Variational optimal power flow and dispatch problems and their approximations Anna Scaglione Anna.Scaglione@asu.edu Arizona State University PSERC Webinar April 18 2107 1 / 42
Motivation ◮ Problem: Shortage of ramping resources in the real-time operation of power systems → ramping is not appropriately represented and incentivized 2 / 42
Motivation ◮ Problem: Shortage of ramping resources in the real-time operation of power systems → ramping is not appropriately represented and incentivized ◮ Flexible ramping products (e.g. CAISO and MISO) 3 / 42
Motivation ◮ Problem: Shortage of ramping resources in the real-time operation of power systems → ramping is not appropriately represented and incentivized ◮ Flexible ramping products (e.g. CAISO and MISO) ◮ Tenet: Better handling of both variability and uncertainty 4 / 42
Modeling errors in time ◮ Load demand is a continuous time random process ◮ Generators have continuous time inter-temporal constraints (ramping, on-off time) Objective Mapping the variational stochastic problems into tractable approximations. 5 / 42
Where is ramping first accounted for? 1. In the Unit Commitment (UC) we schedule a piecewise constant generation trajectory based on a single forecast 2. Trajectory Interpretation: Hourly ramping constraints → piecewise linear generation trajectory 6 / 42
Agenda Information loss → In the conventional practice, continuity and higher order stochastic features are being relaxed ◮ continuous trajectories & derivatives are replaced by samples & finite differences ◮ deterministic approximation: single forecast for the net-load ◮ stochastic approximation: only marginal distributions, Markov chains, discrete time quantized scenario trees/fan In this talk we introduce: ◮ Continuous Time Economic Dispatch (CT-ED), marginal pricing and approximation via Splines of CT DC OPF ◮ Continuous Time Unit Commitment: Deterministic (CT-DUC) and Stochastic Multi-Stage formulations (CT-SMUC) 7 / 42
Nomenclature for Continuous Time Optimization OPF and UC variables, Deterministic case ◮ Generator index g ∈ G : Set of generation units, ◮ Bus index b ∈ B : Set of buses, ◮ ( l , l ′ ) ∈ B × B : Set of transmission lines, ◮ ξ b ( t ) ∈ R + : Net-Load Demand ◮ Schedule for g ∈ G b ◮ x g ( t ) ∈ R + : Scheduled power ◮ ˙ x g ( t ) ∈ R + : Ramping decision ◮ y g ( t ) ∈ { 0 , 1 } : Commitment decision ◮ s g ( t ) switching action from off to on, ◮ s g ( t ) switching action from on to off. ◮ Costs: C g and startup S g , shut-down S g 8 / 42
Continuous Time Economic Dispatch and Marginal Pricing 9 / 42
Economic Dispatch in continuous Time Continuous Time Economic Dispatch: � t 0 + T min � � C g ( x g , t ) dt w.r.t x ( t ) Objective and decision var. g ∈G b b ∈B t 0 � � � � g ∈G b x g ( t ) − ξ b ( t ) = 0 Balance constraint b ∈B G g ≤ x g ( t ) ≤ G g Production capacity − G g ′ ≤ ˙ g ′ x g ( t ) ≤ G Ramping constraint ◮ Note: C g ( x g , t ) is a cost per unit of time (may depend on the x g too, optional) ramp ˙ The DC OPF version simply adds: � � � − L ll ′ ≤ � b ∈B D b g ∈G b x g ( t ) − ξ b ( t ) ≤ L ll ′ Thermal constraints ll ′ 10 / 42
Variational formulation of the CT-ED Lagrangian of the CT-ED: � t 0 + T � � f ( g , b ) ( x g , ˙ x g , t ) dt L = t 0 b ∈B g ∈G b � ξ b ( t ) � f ( g , b ) ( x g , ˙ x g , t ) = C g ( x g , t ) + λ ( t ) |G b | − x g ( t ) g ) + µ g ( t )( G g − ˙ + µ g ( t )( x g ( t ) − G x g ( t )) The variational problem: � t 0 + T � � f ( g , b ) ( x g , ˙ x g , t ) dt min x ( t ) L = min x ( t ) t 0 b ∈B g ∈G b is a special case of the isoperimetric problem in Physics. 11 / 42
Optimum solutions and Euler-Lagrange equations ◮ The optimum trajectories x g o ( t ) are solutions of the Euler-Lagrange partial differential equations: ∂ f ( g , b ) ( x g x g ∂ f ( g , b ) ( x g x g o , ˙ o , t ) o , ˙ o , t ) − d = 0 , ∀ b ∈ B , g ∈ G b ∂ x g ∂ ˙ x g dt plus the remaining KKT conditions... ◮ Hence, the Lagrange multiplier function, the marginal cost and the other Lagrange multipliers functions: λ o ( t ) = ∂ C g ( x g ∂ C g ( x g o , t ) o , t ) − d ∂ x g ∂ ˙ x g dt � �� � = 0 o ( t ) − d γ g d γ g o ( t ) o ( t ) + µ g o ( t ) − µ g + dt dt ∀ t 0 ≤ t ≤ t 0 + T , g ∈ G 12 / 42
