DC Load Flow Visualization

DC Load Flow Visualization

DC Load Flow Visualization Ivan Skokljev, Branko Kovacevic Faculty of Electrical Engineering Belgrade Bulevar Revolucije 73, PO Box 35-54, 11000 Belgrade, Yugoslavia [email protected]

Abstract: The visualization properties offered by contemporary computing tools such as Excel, Matlab or Mathematica, could aid a power system operator in an environment where the decision-making process is still not completely close-loop automated. This is a fact with our local power transmission operating centers. During the normal operating conditions within the system there is still a plenty of time for an operator to study the possibilities of different scenarios and to do his own simulations with relatively simple user-oriented programs, keeping him alert and awake by exercising his creative abilities on the real system conditions. This kind of games could prove far more valuable than checking the sometimes outdated manuals with operating scenarios far from reality. This paper presents the three different computer programs all of them based on the well known idea of the direct, one step method of the DC load flow. By just adopting the built in properties of the computer environment in which this idea is embedded, one could benefit from different possibilities the environment bring up, especially in visualization. The original titles of masks are deliberately maintained with translations given in sequel to pinpoint the general idea that this task is far from being cumbersome to the user with an average interest in programming. Keywords: Visualization, DC Load Flow, Steady State Security 1 Introduction Load flow computer calculations are the most frequently performed of all power system network operation and planning methods. In steady state security e.g., the speed of execution of a single load flow has to be comfortable with the somewhat vague rate-of-change of system state. Steady state security methods are essentially bounded to generating possible solutions and selecting among them by cutting off according to some predetermined criterion. Therefore, the research into the steady state security was directed towards obtaining the fastest load flow. The two load flow algorithms most in use are the fast decoupled load flow [1] and the DC load flow [2]. Let us examine briefly the algorithm of the DC load flow, for which a more complete outline is given in [3]. The method is a trade-off between accuracy and the speed of execution. 2 DC Load Flow Consider the network consisting of c=n+1 nodes, i.e. n nodes and the ground. Lines and transformers are given by their P-equivalents and their node-to-datum branches form shunt (index sh) branches. Their serial parts form serial (index se) branches. Generators are independent current generators and represent generation and/or load in the system, forming generator (index gen) branches supporting all nodes of the system. According to the NA method [4] let us introduce the network bus admittance matrix, Ybus = AYA T , and the vector of net bus current injections, Jg = AIg . The general matrix equation is

YbusV = Jg . After omitting the shunt elements from the incidence matrix A, (1) gives EPSOM’98, Zurich, September 23-25, 1998 Page SKOKLJEV-42-01

(1)

DC Load Flow Visualization

T , J =I Ybus = AseYse Ase g g,gen .

(2)

Yse = − jBse .

(3)

After neglecting the line losses

Bse is a matrix of real elements, minus sign chosen for convenience. Then, (1) becomes T V = Ig,gen − jAseBseAse

(4)

Here, V is a nx1 vector of bus nodal voltages. To make a compact matrix formulation of the DC load flow method the diagonal matrix of the bus voltages is introduced

Vd = diag(V 1,V 2,...,V n ) .

(5)

The complex power vector of the current sources is T * * Sg,gen = Pg,gen + jQg,gen = VdIg,gen = jVd AseBse Ase V .

(6)

Assume that the per unit system is used , then

cos(θi − θm ) ≈ 1, sin(θi − θm ) ≈ θi − θm

Vi = 1, i = 12 , ,..., n

i = 12 , ,..., n , m = 12 , ,..., n , i ≠ m .

(7)

Expanding (6) the i-th row of the Sg,gen vector n

n

m=1 m≠ i

m=1 m≠ i

S i = jVi2 ∑ Bim − V i ∑ V *mBim .

(8)

* ≈ 1 + j(θ − θ ) V iV m i m

(9)

Taking into account that

(8) simplifies to a real-valued expression n

n

m=1 m≠ i

m=1 m≠ i

S i = θi ∑ Bim − ∑ θmBim = Pi + j0 .

(10)

By inspection into (10) the general matrix form can be identified

BseΘ = P

(11)

where Θ = [θ1 θ2 ... θn ] , P = [P1 P2 ... Pn ] . For the given network, i.e. Bse , and power injections, P , the bus voltage angles, Θ , are determined. Since the DC load flow method assumes the lossless case, the admittances are purely imaginary and the active power conservation (Tellegen’s theorem) is satisfied by the sum T

T

n

∑ Pi = 0 .

i =1

(12)

The condition (12) implies that the equations of the system (11) are linearly dependent, i.e. one variable, θi 0 , can be expressed in terms of the others, θi , i = 12 , ,..., n ; i ≠ i0 . EPSOM’98, Zurich, September 23-25, 1998 Page SKOKLJEV-42-02

