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An augmented Newton–Raphson power flow formulation based on current injections
JEPE411
Electrical Power and Energy Systems 23 (2001) 305±312
www.elsevier.com/locate/ijepes
An augmented Newton±Raphson power ¯ow formulation based on current injections V.M. da Costa a, J.L.R. Pereira a,*, N. Martins b a
U.F.J.F., Faculdade de Engenharia, Juiz de Fora, MG, Brazil CEPEL, P.O. Box 68007, CEP 21944-970, Rio de Janeiro, RJ, Brazil
b
Received 21 December 1998; accepted 13 March 2000
Abstract This paper presents a new procedure for the solution of power ¯ow problem, by using the current injection equations written in rectangular coordinates. From this formulation, it is possible to obtain the same convergence characteristics of the conventional power ¯ow expressed in terms of power mismatches and written in polar coordinates. By using this methodology, a highly sparse augmented formulation suited to the incorporation of FACTS devices and control of any kind is also obtained. The results presented validate the proposed method. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Current injection formulation; Power ¯ow; Power ¯ow controls
1. Introduction The power ¯ow problem involves determining voltages and ¯ows throughout the network, represented as fundamental frequency phasors. The problem formulation assumes that the network con®guration, the loads, the voltages and active power output of most generators are known quantities. Ref. [1] presents a power ¯ow formulation based on current injections, in which a PV bus is represented by a single equation expressed in terms of active power mismatches and angle deviations, and for a PQ bus, the current injection equations are written in rectangular coordinates. Control adjustments on the power ¯ow solution is a very important tool to match the required operating conditions on an energy management system. The LTC transformer, the phase shifter transformer and generator excitation are the most common devices to locally control the voltage at a PQ bus [1], the active power ¯ow [2], and the voltage at a PV bus [3±5], respectively. Control adjustments of a single device are usually used to control a single variable. Multiple devices can, however, be used to simultaneously control a single variable [6,7], such as multiple generators or LTC transformers controlling the voltage of a single PQ bus. * Corresponding author. E-mail addresses: [email protected] (V.M. da Costa), [email protected] (J.L.R. Pereira), [email protected] (N. Martins).
The solution of adjustment interactions in the fast decoupled load ¯ow is presented in Ref. [7]. Alternating adjustments between iterations are used to analyze the nature of the control interactions and to propose solutions to overcome these problems. In this paper, the current injection equations written in rectangular coordinates are used for both PQ and PV buses. For each PV bus is introduced a new dependent variable (DQ) and an equation that imposes the bus voltage constraint. Thus, the Jacobian matrix has the elements of the 2 £ 2 off-diagonal blocks equal to those of the bus admittance matrix, independent of the bus type to be considered. The elements of the 2 £ 2 diagonal blocks are modi®ed at each iteration according to the load model being considered. This paper describes an augmented formulation suitable for the incorporation of FACTS devices and power ¯ow controls of any kind. The set of controls and devices studied in this paper includes the representation of the voltage dependent load, load tap changing transformer and phase shifter transformer.
2. Current injection power ¯ow modeling 2.1. Basic equations
0142-0615/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(00)00045-4
The complex current mismatch at a given bus k is given
306
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
DImk
Nomenclature h iteration counter n number of buses complex conjugated voltage phasor at bus k Ekp DPk 1 jDQk complex power mismatch at bus k PG k 1 jQG k generated complex power at bus k PL k 1 jQL k complex load at bus k Pk 1 jQk net complex injected power at bus k sp Psp k 1 jQk scheduled complex power at bus k calc calculated complex power at bus k Pk 1 jQcalc k DIrk 1 jDImk complex current mismatch at bus k Irspk 1 jImspk scheduled injected current at bus k 1 jImcalc calculated injected current at bus k Ircalc k k Vrk 1 jVmk complex voltage at bus k uk ; Vk voltage angle and magnitude at bus k scheduled voltage magnitude at bus k Vksp gkj 1 jbkj series admittance of line k 2 j Gkj 1 jBkj (k,j)th element of bus admittance matrix Du , DV voltage angle, magnitude corrections transformer tap from bus k to bus j akj Dakj transformer tap correction w kj phase shifter angle from bus k to bus j phase shifter angle correction Dw kj Matrices are shown in bold and vectors are shown in bold underlined. by [1]: n X Psp 2 jQsp Yki Ei 0 DIk k p k 2 Ek i1
9
and written in compact form: DIrk Irspk 2 Ircalc k
10
DImk Imspk 2 Imcalc k
11
The Newton±Raphson solution algorithm, applied to Eqs. (8) and (9), is given by:
12
which can be written in a compact form: 1
where
DImr Yp DVrm
13
where
Psp k PG k 2 PL k
2
Qsp k QG k 2 QL k
3
The voltage dependence of the load powers is modeled in polynomial form: PL k P0k ap 1 bp Vk 1 cp Vk2
4
cq Vk2
5
QL k Q0k aq 1 bq Vk 1 where a p 1 bp 1 c p 1
6
aq 1 bq 1 c q 1
7
Eq. (1) can be expanded into its real and imaginary components: DIrk
sp n X Psp k Vmk 2 Qk Vrk 2 Gki Vmi 1 Bki Vri 0 Vr2k 1 Vm2 k i1
sp n X Psp k Vr k 1 Q k Vm k 2 Gki Vri 2 Bki Vmi 0 Vr2k 1 Vm2 k i1
8
" p Ykk
B 0kk G 0kk
#
G 00kk B 00kk
" p Ykm
Bkm Gkm
#
Gkm 2 Bkm
B 0kk Bkk 2 ak
14
G 0kk Gkk 2 bk
15
G 00kk Gkk 2 ck
16
B 00kk 2Bkk 2 dk
17
The parameters ak, bk, ck and dk are presented in Appendix A. 2.2. Calculation of current mismatches The active and reactive power mismatches for bus k is
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
whose linearized form is:
given by: calc DPk Psp k 2 Pk
18
calc DQk Qsp k 2 Qk
19
where Vrk Ircalc 1 Vmk Imcalc Pcalc k k k
20
Vmk Ircalc 2 Vrk Imcalc Qcalc k k k
21
By simple manipulation of the above equations, the current mismatches in Eqs. (10) and (11) can be expressed only in terms of power mismatches and voltages at bus k: Vrk DPk 1 Vmk DQk Vr2k 1 Vm2 k
22
Vmk DPk 2 Vrk DQk Vr2k 1 Vm2 k
23
DIrk
DImk
DImk
Vmk DPk Vr2k 1 Vm2 k
24
DIrk
Vrk DPk Vr2k 1 Vm2 k
25
Assuming that the reactive power mismatch is a dependent variable, then an additional equation is introduced in order to set the over-determination of the system of equations as follows: Vk2 Vr2k 1 Vm2 k
26
By linearizing Eq. (26) yields: Vr k Vmk DVrk 1 DVmk Vk Vk
Eqs. (12), (27) and (29) are written in the following matrix form: #" " # " p # DVrm 0 Y B 30 DPQ DuV C 0 The matrices B and C have a block-diagonal structure: 2 6 6 6 B6 6 6 4
3
B1
7 7 7 7 7 7 5
B2 ]
2 6 6 6 C6 6 6 4
3
C1
7 7 7 7 7 7 5
C2 ] Cn
where 2
3 Vr i Vi2 7 7 7 2Vmi 7 5 2 Vi
2
3 Vri Vi2 7 7 7 7 Vm i 5 Vi
2Vmi 6 V2 6 i Bi 6 6 2V 4 ri Vi2 2Vmi 6 V2 6 i Ci 6 6 4 Vr i Vi DuV Du1
DV1
DPQ DP1
DQ1
D u2
DV2
¼ D un
DVn t
DVrm DVr1
DVm1
DP2
DQ2
¼ DVr n
¼
DPn
DQn t
DVmn t
This formulation is equivalent to and has the same convergence characteristics of the conventional Newton±Raphson power ¯ow in polar coordinates [3]. The elimination of DVrm in Eq. (30) yields:
3. Augmented formulation The voltage angle at a bus k can be expressed by: Vm k Vrk
29
27
For a PV bus DVk 0: Thus, for each PV bus, there are three equations and variables DVrk , DVmk and DQk . This formulation has been accepted for publication in Ref. [8].
