DC Optimal Power Flow Proxy Limits

Transcription

1 DC Optimal Power Flow Proxy Limits Bernard C. Lesieutre Univ. of Wisconsin-Madison Michael Schlindwein Univ. of Wisconsin-Madison Edward E. Beglin Navigant Consulting Abstract Proxy limits are used to represent voltage constraints from an AC power flow as line flow constraints in a DC power flow. When a DC power flow is used to settle markets, the use of proxy limits can introduce errors in LMPs, dispatch, or both. his paper describes a method to obtain the optimal set of proxy limits for known voltage constraints. We discuss the general properties of these proxy limits. 1. Introduction A key feature of efficient electric energy markets is a transparent valuation of energy. Locational Marginal Prices (LMPs) are often used to settle markets, aid in congestion management, and provide feedback to market participants. he calculation of LMPs involves sophisticated optimization routines that combine economic information concerning supplier offers to sell and various technical constraints and limits that govern the operation of the electric power grid. wo important network constraints that we consider in this report are transmission capacity constraints and voltage constraints. In operation the allowed power flow along a line may be limited, increasing the overall cost to supply the energy demand. Likewise, the most efficient production may produce power flows that result in unacceptable voltage levels. Honoring voltage constraints can also increase overall costs. A traditional AC optimal power flow (ACOPF) will simultaneously consider transmission line and voltage constraints and produce the least cost total dispatch required to meet demand. ACOPF programs typically run well under most conditions, but under unusual circumstances can have difficulty converging to a solution, or take a long time to do so. he ACOPF model s nonlinearity also makes it susceptible to non-global solutions. hese pitfalls are unacceptable for producing market clearing prices. Market managers must have consistent and justifiable solutions. he markets, therefore, commonly employ a linear approximation of the ACOPF, termed DC optimal power flow (DCOPF). Linear techniques abound, solving models quickly and with confidence in optimality. he trade-off is a simplification of the model: voltage characteristics are omitted. Because the DC approximation of the ACOPF has no voltage magnitude variables, real power flow limits are imposed as proxies to enforce voltage limits. In the absence of a scientific method for choosing proxy limits, the error in nodal prices can be significant even if the power dispatch closely matches the ACOPF solution. Previous research has shown examples of different proxy limits producing identical dispatch solutions yet with greatly varying prices[1,4]. In [2] the problem of finding proxy limits was cast as Mixed Integer Program (MIP). he method was successful in finding optimal limits, but suffered from scaling issues. In general it cannot be applied to realistically sized systems. Furthermore analysis of results did not offer obvious ways to pare the set of lines to those most likely to be among the optimal solution. Specifically, many of the proxylimited lines were distant from the voltage constraint, and the optimal set changed with system loading. In this paper we revisit the task of finding proxy limits with a sensitivity-based approach and apply it to a 30-bus system. As in the previous work, the problem is cast as an optimization problem with an objective function that weights the difference between DC model and AC model in both dispatch and LMPs. We assess the variability of proxy limits as weight is shifted from minimal dispatch error to minimal LMP error. Results are presented for a 30- bus system. 2. DCOPF Sensitivities and Proxy Limits In this section we develop the models needed to choose optimal proxy limits. We pose the problem as mixed integer problem: /10 $ IEEE 1

