Least Cost System Operation: Economic Dispatch 2OverviewTime Scale SeparationEconomic Dispatch RecapSlide 5Mathematical Formulation of CostsSlide 7Economic DispatchIncremental Cost ExampleSlide 10Economic Dispatch: FormulationUnconstrained MinimizationMinimization with Equality ConstraintEconomic Dispatch LagrangianEconomic Dispatch ExampleEconomic Dispatch Example, cont’dConstrained Optimization & Linear ProgrammingLinear Programming DefinitionFormulating the ProblemSlide 20Slide 21Formulating the LagrangeanSlide 23Lagrangean ExampleEconomic Dispatch & the LagrangeanSlide 26Slide 27Slide 28Slide 29DiscussionPower System Control CenterSlide 32New England Power Grid OperatorRegional Prices and ConstraintsSlide 35Slide 36Summary1Least Cost System Operation: Economic Dispatch 2Smith College, EGR 325March 10, 20062Overview•Complex system time scale separation•Least cost system operation–Economic dispatch first view–Generator cost characteristics•System-level cost characterization•Constrained optimization –Linear programming–Economic dispatch completed3Time Scale Separation1. Decide what to build2. Given the plants that are built  decide which plants to have warmed up and ready to go this month, week...3. Given the plants that are ready to generate  decide which plants to use to meet the expected load today, the next 5 minutes, next hour...4. Given the plants that are generating  Decide how to maintain the supply and demand balance cycle to cycle4Economic Dispatch Recap•Economic dispatch determines the best way to minimize the current generator operating costs•Economic dispatch is not concerned with determining which units to turn on/off (this is the unit commitment problem)•Economic dispatch ignores the transmission system limitations5Constrained Optimization& Economic Dispatch6Mathematical Formulation of Costs•Generator cost curves are not actually smooth•Typically curves can be approximated using –quadratic or cubic functions–piecewise linear functions2( ) $/hr (fuel-cost)( )( ) 2 $/MWh i Gi i Gi Gii Gii Gi GiGiC P P PdC PIC P PdPa b gb g= + += = +7Mathematical Formulation of Costs•The marginal cost is one of the most important quantities in operating a power system•Marginal cost = incremental cost: the cost of producing the next increment (the next MWh)•How do we find the marginal cost?8Economic Dispatch•An economic dispatch results in all the generator generating at a level where they have equal marginal costs (for a lossless system) IC1(PG,1) = IC2(PG,2) = … = ICm(PG,m)9Incremental Cost Example21 1 1 122 2 2 21 11 1 112 22 2 22For a two generator system assume( ) 1000 20 0.01 $ /( ) 400 15 0.03 $ /Then( )( ) 20 0.02 $/MWh( )( ) 15 0.06 $/MWhG G GG G GGG GGGG GGC P P P hrC P P P hrdC PIC P PdPdC PIC P PdP= + += + += = += = +10G1 G2212212If P 250 MW and P 150 MW Then(250) 1000 20 250 0.01 250 $ 6625/hr(150) 400 15 150 0.03 150 $6025/hrThen(250) 20 0.02 250 $ 25/MWh(150) 15 0.06 150 $ 24/MWhCCICIC= == + � + � == + � + � == + � == + � =Incremental Cost Example11Economic Dispatch: Formulation•The goal of economic dispatch is to –determine the generation dispatch that minimizes the instantaneous operating cost– subject to the constraint that total generation = total load + lossesT1mi=1Minimize C ( )Such thatmi GiiGi D LossesC PP P P== +��@Initially we'll ignore generatorlimits and thelosses12Unconstrained Minimization•This is a minimization problem with a single inequality constraint•For an unconstrained minimization a necessary (but not sufficient) condition for a minimum is the gradient of the function must be zero, •The gradient generalizes the first derivative for multi-variable problems:1 2( ) ( ) ( )( ) , , ,nx x x       f x f x f xf x @ K( ) f x 013Minimization with Equality Constraint•When the minimization is constrained with an equality constraint we can solve the problem using the method of Lagrange Multipliers•Key idea is to modify a constrained minimization problem to be an unconstrained problemThat is, for the general problem minimize ( ) s.t. ( ) We define the Lagrangian L( , ) ( ) ( )Then a necessary condition for a minimum is theL ( , ) 0 and L ( , ) 0T    xλf x g x 0xλ f x λ g xxλ x λ14Economic Dispatch LagrangianG1 1GFor the economic dispatch we have a minimization constrained with a single equality constraintL( , ) ( ) ( ) (no losses)The necessary conditions for a minimum areL( , )m mi Gi D Gii iGiC P P PdCP     PP1( )0 (for i 1 to m)0i GiGimD GiiPdPP P   15Economic Dispatch ExampleD 1 221 1 1 122 2 2 21 11What is economic dispatch for a two generator system P 500 MW and( ) 1000 20 0.01 $/( ) 400 15 0.03 $/Using the Largrange multiplier method we know( )20 0G GG G GG G GGGP PC P P P hrC P P P hrdC PdP        12 2221 2.02 0( )15 0.06 0500 0GGGGG GPdC PPdPP P        16Economic Dispatch Example, cont’d121 21212We therefore need to solve three linear equations20 0.02 015 0.06 0500 00.02 0 1 200 0.06 1 151 1 0 500312.5 MW187.5 MW26.2 $/MWGGG GGGGGPPP PPPPPllll+ - =+ - =- - =- -� �� � � �� �� � � �- = -� �� � � �- - -� �� � � �� �� � � �� �� �=� �� �� �h� �� �� �� �� �17Constrained Optimization & Linear Programming18Linear Programming Definition•Optimization is used to find the “best” value–“Best” defined by us, the analysts and designers•Constrained opt  Linear programming–Linear constraints–Complicates the problem•Some binding, some non-binding•Visualize via a ‘feasible region’19Formulating the Problem•Objective function•Constraints•Decision variables•Variable bounds•Standard form–min cx–s.t. Ax = b xmin <= x <= xmax20Formulating the Problem•For power systems:min CT = ΣCi(PGi)s.t. Σ(PGi) = PL PGi min <= PGi <= PGi max21Constrained Optimization& Economic DispatchThe Lagrangean22Formulating the Lagrangean•Rewrite the constrained optimization problem as an unconstrained optimization problem !–Then we can use the simple derivative (unconstrained optimization) to solve•The task is to interpret the results correctly23•We are minimizing gradients of both multivariate equations–CT & ΣPGi = PL•For