uk tan21
Vrk Vm DVmk 2 2k DVrk Vk2 Vk
D uk
Bn
The calculation of real and imaginary current mismatches is straightforward for PQ buses, because of real and reactive power mismatches are known. For PV buses, the reactive power mismatch is unknown and it is treated in this formulation as a dependent variable. Then the current mismatch equations become:
DVk
307
28
DuV J21 red DPQ
31
308
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
where
where 21 B J21 red 2CY p
32
Jred being numerically identical to the conventional polar form power ¯ow Jacobian matrix. 3.1. Solution algorithm Eq. (30) shows that the linear system of equations to be solved at every iteration is hybrid. In other words, it has on the right-hand side of the equations the unknown variables DVr, DVm together with the mismatches DQ related to PV buses and DP and DQ for the PQ buses which are calculated at each iteration. The solution process must therefore be accomplished in two steps: ² Step 1: At the hth iteration, solve the following equation for DVrm : 2 32 3 " # h O DV h B h Yp rm 4 54 5 33 D h D h O I PQ PQ
" D
34
V
u
h11
h11
V u
h
h
1 DV
1 Du
h
Vm j
Vj
Vj
2DImk E 2 ::: 2akj
2DIrk 2akj
¼
#t
:::
2DImj
2DIrj
2akj
2akj
¼
#t
F0 with 2DImk cos wkj gkj Vmj 1 bkj Vrj 1 sin wkj bkj Vmj 2 gkj Vrj 2akj 2 2akj gkj Vmk 1 bkj Vrk
39
2DIrk cos wkj gkj Vrj 2 bkj Vmj 1 sin wkj bkj Vrj 1 gkj Vmj 2akj 1 2akj bkj Vmk 2 gkj Vrk
2DImj 2akj
The new solution point is given by:
Vr j
"
where I is the Identity matrix. ² Step 2: Determine the angle and voltage magnitude corrections using the following expression: h h D h uV C DVrm
¼
40
cos wkj gkj Vmk 1 bkj Vrk 1 sin wkj gkj Vrk 2 bkj Vmk 41
35 2DIrj
h
36
2akj
cos wkj gkj Vrk 2 bkj Vmk 2 sinwkj bkj Vrk 1 gkj Vmk 42
Vectors D and E have, as shown, only two and four non-zero elements, respectively. Note the LTC transformer introduces a mismatch equation on the left-hand side:
3.2. The on-load tap changing transformer (LTC) Let a LTC transformer be connected between buses k and j where the tap akj controls the voltage at bus j which is given by: Vj Vr2j 1 Vm2 j 1=2
37
Eq. (37) is linearized and introduced into the Jacobian matrix equation as follows: 2
0
3
2
Yp
6 7 6 6 DuV 7 6 C 4 5 4 DVj Dt
B 0 0
t
E
32
DVrm
3
76 7 6 7 07 54 DPQ 5 Dakj F
38
DVj Vjsp 2 Vr2j 1 Vm2 j 1=2
43
The tap value for the next iteration is given by: h11 h a h akj kj 1 Dakj
44
3.2.1. Illustration example Suppose a LTC transformer connected from bus 3 to bus 4 in Fig. 1, to control the voltage magnitude at bus 3, say V3sp . The linear system of equations related to the solution algorithm step 1, Eq. (33), is given by Eq. (45), where
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
309
DV3 V3sp 2 V3 . V3 is determined from step 2, Eq. (35): 2
0
6 B 22 6 6 6 6 00 3 6 2 6 G 22 0 6 7 6 6 0 6 0 7 6 7 6 6 7 6 6 6 V 6 DP 7 6 r2 6 27 7 6 6 7 6 V2 6 6 0 7 6 7 6 6 7 6 6 6 0 7 6 7 6 6 7 6 6 7 6 6 6 0 7 6 7 6 6 7 6 6 6 DP3 7 6 7 6 6 7 6 6 6 DQ 7 6 6 6 37 7 6 6 7 6 6 6 0 7 6 7 6 6 7 6 6 6 0 7 6 B42 7 6 6 7 6 6 7 6 6 6 DP4 7 6 7 6 6 7 6 G42 6 6 DQ4 7 6 5 6 4 6 6 6 0 DV3 6 6 6 0 6 6 4
G 022
2Vm2 V22
Vr2 V22
B24
G24
0
0
B 0022
2Vr2 V22
2Vm2 V22
G24
2B24
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0 Vm2 V2
3
B 033
G 033
2Vm3 V32
Vr3 V32
B34
G34
0
0
2
2DIm3 2a34
G 0033
B 0033
2Vr3 V32
2Vm3 V32
G34
2B34
0
0
2
2DIr3 2a34
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0 2Vm4 V42
0 Vr4 V42
0 2DIm4 2 2a34
G42
0
0
B43
G43
0
0
B 044
G 044
2B42
0
0
G43
2B43
0
0
G 0044
B 0044
2Vr4 V42
2Vm4 V42
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0 Vr3 V3
0 Vm3 V3
0
0
0
0
0
1
0
0
0
2
2DIr4 2a34
7 7 7 7 7 7 2 3 7 DVr2 7 7 6 7 7 6 7 6 DVm2 7 7 7 6 7 7 6 7 6 DP2 7 7 7 6 7 7 6 7 6 DQ 7 7 7 6 2 7 7 6 7 7 6 7 6 DVr3 7 7 7 6 7 7 6 7 6 DV 7 7 6 m3 7 7 7 6 7 7 6 7 £ 6 DP3 7 7 7 6 7 7 6 7 6 DQ3 7 7 7 6 7 7 6 7 6 DV 7 7 6 r4 7 7 7 6 7 7 6 7 6 DVm4 7 7 7 6 7 7 6 7 6 DP4 7 7 7 6 7 7 6 7 6 DQ 7 7 7 4 4 5 7 7 7 Da34 7 7 7 7 7 5
45 3.3. The phase shifter transformer Assume that a phase shifter transformer is connected between buses k and j to control the active power ¯ow Pkj through changes in the phase angle w kj. In this case the linearized equation of Pkj needs be introduced into the Jacobian matrix as follows: 32 2 3 2 p 3 0 DVrm Y B E 76 6 7 6 7 6 DuV 7 6 C 0 0 76 DPQ 7 46 54 4 5 4 5 DPkj Dwkj Dt 0t F where" D
2Pkj 2Vrk
¼ "
E2 ¼
F
2DImk 2wkj