Observations ◮ Due to complementarity slackness if constraints are not tight µ g o ( t ) = 0 and/or γ g o ( t ) = µ g o ( t ) = γ g o ( t ) = 0. ◮ For feasibility each time instant t 0 ≤ t ≤ t 0 + T the always exist an extra unit to meet demand ◮ The marginal unit is the unit g ∗ for which at time t and so o ( t ) = 0 and/or γ g ∗ µ g ∗ o ( t ) = 0 λ o ( t ) = ∂ C g ∗ ( x g ∗ o , t ) ∂ x g ◮ Note that since the marginal unit in general will be different at different times, λ o ( t ) is naturally a discontinuous function (piece-wise constant if costs are linear in x g ( t ) ) 13 / 42
Marginal Price ◮ Suppose we increase the entire load trajectory at an arbitrary bus by a constant ξ b ( t ) → ˜ ξ b ( t ) = ξ b ( t ) + ǫ without any change in ramp ◮ It is not difficult to see that the rate of change of the objective w.r.t. ǫ is: � t 0 + T L ∗ ( ǫ ) − L ( ǫ ) lim = λ o ( t ) dt ǫ ǫ → 0 t 0 which in turn implies that λ o ( t ) could be interpreted as a shadow price per unit of time. 14 / 42
Approximation of the CT DC-OPF ◮ Without loss of generality let t 0 = 0 and T = 1 ◮ Suppose also that C g ( x g , t ) = C g ( x g ) = Λ g x g + const . ◮ If the net-load lies approximately in an n + 1 dimensional signal space, spanned by the linearly independent functions { b ( n ) ( t ) } n i = 0 can we approximate the variational solution? i n n � � x g ξ b ( t ) ≈ ξ b x g ( t ) ≈ i b i , n ( t ) → i b i , n ( t ) i = 0 i = 0 There are uncountable constraints i − � g ∈G b x g ξ b ◮ Balance: OK if ∀ b ∈ B , i = 0 , .., n , i = 0 ◮ Inequalities: Capacity and ramping constraints, flows need attention → this goal guides the choice of { b ( n ) ( t ) } n i i = 0 15 / 42
Bernstein Polynomials Bernstein polynomials of degree n are defined as � n � t i ( 1 − t ) n − i Π( t ) , b i , n ( t ) = i ∈ [ 0 , n ] i � 0 ≤ t ≤ 1 1 Π( t ) = 0 else And the vector of polynomials of degree n is denoted by b n ( t ) . 1 b 0,4 (t) 0.8 b 1,4 (t) b 2,4 (t) b 3,4 (t) 0.6 b 4,4 (t) 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Figure: Bernstein polynomial basis for n = 4 16 / 42
Convex Hull Property ◮ The coefficients of the Bernstein polynomial expansion define control points for the corresponding curves are called Bérzier curves ◮ A Bérzier curve is always contained in the convex hull of the control points ◮ For a 1 D function: x i ≤ x ( t ) ≤ max min x i i i ◮ The derivative is also a Bérzier curve of order n − 1 such that n − 1 � x ( t ) = ˙ n ( x i + 1 − x i ) b i ( n − 1 ) ( t ) � �� � i = 0 ˙ x i 17 / 42
Approximation of DC -OPF Indicating by x the ( n + 1 ) × |G| matrix of all coefficients: � 1 � 1 n � � � x g C g ( x g ) dt = min Λ g min b in ( t ) dt i x ( t ) X 0 0 i = 0 g ∈G g ∈G � �� � 1 n + 1 � x g ξ b s.t. Balance ∀ b ∈ B , i = 0 , .., n , i − i = 0 g ∈G b g & min i x g i ≥ G g imply ◮ Capacity: max i x g i ≤ G G g ≤ x g ( t ) ≤ G g i ) ≥ G ′ g / n & ◮ Ramping: Similarly max i ( x g i + 1 − x g i ) ≥ − G ′ g / n imply − G ′ g ≤ ˙ x g ( t ) ≤ G ′ g min i ( x g i + 1 − x g ◮ Flow constraints : Analogously, sufficient conditions are: � � � � �� � � � � �� D b x g i − ξ b D b x g i − ξ b ≥ − L ll ′ ≤ L ll ′ min max ll ′ ll ′ i i i i b ∈B g ∈G b b ∈B g ∈G b 18 / 42
Cost and price ◮ In the approximate solution constraints become tight 1 / ( n + 1 ) earlier than in reality ◮ It forces C 1 continuity of generation trajectory → it imposes the generators to go smoothly towards their limits 19 / 42
Continuous Time Deterministic and Stochastic Multi-Stage Unit Commitment 20 / 42
Deterministic CT UC ◮ In principle the commitment function y g ( t ) ∈ { 0 , 1 } could switch units at any time → UC then non-linear problem ◮ State of the art MILP approximation: switching only at the beginning of hour h : � t − t h − 1 H � � y g y g ( t ) = h Π t h − t v − 1 h = 1 i.e. one hourly variable y g h ∈ 0 , 1 describes the degrees of freedom for y g ( t ) in ( t h − 1 , t h ] The idea of the CT UC: ◮ Allow the scheduled trajectories x g ( t ) within each ( t h − 1 , t h ] to be a Bérzier curve of the order needed to represent accurately the Bérzier curve of net-load ξ b ( t ) ◮ Keep things continuous from one hour to the next 21 / 42
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