2 min P PLi, S PL ( ) is PL AC DC (P PLi ) + P AC P DC (P PLi ) (1) where P PLi are the values for the proxy limits and S PL is the set of lines that have proxy limits. he values and weight LMP differences against dispatch errors. We take the ACOPF LMPs and dispatch as known, the values AC and P AC respectively. he values for DC and P DC correspond to LMPs and dispatches obtained using a proxy-limited constrained DCOPF. Our task in solving (1) is to identify lines to have binding proxy-limits, and to specify the values for these limits. We use a sensitivity analysis of a DCOPF to determine these values. o show how to calculate the relevant sensitivities we first develop the DC optimal power flow model. hen we derive its useful sensitivities. In the DCOPF, voltage variations, reactive power and losses are neglected (or approximated in various ways). Under these conditions the power flowing in the branch elements is described by P branch = diag(b)a (2) where diag(b) is a diagonal matrix of branch susceptances, is a vector of bus voltage angles, and A is a node-branch incidence matrix [3]. Each column of A serves as an indicator vector describing the connection of a line in the system. he column contains a single +1 entry and a single -1 entry in rows corresponding to the terminal buses of the element. All other entries are zero. he power injected into the network at each bus is given by P inj = Adiag(b)A (3) he DCOPF problem may then be described as min P g C i (P gi ) i subject to Adiag(b)A P inj = 0 P min diag(b)a P max (4) We then replace the inequality constraints with chosen equality constraints and corresponding limits: diag(b PL )A PL P PL = 0 (5) where the subscript PL denotes a set of proxylimited lines. In this work we consider generators having quadratic cost functions (and ignore generator limits): 2 C(P gi ) = C 0i + C 1i P gi + C 2i P gi (6) he Lagrangian formulation for this problem is given by L = C i (P gi ) + ( Adiag(b)A P inj ) i (7) + μ PL ( diag(b PL )A PL P PL ) he first-order conditions are 2diag(C 2 ) 0 E g 0 P g 0 0 Adiag(b)A A PL diag(b PL ) E g Adiag(b)A diag(b PL )A PL 0 0 μ PL C 1 0 = P D P PL (8) where P inj = P g +P D separates the injected powers into generator and load (demand) portions. E g is a matrix that indicates generator buses. Each row has a single nonzero +1 entry in the column corresponding to the generator bus. Our task is to find lines and line limits that move the solution to minimize our objective described in (1), a weighted norm of dispatch and LMPs. o accomplish this we find it convenient to calculate sensitivities of prices, dispatch, and line flows to values of the lagrange multipliers, μ, associated with line constraints. his sensitivity is the solution to 2diag(C 2 ) 0 E g 0 P g 0 0 Adiag(b)A 0 E g Adiag(b)A 0 0 (9) 0 diag(b)a 0 I P branch 0 0 = μ Adiag(b) 0 he solution of this equation is a matrix of sensitivities, each column corresponding to a different line. In (9) (and in contrast to (8)), we include all lines as potentially constrained. hose line that are not considered for proxy limits are eliminated by setting the corresponding value of μ=0. Let the sensitivity solution be described by 2