2Pkj 2Vmk
¼
2DIrk 2wkj
2Pkj 2Vrj ¼
2Pkj 2Vmj
¼
2DImj
2DIrj
2wkj
2wkj
2DIrk sin wkj akj bkj Vmj 2 akj gkj Vrj 2wkj 1 cos wkj akj bkj Vrj 1 akj gkj Vmj 2DImj 2wkj
2wkj ¼
#t
2Pkj 2wkj
sin wkj 2akj bkj Vrk 2 akj gkj Vmk 1 cos wkj akj gkj Vrk 2 akj bkj Vmk
2DIrj
#t
1 cos wkj akj bkj Vmj 2 akj gkj Vrj
49
sin wkj 2akj gkj Vrk 1 akj bkj Vmk 1cos wkj 2akj bkj Vrk 2 akj gkj Vmk
50
2Pkj 2a2kj gkj Vrk 1 akj cos wkj bkj Vmj 2 gkj Vrj 2Vrk 2 akj sin wkj gkj Vmj 1 bkj Vrj
with 2DImk sin wkj 2akj bkj Vrj 2 akj gkj Vmj 2wkj
48
51
2Pkj 2a2kj gkj Vmk 2 akj cos wkj gkj Vmj 1 bkj Vrj 2Vmk 47
1 akj sin wkj gkj Vrj 2 bkj Vmj
52
310
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
2Pkj akj sin wkj gkj Vrk Vrj 1 Vmk Vmj 1 bkj Vmk Vrj 2wkj Slack 1
PV
2
2 Vrk Vmj 1 akj cos wkj gkj Vmk Vrj 2 Vrk Vmj 2 bkj Vrk Vrj 1 Vmk Vmj The phase shift value for the next iteration is given by:
3
w kjh11 w kjh 1 Dw kjh
4 PQ
PQ Fig. 1. Topology for tap control.
2Pkj 2akj cos wkj gkj Vrk 1 bkj Vmk 2Vrj 1akj sin wkj gkj Vmk 2 bkj Vrk
53
2Pkj akj cos wkj bkj Vrk 2 gkj Vmk 2Vmj 2akj sin wkj gkj Vrk 1 bkj Vmk 2
0
6 B 22 6 6 6 6 00 G 2 3 6 6 22 0 6 6 7 6 0 6 0 7 6 6 7 6 6 7 6 Vr 6 DP 7 6 6 2 6 2 7 6 7 6 V2 6 7 6 6 0 7 6 6 6 7 6 6 7 6 0 7 6 6 7 6 6 7 6 6 7 6 6 0 7 6 6 7 6 6 7 6 6 DP3 7 6 6 7 6 6 7 6 6 DQ 7 6 6 6 3 7 6 7 6 6 7 6 6 0 7 6 6 7 6 6 7 6 6 0 7 6 B42 6 7 6 6 7 6 6 7 6 6 DP4 7 6 6 7 6 6 7 6 G42 6 DQ4 7 6 4 5 6 6 6 6 0 DP34 6 6 6 0 6 6 4
56
54
3.3.1. Illustration example Now, suppose a phase shifter transformer connected from bus 3 to bus 4 in Fig. 1, to control its active power ¯ow in Psp 34 . The linear system of equations related to the solution algorithm step 1, Eq. (33), is given by Eq. (57), where DP34 Psp 34 2 P34 V; u; w34 : From the equations in the proposed formulation, it is seen that the overhead in computer time is due to the representation of PV buses, when compared to the conventional power ¯ow formulation [3]. In other words, a PV bus is treated as if it were a control device, requiring an additional equation. In the illustration example of this section, the control equation representing the PV bus number 2, is given by the fourth line of Eq. (57).
G 022
2Vm2 V22
Vr 2 V22
B24
G24
0
0
B 0022
2Vr2 V22
2Vm2 V22
G24
2B24
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0 Vm 2 V2
3
B 033
G 033
2Vm3 V32
Vr 3 V32
B34
G34
0
0
2
2DIm3 2w34
G 0033
B 0033
2Vr3 V32
2Vm3 V32
G34
2B34
0
0
2
2DIr3 2w34
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0 2Vm4 V42
0 Vr 4 V42
0 2DIm4 2 2w34
G42
0
0
B43
G43
0
0
B 044
G 044
2B42
0
0
G43
2B43
0
0
G 0044
B 0044
2Vr4 V42
2Vm4 V42
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0 2P34 2Vr3
0 2P34 2Vm3
0
0
1
0
0 2P34 2Vm4
0
0
0 2P34 2Vr4
0 2P34 2w34
2
2DIr4 2w34
7 7 7 7 7 7 2 3 7 DVr2 7 7 6 7 7 6 7 6 DVm2 7 7 7 6 7 7 6 7 6 DP2 7 7 7 6 7 7 6 7 6 DQ 7 7 7 6 2 7 7 6 7 7 6 7 6 DVr3 7 7 7 6 7 7 6 7 6 DV 7 7 6 m3 7 7 7 6 7 7 6 7 £ 6 DP3 7 7 7 6 7 7 6 7 6 DQ3 7 7 7 6 7 7 6 7 6 DV 7 7 6 r4 7 7 7 6 7 7 6 7 6 DVm4 7 7 7 6 7 7 6 7 6 DP4 7 7 7 6 7 7 6 7 6 DQ 7 7 7 4 4 5 7 7 7 Dw34 7 7 7 7 7 5
57
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312 Table 5 Control devices convergence characteristics
Table 1 Test system characteristics Test system
730-bus
1653-bus
Circuits PV buses Transformers Load (MW) Load (MVAr)
1146 103 277 28565 6574
2382 121 472 26703 13436
Iteration
0 1 2 3 4 5
Table 2 Convergence characteristics Iteration
0 1 2 3 4 5
Maximum mismatch (p.u.) 730-Bus
1653-Bus
48.876430 0.354070 20.006990 20.000329 ± ±
82.506656 16.926767 4.226183 0.858707 0.065671 20.000759
Table 3 Load modeling-730-bus system Iteration
0 1 2 3
311
Maximum mismatch (p.u.) Z cte. Load
I cte. load
Mixed load
48.876430 20.403327 20.007018 20.000493
48.876430 0.374413 20.007093 0.000260
248.876430 0.373643 20.007110 20.000291
4. Results The proposed power ¯ow control models were validated through tests with systems with 730 and 1653 buses related to the interconnected south±southeastern Brazilian system, whose main characteristics are described in Table 1. The active and reactive loads for the 1653 bus system are modeled as 50% constant impedance plus 50% constant current. The maximum power mismatch at each iteration is shown in Table 2, when the augmented formulation model is processed. The convergence tolerance considered is 0.001 p.u. In the 730-bus system the constant power load model is adopted.
