flat price curve segments
proportionally to segment sizes.
SCED Implementation:
•A small Δ price value is added at the end points of tie
segments to create non-zero segment slope
SCED Tie Breaking Properties:
•Both units will leave beginning points simultaneously
•Units will be dispatched within tie segments in
proportion of tie segment sizes
•Both units will achieve segment end points
simultaneously
•Impact on unit costs is neglectable (small Δ value)