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

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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)
CHEN et al.: COMPONENTS OF NODAL PRICES FOR ELECTRIC POWER SYSTEMS 43By solving OPF (1)–(3), we have an optimal solution where,and line flow from bus 3 to bus 2 reaches its limit – .In addition, the voltage values of bus 2 and bus 4 also reachtheir lower and upper bounds, respectively, i.e., and. Therefore, there are only a few Lagrangian multipliersnonzero (related to four equalities and three active inequalities)which areFig. 1.Four-bus test system.TABLE ITRANSMISSION LINE DATA OF A FOUR-BUS TEST SYSTEMExample 1: For the system shown in Fig. 1 and Table I, upperand lower bounds for generators G1 and G4 are, ; , .The voltage values for all buses are bounded between 0.95 and1.05. Besides, the real power flow in line 2–3 is also restrictedbetween 0.3 and 0.3. All of the values are indicated by p.u.The fuel cost function for generators G1 and G4 is expressed as.For this four-bus system, there are four equalities for (2) correspondingto their respective real and reactive power balances(or Kirchoff’s laws) of load-buses 2 and 3, and 18 inequalitiesfor (3) corresponding to four pairs of voltage, 2 2 pairs ofgeneration output, and one pair of line flow upper and lowerbounds, respectively. In this paper, we take real and imaginaryparts of bus voltages as state variable which has 2 4 elements.Therefore, and in ,and can be represented in terms of andby using real power balances of their respective buses.Regardless of the solution, according to (5), the nodal prices ofreal power have the forms aswhere and are the Lagrangian multipliers related toreal power balances of the respective buses 2 and 3. andare the Lagrangian multipliers related to the upper andlower bounds of real power generation for generator , respectively,while and are for generator .Obviously, nodal price at demand bus- is expressed onlyby the Lagrangian multiplier corresponding to the powerflow balance of bus- , and all other terms in (5) vanish due totheir relation independent of . It is also true for the nodalprices of reactive power.(7)(8)(9)(10)where and are the Lagrangian multipliers related tothe lower and upper bounds of buses 2 and 3, and – correspondsto the line flow constraint from bus 3 to bus 2. Since thegenerator at bus 4 is cheaper than the generator , electricpower is mainly transferred from through bus 3 toward bus2. As a result, power flow in line 3-2 reaches its limit (0.3 p.u.)which causes the congestion problem, although the cheaper generatorstill has generation capability.Substituting the values of the Lagrangian multipliers and( ) into (7), we have the nodal prices of real power(11)where .Hence, even although we can obtain the nodal pricesaccording to (5) and (6), it is still unknown exactly whatcomponents of each nodal price are and how the constraints(congestion or other limits) and generators influence the valueof each nodal price.III. COMPONENTS OF NODAL PRICEThere are many factors or constraints affecting the operationof power systems, e.g., generators, voltage limits, line flowlimits, power flow balance conditions (Kirchoff’s laws). Someof them (e.g., voltage limits) have market values which maybe relaxed (e.g., from 1.00 1.05 to 0.95 1.05) and taken astradable goods depending on market needs. The relaxation forthese limits may be realized by technology innovations or facilitiesinvestments, etc. But some of them actually cannot betraded, e.g., for real power flow balance condition at each bus,the summation of all injected real power at each bus must bezero which cannot be relaxed or violated because it is a physicallaw. Therefore the evaluation or pricing for the factors with nomarket value is meaningless, even although we can theoreticallytrace the contributions of all factors involving in the operationsof power systems to each nodal price. Hence, before breakingdown the nodal prices, we have to classify all constraints in theoperations of power systems into two groups, i.e., tradable constraintswhich should be components of each nodal price, andnontradable constraints which are mandatory constraints duringthe operation and are not components of nodal prices.In this section, we theoretically propose a method to breakdown the nodal prices into a variety of components by usingthe marginal conditions of operation, which are derived from
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