Maximum mismatch (p.u.) Tap control
Phase shifter
2 48.876430 24.444066 4.394440 0.562881 0.011850 0.000418
2 48.876430 0.353983 2 0.006986 0.000245 ± ±
Table 3 shows the 730-bus convergence characteristics using the augmented formulation and considering the loads modeled as constant impedance (Z cte.), constant current (I cte.) and a mix of 40% of constant power plus 30% of constant current plus 30% of constant impedance (mixed). Table 4 shows the results related to LTC transformer control, considering the loads modeled as constant power. The phase shifter transformer control has the objective of controlling the active power ¯ow from bus 461 to bus 252 in 2.44 p.u., by using the 730-bus system. The initial and ®nal phase shifter angles are 30.00 and 46.968, respectively. Table 5 shows the convergence characteristics in terms of number of iterations, when employing the control devices cited previously. Convergence characteristics are achieved when the absolute value of the difference between the speci®ed and calculated parameters (voltage magnitude and active power ¯ow) is less than 0.001 p.u. Note that in this formulation the dependent variables DP and DQ allow an easy way to represent control devices in which the control variables can either be DP or DQ components. For example, on the steady-state secondary voltage control, multiple reactive power sources (dependent variables after linearization) participate to control a voltage at a given bus.
5. Conclusions This paper proposes a novel approach for solving the power ¯ow problem based on nodal current injections and a highly sparse augmented Jacobian. The main advantage of this formulation lies on the calculation of matrix Y p, because its off-diagonal elements are exactly the terms of
Table 4 Automatic tap control-730-bus system Terminal buses (k±m)
Control bus #
Voltage (p.u.)
Initial tap (p.u.)
Final tap (p.u.)
61±60 120±121 161±130 180±181 367±368
61 120 161 180 368
1.000 1.020 1.030 0.990 1.050
1.050 0.990 1.065 0.956 0.975
0.866 0.925 0.990 0.934 1.001
312
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
admittance matrix bus and the diagonal elements are calculated using non-transcendental functions, even if load models other than constant power are included. The augmented formulation is equivalent to the conventional Newton±Raphson power ¯ow regarding convergence characteristics, but allows an easier incorporation of control device models and power ¯ow controls of any kind. This formulation also directly incorporates more realistic modeling of power system components, such as, static var compensators, TCSC and voltage control through multiple reactive sources. The studies performed so far indicate the proposed formulation may become a valuable tool for solving present day power ¯ow problems, where the proper consideration of controls is becoming a key issue.
ck
Pk Vm2 k 2 Vr2k 2 2Vrk Vmk Qk Vrk Vmk Q1 k 2 P1k Vr2 k 1 Vk4 Vk3 2 P2 k
dk
Qk Vr2k 2 Vm2 k 2 2Vrk Vmk Pk Vrk Vmk P1k 2 Q1k Vr2k 1 Vk4 Vk3 2 Q2k
For the load at bus k of constant power type then ak dk
Qk Vr2k 2 Vm2 k 2 2Vrk Vmk Pk Vk4
6. Future work The authors are currently working on the representation of FACTS devices, remote voltage control by multiple reactive sources and on the implementation of user de®ned control strategies. These issues are a subject of another paper, which is under preparation by the authors.
bk 2ck
Pk Vr2k 2 Vm2 k 1 2Vrk Vmk Qk Vk4
These parameters are calculated at every iteration of the power ¯ow solution and are used for updating the diagonal blocks of matrix Y p.
Acknowledgements The ®rst two authors would like to thank the Brazilian agencies CNPQ and FAPEMIG for ®nancial support. Appendix A. Load models parameters With respect to load models to be represented in the power ¯ow problem, the parameters ak, bk, ck and dk shown in the Eqs. (14)±(17) are given by the following expressions: ak
Qk Vr2k 2 Vm2 k 2 2Vrk Vmk Pk Vrk Vmk P1 k 1 Q1k Vm2 k 1 Vk4 Vk3 1 Q2 k
bk
Pk Vr2k 2 Vm2 k 1 2Vrk Vmk Qk Vrk Vm k Q1 k 1 P1k Vr2k 2 Vk4 Vk3 2 P2 k
References [1] Dommel HW, Tinney WF, Powell WL. Further developments in Newton's method for power system applications. IEEE Winter Power Meeting, Conference Paper no. 70 CP 161-PWR New York, January 1970. [2] Peterson NM, Meyer WS. Automatic adjustment of transformer and phase shifter in the Newton power ¯ow. IEEE Transactions on Power Systems 1971;January/February:103±8. [3] Tinney WF, Hart CE. Power ¯ow solution by Newton's method. IEEE Transactions on Power Systems 1967;PAS-86:1449±56. [4] Stott B. Review of load-¯ow calculation methods. Proceedings of IEEE 1974;62:916±29. [5] Mamandur KRC, Berg GJ. Automatic adjustment of generator voltages in Newton±Raphson method of power ¯ow solutions. IEEE Transactions on Power Systems 1982;PAS-101(6):1400±9. [6] Maria GA, Yuen AH, Findlay JA. Control variable adjustment in load ¯ows. IEEE Transactions on Power Systems 1988;3(3):858±64. [7] Chang S, Brandwajn V. Solving the adjustments interactions in fast decoupled load ¯ow. IEEE Transactions on Power Systems 1991; 6(2):801±5. [8] Da Costa VM, Martins N, Pereira JLR. Developments in the Newton± Raphson power ¯ow formulation based on current injections. IEEE Transactions on Power Systems 1999;14(4):1320±6.
