Components of nodal prices for electric power systems - Power ...

42 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002Therefore, the operation problem of a power system forthe given loads ( , ) can be formulated as an OPF problem[8]–[10]Min for (1)s.t. (2)(3)whereandhave andequations, respectively, and are column vectors whilestands for the transpose of a vector .: scalar, short-term operating cost, such as fuelcost;: vector, equality constraints, such as buspower flow balances (Kirchoff’s laws);: vector, inequality constraints includinglimits of all variableswhereincludes all variable limits and functionlimits, such as upper and lower bounds of transmission lines,generation outputs, stability or security limits [10], etc. Whenis a fuel cost function of the system, can be expressed aswhere is the fuel cost ofthe th generator. Notice that can also be a benefit functionalthough this paper takes as a cost function.Obviously, (1)–(3) are a typical OPF problem as far as thedemands ( , ) are given. There are many efficient approacheswhich can be used to obtain an optimal solution, such assuccessive linear programming, successive quadratic programming,the Newton method, the – decomposition approach[6], [7], the surrogate constraint and functional transformationapproaches [8]–[10].B. Nodal PriceDefine the Lagrangian function (or system cost) of (1)–(3) as, thenwhereandare the Lagrangian multipliers (or dual variables) associatedto (2) and (3), respectively, and are usuallyexplained as shadow prices from the viewpoint of economics.Hence,and. Actually, the Lagrangianfunction can also be viewed as an equivalentsystem cost.Then at an optimal solution ( ) and for a set of given( , ), the nodal prices of real and reactive power for each busare expressed below for(4)(5)where and are nodal prices of real and reactive powerat bus- , respectively. and are usuallyconsidered as the real and reactive power transaction chargesfrom bus- to bus- .Notice that the nodal prices derived by differentiating themaximized social welfare function with respect to the real andreactive demands, yield the same results as (5) and (6). That is,first replace (1) with “for ” subject to the same constraints as (2) and (3),whereis short-term value-added function of thecustomers connected to bus- . Then we can prove that nodalprices are the same as (5) and (6), by letting the marginalbenefitequal to the nodal prices of realand reactive power at the optimal solution, respectively [5].Therefore according to (5), the nodal price of realpower at bus- can be viewed as the system marginal cost(the sum of a marginal generation cost plus a setof premiums corresponding to their respective constraints)created by an increment of real power load at bus- . Thesame explanation is also applicable to the nodal price ofreactive power .A significant property for nodal prices is that each nodal priceis actually defined simply as a linear summation of all factorsaccording to (5) and (6) because each nodal price, e.g., canbe rewritten asThis property is completely different from that of AC powerflow which is generally nonlinear to each source or route, andis also fundamental to the decomposition or coloring of nodalprices in this paper. Therefore, theoretically it is possible to tracethe contributions of all factors involving in the operations ofpower systems to each nodal price.C. Problem and ExampleEquation (5) or (6) seems to have given a full descriptionof each component for a nodal price even without any furtheranalysis. However, in contrary to intuitive observation, we willshow late that it is incorrect. Actually there are generally onlyone or two terms remaining nonzero at an optimal solutionfor (5) and (6) depending on the formulation, even thoughmany constraints become active (or binding) and many generatorscontribute to this nodal price. The main reason is thatthe variables () in the Lagrangian function (4) [orOPF equations (1)–(3)] are all independent variables and only afew equations among (2) and (3) or have direct relation withcertain bus demand or . Therefore, most of the terms in (5)or (6) are eliminated after differentiating with certain demandor , irrespective of any solution. We take a four-bus systemshown in Fig. 1 and Table I, as an example to show this point.(6)