3 P g P branch = M μ (10) Because we have specified quadratic cost curves, (10) is an exact incremental model of the DCOPF 1. For specified values of μ, the DCOPF solution is given by P gdc DC DC P branch = P g0 0 0 P branch 0 + M μ (11) where subscript 0 is used to denote the solution to the unconstrained DCOPF. In terms of μ it is straightforward to set up and solve a weighted least squares problem: min μ i ( AC DC + P AC P DC ) is PL = min AC 0 + P AC P g 0 P g μ i is PL ( ) (12) where a 2-norm is used (to allow least-squares analysis), and all variable quantities in (12) are functions of μ as described in (11). his approach is conceptually and quantitatively easy once the set S i is specified. In this paper we cycle through all possible combinations of lines up to a specified number of lines. Specifically, we perform the analysis to find the best single line proxy limit, then we find the best two-line proxy limits, etc. In the analysis that follows in the next section for the 30-bus, 41 line system, the complete enumeration of choosing 5 lines among 41 line, find the values for the proxy limits and calculating the errors, for 11 different weightings takes less than six minutes on an ordinary laptop computer. his involves solving (12) more than 10,000,000 times. his brute-force method is practical for examining a handful of lines on a small system, but an appropriate branch-andbound technique would be better suited for larger systems. 1 hat is, it is exact to the extent that the DCOPF admits unique solutions. his excludes load-only subgraphs. 3. Analysis of 30-bus System In this section we apply the method described in the previous section to the task of assigning proxylimits for voltage constraints in a power system. For our example here we use the 30-bus system model with parameters and costs functions obtained in the MatPower distribution [5], an open source, Matlab based power system solver developed at Cornell University. Aside from the ease of developing in Matlab, MatPower is useful because it enforces voltage constraints. We refer the reader to the Matpower distribution for the model parameters. In the Appendix we list the lines numbers and the buses they connect. We modify the 30-bus model in two ways. We decrease the maximum voltage limit at bus 1 to 1.0 per unit, and we increase the minimum voltage limit at bus 19 to 1.0 per unit. his creates two binding voltage constraints. We note that any lossy network will have a binding maximum voltage limit since uniformly raising voltage will serve to reduce losses. Since the system is lossy, we accommodate the losses in the DCOPF by distributing the losses in each line equally to loads at the terminals of the line. his is a common technique. In the following tables we present results for one-line, two-line, three-line, four-line, and five-line proxy limits. For each of these we calculate the proxy limits for 11 different weightings, varying between 0 and 1 in increments of 0.1 (correspondingly varies between 1 and 0 in 0.1 increments). We pause to note that at extreme points, = 0, or = 0, the optimal answers may not be unique. herefore, we perturb the weight by 10-4 to seek a solution in line with the non-zero weighted answers. For example, there are many five-line solutions that match the ACOPF dispatch exactly (which we discuss more later). By perturbing the weights, we find a result in the direction of minimizing LMPs errors. In able 1 we show the results for the best weight solution for proxy limits for various values of and. he top row corresponds to a strongly LMP weighted solutions and the last row to a dispatchbiased solution. he error (2-norm) in LMPs and dispatch are provided, and the best proxy-limited line is given. In ables 2, 3, 4, and 5, we provide similar results for the 2, 3, 4, and 5 line proxy limit solutions. Note that in these tables that while identical lines 3

4 may be listed for different weights, the errors may differ. (Compare rows 2-8 in able 2.) his is due to the different values for the line limits. We would like to highlight several features of these results. he lines that best match the dispatch are not generally the best lines to match prices. Compare the first and last rows in ables 1-5. here is only one line in common (able 5). A pattern of best lines is not apparent. It seems that the some lines appear near generator buses and some appear near the constraint. Choosing lines to match LMPs does improve dispatch as more lines are added, yet the error remains large. See the first rows of ables 1-5. Choosing lines to match dispatch has no impact on bettering LMPs. he last rows in ables 1-4 show no decrease in LMP error. Choosing five lines, there are enough degrees of freedom to exactly match the generator dispatch. here is not a unique solution. We find a solution that also decreases LMP error. able 1. Results for one-line proxy limits LMP Dispatch Proxy Lines able 2. Results for two-line proxy limits LMP Dispatch Proxy Lines , , , , , , , , , , , 15 able 3. Results for hree-line Proxy Limit LMP Dispatch Proxy Lines , 22, , 22, , 6, , 6, , 6, , 14, , 14, , 14, , 14, , 14, , 14, 35 able 4. Results for four-line proxy limits LMP Dispatch Proxy Lines , 25, , 22, 35, , 3, 22, , 6, 32, , 6, 32, , 6, 32, , 6, 32, , 6, 32, , 6, 32, , 6, 32, , 16, 35, 41 able 5. Results for five-line proxy limits LMP Dispatch Proxy Lines , 25, 35, , 3, 22, 36, , 3, 22, 36, , 3, 22, 36, , 6, 32, 36, , 6, 32, 36, , 6, 32, 36, , 6, 32, 36, , 6, 32, 36, , 3, 22, 30, , 3, 22, 30, Discussion and Conclusions In this paper we have presented a mathematical formulation to determine optimal proxy limits for a DC power flow to best match dispatches and prices 4