Electrical Power and Energy Systems 23 (2001) 305±312
www.elsevier.com/locate/ijepes
An augmented Newton±Raphson power ¯ow formulation based on current injections V.M. da Costa a, J.L.R. Pereira a,*, N. Martins b a
U.F.J.F., Faculdade de Engenharia, Juiz de Fora, MG, Brazil CEPEL, P.O. Box 68007, CEP 21944-970, Rio de Janeiro, RJ, Brazil
b
Received 21 December 1998; accepted 13 March 2000
Abstract This paper presents a new procedure for the solution of power ¯ow problem, by using the current injection equations written in rectangular coordinates. From this formulation, it is possible to obtain the same convergence characteristics of the conventional power ¯ow expressed in terms of power mismatches and written in polar coordinates. By using this methodology, a highly sparse augmented formulation suited to the incorporation of FACTS devices and control of any kind is also obtained. The results presented validate the proposed method. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Current injection formulation; Power ¯ow; Power ¯ow controls
1. Introduction The power ¯ow problem involves determining voltages and ¯ows throughout the network, represented as fundamental frequency phasors. The problem formulation assumes that the network con®guration, the loads, the voltages and active power output of most generators are known quantities. Ref. [1] presents a power ¯ow formulation based on current injections, in which a PV bus is represented by a single equation expressed in terms of active power mismatches and angle deviations, and for a PQ bus, the current injection equations are written in rectangular coordinates. Control adjustments on the power ¯ow solution is a very important tool to match the required operating conditions on an energy management system. The LTC transformer, the phase shifter transformer and generator excitation are the most common devices to locally control the voltage at a PQ bus [1], the active power ¯ow [2], and the voltage at a PV bus [3±5], respectively. Control adjustments of a single device are usually used to control a single variable. Multiple devices can, however, be used to simultaneously control a single variable [6,7], such as multiple generators or LTC transformers controlling the voltage of a single PQ bus. * Corresponding author. E-mail addresses: [email protected] (V.M. da Costa), [email protected] (J.L.R. Pereira), [email protected] (N. Martins).
The solution of adjustment interactions in the fast decoupled load ¯ow is presented in Ref. [7]. Alternating adjustments between iterations are used to analyze the nature of the control interactions and to propose solutions to overcome these problems. In this paper, the current injection equations written in rectangular coordinates are used for both PQ and PV buses. For each PV bus is introduced a new dependent variable (DQ) and an equation that imposes the bus voltage constraint. Thus, the Jacobian matrix has the elements of the 2 £ 2 off-diagonal blocks equal to those of the bus admittance matrix, independent of the bus type to be considered. The elements of the 2 £ 2 diagonal blocks are modi®ed at each iteration according to the load model being considered. This paper describes an augmented formulation suitable for the incorporation of FACTS devices and power ¯ow controls of any kind. The set of controls and devices studied in this paper includes the representation of the voltage dependent load, load tap changing transformer and phase shifter transformer.
2. Current injection power ¯ow modeling 2.1. Basic equations
0142-0615/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(00)00045-4
The complex current mismatch at a given bus k is given
306
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
DImk
Nomenclature h iteration counter n number of buses complex conjugated voltage phasor at bus k Ekp DPk 1 jDQk complex power mismatch at bus k PG k 1 jQG k generated complex power at bus k PL k 1 jQL k complex load at bus k Pk 1 jQk net complex injected power at bus k sp Psp k 1 jQk scheduled complex power at bus k calc calculated complex power at bus k Pk 1 jQcalc k DIrk 1 jDImk complex current mismatch at bus k Irspk 1 jImspk scheduled injected current at bus k 1 jImcalc calculated injected current at bus k Ircalc k k Vrk 1 jVmk complex voltage at bus k uk ; Vk voltage angle and magnitude at bus k scheduled voltage magnitude at bus k Vksp gkj 1 jbkj series admittance of line k 2 j Gkj 1 jBkj (k,j)th element of bus admittance matrix Du , DV voltage angle, magnitude corrections transformer tap from bus k to bus j akj Dakj transformer tap correction w kj phase shifter angle from bus k to bus j phase shifter angle correction Dw kj Matrices are shown in bold and vectors are shown in bold underlined. by [1]: n X Psp 2 jQsp Yki Ei 0 DIk k p k 2 Ek i1
9
and written in compact form: DIrk Irspk 2 Ircalc k
10
DImk Imspk 2 Imcalc k
11
The Newton±Raphson solution algorithm, applied to Eqs. (8) and (9), is given by:
12
which can be written in a compact form: 1
where
DImr Yp DVrm
13
where
Psp k PG k 2 PL k
2
Qsp k QG k 2 QL k
3
The voltage dependence of the load powers is modeled in polynomial form: PL k P0k ap 1 bp Vk 1 cp Vk2
4
cq Vk2
5
QL k Q0k aq 1 bq Vk 1 where a p 1 bp 1 c p 1
6
aq 1 bq 1 c q 1
7
Eq. (1) can be expanded into its real and imaginary components: DIrk
sp n X Psp k Vmk 2 Qk Vrk 2 Gki Vmi 1 Bki Vri 0 Vr2k 1 Vm2 k i1
sp n X Psp k Vr k 1 Q k Vm k 2 Gki Vri 2 Bki Vmi 0 Vr2k 1 Vm2 k i1
8
" p Ykk
B 0kk G 0kk
#
G 00kk B 00kk
" p Ykm
Bkm Gkm
#
Gkm 2 Bkm
B 0kk Bkk 2 ak
14
G 0kk Gkk 2 bk
15
G 00kk Gkk 2 ck
16
B 00kk 2Bkk 2 dk
17
The parameters ak, bk, ck and dk are presented in Appendix A. 2.2. Calculation of current mismatches The active and reactive power mismatches for bus k is
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
whose linearized form is:
given by: calc DPk Psp k 2 Pk
18
calc DQk Qsp k 2 Qk
19
where Vrk Ircalc 1 Vmk Imcalc Pcalc k k k
20
Vmk Ircalc 2 Vrk Imcalc Qcalc k k k
21
By simple manipulation of the above equations, the current mismatches in Eqs. (10) and (11) can be expressed only in terms of power mismatches and voltages at bus k: Vrk DPk 1 Vmk DQk Vr2k 1 Vm2 k
22
Vmk DPk 2 Vrk DQk Vr2k 1 Vm2 k
23
DIrk
DImk
DImk
Vmk DPk Vr2k 1 Vm2 k
24
DIrk
Vrk DPk Vr2k 1 Vm2 k
25
Assuming that the reactive power mismatch is a dependent variable, then an additional equation is introduced in order to set the over-determination of the system of equations as follows: Vk2 Vr2k 1 Vm2 k
26
By linearizing Eq. (26) yields: Vr k Vmk DVrk 1 DVmk Vk Vk
Eqs. (12), (27) and (29) are written in the following matrix form: #" " # " p # DVrm 0 Y B 30 DPQ DuV C 0 The matrices B and C have a block-diagonal structure: 2 6 6 6 B6 6 6 4
3
B1
7 7 7 7 7 7 5
B2 ]
2 6 6 6 C6 6 6 4
3
C1
7 7 7 7 7 7 5
C2 ] Cn
where 2
3 Vr i Vi2 7 7 7 2Vmi 7 5 2 Vi
2
3 Vri Vi2 7 7 7 7 Vm i 5 Vi
2Vmi 6 V2 6 i Bi 6 6 2V 4 ri Vi2 2Vmi 6 V2 6 i Ci 6 6 4 Vr i Vi DuV Du1
DV1
DPQ DP1
DQ1
D u2
DV2
¼ D un
DVn t
DVrm DVr1
DVm1
DP2
DQ2
¼ DVr n
¼
DPn
DQn t
DVmn t
This formulation is equivalent to and has the same convergence characteristics of the conventional Newton±Raphson power ¯ow in polar coordinates [3]. The elimination of DVrm in Eq. (30) yields:
3. Augmented formulation The voltage angle at a bus k can be expressed by: Vm k Vrk
29
27
For a PV bus DVk 0: Thus, for each PV bus, there are three equations and variables DVrk , DVmk and DQk . This formulation has been accepted for publication in Ref. [8].