DC Load Flow Visualization

The angle at one bus can be specified arbitrarily. Usually, θ1 = 0 which is known as specifying the slack bus. Therefore, the order of the system of equations (11) is reduced by 1. The line power is

Pim = Bim (θi − θm )

(13)

i = 12 , ,..., n ; m = 12 , ,..., n ; i ≠ m . An isomorphism exists between (1) and (11). There is an one-to-one correspondence among the variables: a) the voltages in NA stand for the angles in DC load flow, and b) the currents in NA are substituted by the real powers in DC load flow. 3 DCXLS: Steady State Security Planning - Numerical Example DCXLS is written in Visual Basic, Excel 5.0 for Windows. Hardware assumes the Intel PC platforms. DCXLS is made for interactive use and communicates with the user by the standard Workbook menu dialogues. When started, the program displays the introductory mask and the standard Excel menu is substituted by the users menu. The user is offered the five working spreadsheets and two spreadsheets for graphing. The program also makes use of five auxiliary tables (spreadsheets). DCXLS creates its own system of menus defined for seven different purposes. This system enables the input of data and calculations from them. For example, the data could be transferred and filtered out from the regular AC load flow report. The classical AC load flow is used for comparison with the DC load flow results [5]. On request, the data could be assembled from the so-called primitive admittance matrix (element by element) or they could be compiled as a bus impedance matrix, as in (1). Fig.1 shows the introductory mask enabling the selection of program inputs. The program was also meant for educational purposes and therefore some accuracy double-checks. Fig. 1 DCXLS introductory mask (Fajl= File; Podaci= Data; Tabele= Tables; Stampa= Print; Importovanje ulaznih podataka iz AC izve{taja= AC Report Input Data; Prora~un iz podataka o granama= Network Data Compiled by Branches; Prora~un podataka iz tabela...= Network Data Compiled from AC Load Flow Data; Prora~un iz tabela...= Explicit Inverse of the Ybus; Ra~un standardne devijacije...= Line flow (%) error diagram and standard deviation (s))

On request, DCXLS compares results of the line flows calculated from the DC with respective ones from the AC load flow (Fig. 2). This feature should assure the user of “accuracy”. The program evaluates the line flow relative errors, calculates the standard deviation (σ) and scales the results on the relative error (%) v.s. line MW flow, plotting the 3σ envelope, a standard measure for screening the results for such an evaluation. For the IEEE RTS network [6], σ=12.65 MW, 3σ=37.95 MW. All AC line power flows range between 4.48 MW and 327 EPSOM’98, Zurich, September 23-25, 1998 Page SKOKLJEV-42-03

DC Load Flow Visualization

MW. Even 100% DC calculated line flows are found within the 3σ area. Larger relative errors are noticeable for smaller line MW flows, as expected. Next, consider the data [6] as related to 1998 and let us assume the 7% per year rise of load and generation in the system. One could be interested in the year when the line flow long term steady-state security constraints are to be violated. It is relatively easy with the DCXLS to find the year when the first violation occurs, or the most severe violation if the topology changes, the bottlenecks of the system, and so on. DC line flows (actual/limits) v.s. line numbers in 2008 with limit violations labeled, are shown on Fig. 3. DCXLS also produces (n-1)-security analysis reports and statistics via piecharts and line diagrams, on request. MW

(%)

1000 900 DC (MW)

600

800 MAX (MW)

543

700 600

400

500 400

200

182

200

195

300

100

0 50

100

150

200

250

(MW)

Fig. 2 IEEE RTS AC/DC line flow error

300

0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

0

LINE NO.

Fig. 3 Long term security breaches

4 DCMAT: Corrective Switching - Numerical Example DCMAT embeds the DC load flow into Matlab environment. Matlab is known for efficient matrix manipulations and powerfull visualizations. Here, two of the steady-state security related functions, contingency analysis and corrective switching, are excersised on the IEEE RTS network. The program is fast, easy to handle and aimed at users interested in security analysis in research, design and teaching of power systems. The first step with DCMAT is contingency analysis on the list of lines/transformers and generators. In the second step, active cases are treated with corrective switching. Regarding the DC model, active cases are outages resulting in excessive flows on the remaining network. Corrective switching is the time and money saving method of load flow control in power networks [7,8]. With DCMAT a set of heuristic rules is applied to guide the actions. The timer is set on ten minutes to fulfill the saving action, before the assumed relay trips out the endangered section. ESOs (Elementary Switching Operations, [8]) comprise the switching out of a (1) overloaded line, or (2) a least loaded neighboring line, or (3) any neighboring line [9]. Switching-in is also an option. Contingency analysis performed on the IEEE RTS network revealed the active case on line 23 after the outage of line 7 (Fig. 4). After attempting ESO (1) which failed, successful ESO was found in switching out of line no.19, which produced the (n-2)-state of no limit violations according to the DC model (Fig. 5). Visualization of flows is a very powerfull tool and prior to Excel, Matlab and similar applications the analyst had to scan arrays of numbers “manually”. Otherwise, plotting was also time consuming and therefore obsolete. For a system operator it becomes easy now to grasp the notion of “above the threshold”, for the flows which stand taller like the catcher in the rye. EPSOM’98, Zurich, September 23-25, 1998 Page SKOKLJEV-42-04

DC Load Flow Visualization

(Legend: Ispad= Outage; Grana= Branch (Line/Transformer); Snaga (r.j.)= Power (p.u.); Bazni= Base case; Korekcija= Correction; Zatvori= Close).

