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 MW P min Real Price Curve Piecewise Linear Approximation Stepwise Approximation P max

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 CurveEnergy Cost Curve (Objective Function) Piecewise Linear: P price = a slope ∙ P unit + b const Piecewise Quadratic: C cost = ½ ∙ a slope ∙ P 2 unit + P unit ∙ b const + c mincost Stepwise: P price = b const Piecewise Linear: C cost = b const ∙ P unit + c mincost The difference is quadratic term marked by red color P unit is the delta MW output from the start of the given price curve segment.

SCED – Mathematical Formulation Minimize Sum seg&unit { C cost = ½ ∙ a slope ∙ P 2 unit + P unit ∙ b const + c mincost } Subject to: sum seg&unit { P unit } = P load - Power balance sum seg&unit { SF unit/line ∙ P unit } ≤ Limit line - Transmission limits P min ≤ sum seg { P unit } ≤ P max - Unit limits Note: Optimization Objective is bounded, continuous and convex function (a slope ≥ 0) for each segment. For QP SCED a slope > 0 and for LP SCED a slope = 0. All constraints are linear, i.e. determine a convex set.

SCED – Optimality Conditions Lagrange Function: £ = sum seg&unit { ½ ∙ a slope ∙ P 2 unit + P unit ∙ b const + c mincost } + λ ∙ (P load – sum seg&unit { P unit }) + sum line { η line ∙ (Limit line – sum seg&unit { SF unit/line ∙ P unit } ) } Optimality Conditions: d£/dP unit = a slope ∙ P unit + b const - λ - sum line { η line ∙ SF unit/line } = 0 sum seg&unit { P unit } = P load sum seg&unit { SF unit/line ∙ P unit } + F slack = Limit line η line ∙ F slack = 0- complementary slackness P min ≤ sum seg {P unit }≤ P max ; F slack ≥ 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)