uk tan21
Vrk Vm DVmk 2 2k DVrk Vk2 Vk
D uk
Bn
The calculation of real and imaginary current mismatches is straightforward for PQ buses, because of real and reactive power mismatches are known. For PV buses, the reactive power mismatch is unknown and it is treated in this formulation as a dependent variable. Then the current mismatch equations become:
DVk
307
28
DuV J21 red DPQ
31
308
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
where
where 21 B J21 red 2CY p
32
Jred being numerically identical to the conventional polar form power ¯ow Jacobian matrix. 3.1. Solution algorithm Eq. (30) shows that the linear system of equations to be solved at every iteration is hybrid. In other words, it has on the right-hand side of the equations the unknown variables DVr, DVm together with the mismatches DQ related to PV buses and DP and DQ for the PQ buses which are calculated at each iteration. The solution process must therefore be accomplished in two steps: ² Step 1: At the hth iteration, solve the following equation for DVrm : 2 32 3 " # h O DV h B h Yp rm 4 54 5 33 D h D h O I PQ PQ
" D
34
V
u
h11
h11
V u
h
h
1 DV
1 Du
h
Vm j
Vj
Vj
2DImk E 2 ::: 2akj
2DIrk 2akj
¼
#t
:::
2DImj
2DIrj
2akj
2akj
¼
#t
F0 with 2DImk cos wkj gkj Vmj 1 bkj Vrj 1 sin wkj bkj Vmj 2 gkj Vrj 2akj 2 2akj gkj Vmk 1 bkj Vrk
39
2DIrk cos wkj gkj Vrj 2 bkj Vmj 1 sin wkj bkj Vrj 1 gkj Vmj 2akj 1 2akj bkj Vmk 2 gkj Vrk
2DImj 2akj
The new solution point is given by:
Vr j
"
where I is the Identity matrix. ² Step 2: Determine the angle and voltage magnitude corrections using the following expression: h h D h uV C DVrm
¼
40
cos wkj gkj Vmk 1 bkj Vrk 1 sin wkj gkj Vrk 2 bkj Vmk 41
35 2DIrj
h
36
2akj
cos wkj gkj Vrk 2 bkj Vmk 2 sinwkj bkj Vrk 1 gkj Vmk 42
Vectors D and E have, as shown, only two and four non-zero elements, respectively. Note the LTC transformer introduces a mismatch equation on the left-hand side:
3.2. The on-load tap changing transformer (LTC) Let a LTC transformer be connected between buses k and j where the tap akj controls the voltage at bus j which is given by: Vj Vr2j 1 Vm2 j 1=2
37
Eq. (37) is linearized and introduced into the Jacobian matrix equation as follows: 2
0
3
2
Yp
6 7 6 6 DuV 7 6 C 4 5 4 DVj Dt
B 0 0
t
E
32
DVrm
3
76 7 6 7 07 54 DPQ 5 Dakj F
38
DVj Vjsp 2 Vr2j 1 Vm2 j 1=2
43
The tap value for the next iteration is given by: h11 h a h akj kj 1 Dakj
44
3.2.1. Illustration example Suppose a LTC transformer connected from bus 3 to bus 4 in Fig. 1, to control the voltage magnitude at bus 3, say V3sp . The linear system of equations related to the solution algorithm step 1, Eq. (33), is given by Eq. (45), where
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
309
DV3 V3sp 2 V3 . V3 is determined from step 2, Eq. (35): 2
0
6 B 22 6 6 6 6 00 3 6 2 6 G 22 0 6 7 6 6 0 6 0 7 6 7 6 6 7 6 6 6 V 6 DP 7 6 r2 6 27 7 6 6 7 6 V2 6 6 0 7 6 7 6 6 7 6 6 6 0 7 6 7 6 6 7 6 6 7 6 6 6 0 7 6 7 6 6 7 6 6 6 DP3 7 6 7 6 6 7 6 6 6 DQ 7 6 6 6 37 7 6 6 7 6 6 6 0 7 6 7 6 6 7 6 6 6 0 7 6 B42 7 6 6 7 6 6 7 6 6 6 DP4 7 6 7 6 6 7 6 G42 6 6 DQ4 7 6 5 6 4 6 6 6 0 DV3 6 6 6 0 6 6 4
G 022
2Vm2 V22
Vr2 V22
B24
G24
0
0
B 0022
2Vr2 V22
2Vm2 V22
G24
2B24
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0 Vm2 V2
3
B 033
G 033
2Vm3 V32
Vr3 V32
B34
G34
0
0
2
2DIm3 2a34
G 0033
B 0033
2Vr3 V32
2Vm3 V32
G34
2B34
0
0
2
2DIr3 2a34
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0 2Vm4 V42
0 Vr4 V42
0 2DIm4 2 2a34
G42
0
0
B43
G43
0
0
B 044
G 044
2B42
0
0
G43
2B43
0
0
G 0044
B 0044
2Vr4 V42
2Vm4 V42
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0 Vr3 V3
0 Vm3 V3
0
0
0
0
0
1
0
0
0
2
2DIr4 2a34
7 7 7 7 7 7 2 3 7 DVr2 7 7 6 7 7 6 7 6 DVm2 7 7 7 6 7 7 6 7 6 DP2 7 7 7 6 7 7 6 7 6 DQ 7 7 7 6 2 7 7 6 7 7 6 7 6 DVr3 7 7 7 6 7 7 6 7 6 DV 7 7 6 m3 7 7 7 6 7 7 6 7 £ 6 DP3 7 7 7 6 7 7 6 7 6 DQ3 7 7 7 6 7 7 6 7 6 DV 7 7 6 r4 7 7 7 6 7 7 6 7 6 DVm4 7 7 7 6 7 7 6 7 6 DP4 7 7 7 6 7 7 6 7 6 DQ 7 7 7 4 4 5 7 7 7 Da34 7 7 7 7 7 5
45 3.3. The phase shifter transformer Assume that a phase shifter transformer is connected between buses k and j to control the active power ¯ow Pkj through changes in the phase angle w kj. In this case the linearized equation of Pkj needs be introduced into the Jacobian matrix as follows: 32 2 3 2 p 3 0 DVrm Y B E 76 6 7 6 7 6 DuV 7 6 C 0 0 76 DPQ 7 46 54 4 5 4 5 DPkj Dwkj Dt 0t F where" D
2Pkj 2Vrk
¼ "
E2 ¼
F
2DImk 2wkj