Fig. 4: Outage on 7: active case on 23

Fig. 5: Solution: switching out on 19

5 SADCLF: Generation Rescheduling and Network Planning - Numerical Example SADCLF is a new computer program comprising symbolic analysis of the DC load flow [3]. The next section introduces a new approach to the DC load flow method: automated, computer-aided symbolic analysis of power networks described by parameters given by symbols [10]. The active line power flows are found as symbolic expressions (analytical solutions) for the most general case of given topology and structure. Symbolics enables new and completely different analysis like sensitivity and quality analysis, besides the traditional numerical analysis. Here, the program is demonstrated on the unconstrained generation rescheduling problem and the small scale network planning problem. A power system to be analyzed is specified as a list of components: PS = {PSC[1], PSC[2], ... , PSC[NPSC]} which are, also, lists of the form PSC[k] = {identifier, connection, parameters}. The identifier consists of two strings required to uniquely identify the type and the individual name of a component. identifier = "type", "name" The connection is a list of bus labels the component is attached to. The parameters is a list of symbols or a single symbolic value representing component parameters. One of the features is that the slack bus is a component easily attached to an arbitrary node thus leaving obsolete the renumeration when such a “surgery” is performed with traditional load flows. One of the mishaps is the length of expressions. On the other hand, they do not always have to be visible, it is more important that they do exist at all. The generated symbolic expressions could also be dragged into some conventional programs. Let us consider a network transfer function in which a line power P23 is monitored v.s. the two critical injections, as on Fig. 6. All symbols are set to predetermined values except the line power P23, the generation Pin2 and the load Pout7 which are left as symbols and then analyzed for a span of their values. The visualization is rather suggestive for an operator of the system. Such charts could be easily derived at the spot for all critical routes, from the predetermined transfer functions. Or, consider the planning routine where a generation/consumer node (no.5, Fig.7) is “clicked and dragged” with lines attached to it and to the rest of the network (nodes no.s 6 EPPSOM’98, Zurich, September 23-25, 1998 Page SKOKLJEV-42-05

DC Load Flow Visualization

and 8, Fig. 7). How much power is flowing on the monitored line with nothing else changed? The answer is found on a chart similar to the one on Fig. 7.

Fig. 7: Monitoring line flow for two changing line lengths Fig. 6: Monitoring line flow for two changing injections 6 Conclusions Visualization of processes is important in a man-machine interactive environment such as power system planning and operation. Visualization properties are at hand with the contemporary computing tools such as Excel, Matlab or Mathematica. They could aid a power system operator in the decision-making processes where the task of close-loop automation is not fulfilled to an extent. Furthermore, the teaching of power systems becomes easier with such tools. This paper presents three original programs featuring the familiar DC load flow method. The built-in properties of the tools could inspire the use of the DC method. 7 References [1] Stott, B.; Alsac, O.: Fast decoupled load flow. IEEE Trans. PAS 93 (1974) 859-869. [2] Wood, A. J.; Wollenberg, B. F.: Power generation, operation and control. New York: John Wiley&Sons 1984. [3] [kokljev, I.; To{i}, D.: A New Symbolic Analysis Approach to the DC Load Flow Method. Electric Power Systems Research 40 (1997) 127-135. [4] Pai, M. A.: Computer techniques in power system analysis. New Delhi: Tata McGraw-Hill 1980. [5] Tinney, W. F.; Hart, C. E.: Power flow solution by Newton’s method. IEEE Trans. PAS 86 (1967) 1449-1460. [6] Reliability System Task Force: IEEE Reliability Test System. IEEE Trans. PAS 98 (1979) 20472054. [7] Glavitsch, H.: Switching as means of control in the power system - State of the Art Review. Electrical Power and Energy Systems 7 (1985) 92-100. [8] Bacher, R.; Glavitsch, H.: Network topology optimization with security constraints. IEEE Trans. PWRS 1 (1986) 103-111. [9] Müller, H.: Korektives Schalten - eine Massnahme zur gezielten Entlastung von Betriebsmitteln in electrischen Energieversorgungsnetzen. Dissertation 1981, Technische Hochschule Darmstat. [10] Wolfram, S.: Mathematica: A system for doing mathematics by computer. Redwood City, CA: Addison - Wesley 1991.

EPPSOM’98, Zurich, September 23-25, 1998 Page SKOKLJEV-42-06