5 obtained in an AC power flow. he results are not without issue. It is troublesome that the best lines that are found differ depending in emphasis (dispatch vs. price), and number of lines chosen. It is not surprising that the dispatch emphasized solutions do not tend to match LMPs. he latter are based on incremental flows. here are many ways to choose limits that will result in the same point of dispatch, but will have strikingly different incremental behavior. he opposite approach, it would seem, would fare better. If we choose constraints that match the LMPs, the dispatches should follow. Specifically, if there is a one-to-one mapping between LMP and dispatch at each generator, then matching either will suffice to get the other at the generator buses. With the LMP emphasis we attempt to match all bus LMPs, and as we add lines to better match these, we should expect the dispatch error to decrease. And it does, but it is still relatively large. We conjecture that this is due to the number of degrees of freedom involved. here are far fewer generator dispatches to match than LMPs. Each imposed limit can potentially have a greater impact on dispatches than LMPs. In this system, five constraints can match the dispatch exactly, but not the price. From the limited results here, it seems that a prudent strategy would be to choose a weighted result, neither optimal for price, or dispatch, but not wildly off either. A practical concern when considering more than a few lines for proxies, is the computational time required to find the optimal solutions. Ideally, one might expect the lines that are chosen would be near the constraint, but this does not appear to be the case. We conjecture the following explanation for this: constraints (line or voltage) effectively impose a limit on the pattern of generation dispatch that is needed. So, it is not inconsistent to note that the best proxy lines are often nearer to generator buses than to the voltage constrained bus. his poses a problem for scaling the analysis to large systems. If we cannot sensibly limit the set of lines from which we seek proxies, the combinatorial nature of the problem will make it difficult or impossible to solve. Future research should investigate how to find good solutions with a predefined set of possible lines to choose from, and how to assign this set. Another issue, unstudied here, is the extent to which proxy limits can be combined. If we determine best proxy limits for different voltage constraints separately, do they necessarily combine well for simultaneous use? It is a desirable property that they do, otherwise proxy limits will be required for all possible combination of limits that might arise. We will examine this issue in future research. We end the paper with a discussion about how these this limits are, and may be used in practice. Power systems are operated over different timescale. On a relatively long operational timescale, transmission path limits are specified seasonally. he calculation of such limits may include voltageproxies. For example, the document [6] qualitatively describes an approach of including voltage constraints in path limits. Since these limits may be used in operation, the resulting prices are correct in the sense that they are consistent with operational practices. Less clear is how well these proxy-limited path constraints describe actual operating conditions. Conditions are likely different in real time than those assumed in off-line seasonal studies, and the off-line studies may not anticipate the range of market offers that will occur in day-ahead and real-time markets. On a faster time scale, to the extent that voltage is closely monitored in operation, it is not clear how the operators/market will adjust resources to maintain voltage levels. Prices should account for specific actions taken in operation, and these actions should minimize cost given a voltage constraint. Real-time tools for voltage-based proxy limits, or voltage-based distribution factors are needed. 5. References [1] Alvarado, F.L., Converting System Limits to Market Signals, IEEE ransactions on Power Systems, Vol. 18, No. 2, May [2] Beglin, E.E., Representing Voltage Constraints in a Proxy-Limited DC Optimal Power Flow, Master s hesis, University of Wisconsin-Madison, Dec [3] Chua, L.O., C.A. Desour, and E.S. Kuh, Linear and Nonlinear Circuits. [4] Lesieutre, B.C,., and F.L. Alvarado, Pondering PAR Pricing, Bulk Power System Dynamics and Control, August 19-24, 2007, Charleston, SC, USA. [5] MatPower, Users Manual and distribution available at [6] NYISO, ransmission Expansion and Interconnection Manual, September planning/tei_mnl.pdf 5

6 Appendix In this appendix we supply a table of line numbers and associated bus connections for the 30 bus system. We also supply a diagram of the system. line buses line buses line buses