2Pkj 2Vmk
¼
2DIrk 2wkj
2Pkj 2Vrj ¼
2Pkj 2Vmj
¼
2DImj
2DIrj
2wkj
2wkj
2DIrk sin wkj akj bkj Vmj 2 akj gkj Vrj 2wkj 1 cos wkj akj bkj Vrj 1 akj gkj Vmj 2DImj 2wkj
2wkj ¼
#t
2Pkj 2wkj
sin wkj 2akj bkj Vrk 2 akj gkj Vmk 1 cos wkj akj gkj Vrk 2 akj bkj Vmk
2DIrj
#t
1 cos wkj akj bkj Vmj 2 akj gkj Vrj
49
sin wkj 2akj gkj Vrk 1 akj bkj Vmk 1cos wkj 2akj bkj Vrk 2 akj gkj Vmk
50
2Pkj 2a2kj gkj Vrk 1 akj cos wkj bkj Vmj 2 gkj Vrj 2Vrk 2 akj sin wkj gkj Vmj 1 bkj Vrj
with 2DImk sin wkj 2akj bkj Vrj 2 akj gkj Vmj 2wkj
48
51
2Pkj 2a2kj gkj Vmk 2 akj cos wkj gkj Vmj 1 bkj Vrj 2Vmk 47
1 akj sin wkj gkj Vrj 2 bkj Vmj
52
310
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
2Pkj akj sin wkj gkj Vrk Vrj 1 Vmk Vmj 1 bkj Vmk Vrj 2wkj Slack 1
PV
2
2 Vrk Vmj 1 akj cos wkj gkj Vmk Vrj 2 Vrk Vmj 2 bkj Vrk Vrj 1 Vmk Vmj The phase shift value for the next iteration is given by:
3
w kjh11 w kjh 1 Dw kjh
4 PQ
PQ Fig. 1. Topology for tap control.
2Pkj 2akj cos wkj gkj Vrk 1 bkj Vmk 2Vrj 1akj sin wkj gkj Vmk 2 bkj Vrk
53
2Pkj akj cos wkj bkj Vrk 2 gkj Vmk 2Vmj 2akj sin wkj gkj Vrk 1 bkj Vmk 2
0
6 B 22 6 6 6 6 00 G 2 3 6 6 22 0 6 6 7 6 0 6 0 7 6 6 7 6 6 7 6 Vr 6 DP 7 6 6 2 6 2 7 6 7 6 V2 6 7 6 6 0 7 6 6 6 7 6 6 7 6 0 7 6 6 7 6 6 7 6 6 7 6 6 0 7 6 6 7 6 6 7 6 6 DP3 7 6 6 7 6 6 7 6 6 DQ 7 6 6 6 3 7 6 7 6 6 7 6 6 0 7 6 6 7 6 6 7 6 6 0 7 6 B42 6 7 6 6 7 6 6 7 6 6 DP4 7 6 6 7 6 6 7 6 G42 6 DQ4 7 6 4 5 6 6 6 6 0 DP34 6 6 6 0 6 6 4
56
54
3.3.1. Illustration example Now, suppose a phase shifter transformer connected from bus 3 to bus 4 in Fig. 1, to control its active power ¯ow in Psp 34 . The linear system of equations related to the solution algorithm step 1, Eq. (33), is given by Eq. (57), where DP34 Psp 34 2 P34 V; u; w34 : From the equations in the proposed formulation, it is seen that the overhead in computer time is due to the representation of PV buses, when compared to the conventional power ¯ow formulation [3]. In other words, a PV bus is treated as if it were a control device, requiring an additional equation. In the illustration example of this section, the control equation representing the PV bus number 2, is given by the fourth line of Eq. (57).
G 022
2Vm2 V22
Vr 2 V22
B24
G24
0
0
B 0022
2Vr2 V22
2Vm2 V22
G24
2B24
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0 Vm 2 V2
3
B 033
G 033
2Vm3 V32
Vr 3 V32
B34
G34
0
0
2
2DIm3 2w34
G 0033
B 0033
2Vr3 V32
2Vm3 V32
G34
2B34
0
0
2
2DIr3 2w34
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0 2Vm4 V42
0 Vr 4 V42
0 2DIm4 2 2w34
G42
0
0
B43
G43
0
0
B 044
G 044
2B42
0
0
G43
2B43
0
0
G 0044
B 0044
2Vr4 V42
2Vm4 V42
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0 2P34 2Vr3
0 2P34 2Vm3
0
0
1
0
0 2P34 2Vm4
0
0
0 2P34 2Vr4
0 2P34 2w34
2
2DIr4 2w34
7 7 7 7 7 7 2 3 7 DVr2 7 7 6 7 7 6 7 6 DVm2 7 7 7 6 7 7 6 7 6 DP2 7 7 7 6 7 7 6 7 6 DQ 7 7 7 6 2 7 7 6 7 7 6 7 6 DVr3 7 7 7 6 7 7 6 7 6 DV 7 7 6 m3 7 7 7 6 7 7 6 7 £ 6 DP3 7 7 7 6 7 7 6 7 6 DQ3 7 7 7 6 7 7 6 7 6 DV 7 7 6 r4 7 7 7 6 7 7 6 7 6 DVm4 7 7 7 6 7 7 6 7 6 DP4 7 7 7 6 7 7 6 7 6 DQ 7 7 7 4 4 5 7 7 7 Dw34 7 7 7 7 7 5
57
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312 Table 5 Control devices convergence characteristics
Table 1 Test system characteristics Test system
730-bus
1653-bus
Circuits PV buses Transformers Load (MW) Load (MVAr)
1146 103 277 28565 6574
2382 121 472 26703 13436
Iteration
0 1 2 3 4 5
Table 2 Convergence characteristics Iteration
0 1 2 3 4 5
Maximum mismatch (p.u.) 730-Bus
1653-Bus
48.876430 0.354070 20.006990 20.000329 ± ±
82.506656 16.926767 4.226183 0.858707 0.065671 20.000759
Table 3 Load modeling-730-bus system Iteration
0 1 2 3
311
Maximum mismatch (p.u.) Z cte. Load
I cte. load
Mixed load
48.876430 20.403327 20.007018 20.000493
48.876430 0.374413 20.007093 0.000260
248.876430 0.373643 20.007110 20.000291
4. Results The proposed power ¯ow control models were validated through tests with systems with 730 and 1653 buses related to the interconnected south±southeastern Brazilian system, whose main characteristics are described in Table 1. The active and reactive loads for the 1653 bus system are modeled as 50% constant impedance plus 50% constant current. The maximum power mismatch at each iteration is shown in Table 2, when the augmented formulation model is processed. The convergence tolerance considered is 0.001 p.u. In the 730-bus system the constant power load model is adopted.
Maximum mismatch (p.u.) Tap control
Phase shifter
2 48.876430 24.444066 4.394440 0.562881 0.011850 0.000418
2 48.876430 0.353983 2 0.006986 0.000245 ± ±
Table 3 shows the 730-bus convergence characteristics using the augmented formulation and considering the loads modeled as constant impedance (Z cte.), constant current (I cte.) and a mix of 40% of constant power plus 30% of constant current plus 30% of constant impedance (mixed). Table 4 shows the results related to LTC transformer control, considering the loads modeled as constant power. The phase shifter transformer control has the objective of controlling the active power ¯ow from bus 461 to bus 252 in 2.44 p.u., by using the 730-bus system. The initial and ®nal phase shifter angles are 30.00 and 46.968, respectively. Table 5 shows the convergence characteristics in terms of number of iterations, when employing the control devices cited previously. Convergence characteristics are achieved when the absolute value of the difference between the speci®ed and calculated parameters (voltage magnitude and active power ¯ow) is less than 0.001 p.u. Note that in this formulation the dependent variables DP and DQ allow an easy way to represent control devices in which the control variables can either be DP or DQ components. For example, on the steady-state secondary voltage control, multiple reactive power sources (dependent variables after linearization) participate to control a voltage at a given bus.
5. Conclusions This paper proposes a novel approach for solving the power ¯ow problem based on nodal current injections and a highly sparse augmented Jacobian. The main advantage of this formulation lies on the calculation of matrix Y p, because its off-diagonal elements are exactly the terms of
Table 4 Automatic tap control-730-bus system Terminal buses (k±m)
Control bus #
Voltage (p.u.)
Initial tap (p.u.)
Final tap (p.u.)
61±60 120±121 161±130 180±181 367±368
61 120 161 180 368
1.000 1.020 1.030 0.990 1.050
1.050 0.990 1.065 0.956 0.975
0.866 0.925 0.990 0.934 1.001
312
V.M. da Costa et al. / Electrical Power and Energy Systems 23 (2001) 305±312
admittance matrix bus and the diagonal elements are calculated using non-transcendental functions, even if load models other than constant power are included. The augmented formulation is equivalent to the conventional Newton±Raphson power ¯ow regarding convergence characteristics, but allows an easier incorporation of control device models and power ¯ow controls of any kind. This formulation also directly incorporates more realistic modeling of power system components, such as, static var compensators, TCSC and voltage control through multiple reactive sources. The studies performed so far indicate the proposed formulation may become a valuable tool for solving present day power ¯ow problems, where the proper consideration of controls is becoming a key issue.
ck
Pk Vm2 k 2 Vr2k 2 2Vrk Vmk Qk Vrk Vmk Q1 k 2 P1k Vr2 k 1 Vk4 Vk3 2 P2 k
dk
Qk Vr2k 2 Vm2 k 2 2Vrk Vmk Pk Vrk Vmk P1k 2 Q1k Vr2k 1 Vk4 Vk3 2 Q2k
For the load at bus k of constant power type then ak dk
Qk Vr2k 2 Vm2 k 2 2Vrk Vmk Pk Vk4
6. Future work The authors are currently working on the representation of FACTS devices, remote voltage control by multiple reactive sources and on the implementation of user de®ned control strategies. These issues are a subject of another paper, which is under preparation by the authors.
bk 2ck
Pk Vr2k 2 Vm2 k 1 2Vrk Vmk Qk Vk4
These parameters are calculated at every iteration of the power ¯ow solution and are used for updating the diagonal blocks of matrix Y p.
Acknowledgements The ®rst two authors would like to thank the Brazilian agencies CNPQ and FAPEMIG for ®nancial support. Appendix A. Load models parameters With respect to load models to be represented in the power ¯ow problem, the parameters ak, bk, ck and dk shown in the Eqs. (14)±(17) are given by the following expressions: ak
Qk Vr2k 2 Vm2 k 2 2Vrk Vmk Pk Vrk Vmk P1 k 1 Q1k Vm2 k 1 Vk4 Vk3 1 Q2 k
bk
Pk Vr2k 2 Vm2 k 1 2Vrk Vmk Qk Vrk Vm k Q1 k 1 P1k Vr2k 2 Vk4 Vk3 2 P2 k
References [1] Dommel HW, Tinney WF, Powell WL. Further developments in Newton's method for power system applications. IEEE Winter Power Meeting, Conference Paper no. 70 CP 161-PWR New York, January 1970. [2] Peterson NM, Meyer WS. Automatic adjustment of transformer and phase shifter in the Newton power ¯ow. IEEE Transactions on Power Systems 1971;January/February:103±8. [3] Tinney WF, Hart CE. Power ¯ow solution by Newton's method. IEEE Transactions on Power Systems 1967;PAS-86:1449±56. [4] Stott B. Review of load-¯ow calculation methods. Proceedings of IEEE 1974;62:916±29. [5] Mamandur KRC, Berg GJ. Automatic adjustment of generator voltages in Newton±Raphson method of power ¯ow solutions. IEEE Transactions on Power Systems 1982;PAS-101(6):1400±9. [6] Maria GA, Yuen AH, Findlay JA. Control variable adjustment in load ¯ows. IEEE Transactions on Power Systems 1988;3(3):858±64. [7] Chang S, Brandwajn V. Solving the adjustments interactions in fast decoupled load ¯ow. IEEE Transactions on Power Systems 1991; 6(2):801±5. [8] Da Costa VM, Martins N, Pereira JLR. Developments in the Newton± Raphson power ¯ow formulation based on current injections. IEEE Transactions on Power Systems 1999;14(4):1320±6.