www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 4th April 2012 Revised on 13th November 2012 Accepted on 13th May 2013 doi: 10.1049/iet-gtd.2012.0466
ISSN 1751-8687
Improved model for tap-changing transformer Carlos Aparecido Ferreira, Ricardo Bernardo Prada Departament of Electrical Engineering, Pontifical Catholic University of Rio de Janeiro, Rua Marquês de São Vicente 225, Gávea 22453-900, Rio de Janeiro, RJ, Brazil E-mail:
[email protected]
Abstract: The model for tap-changing transformers currently in widespread use consists of an impedance, measured in a shortcircuit test with a nominal tap, in series with an ideal transformer. The use of this model in voltage stability studies leads to qualitatively incorrect results. For demonstration purpose, a small circuit and the concepts of maximum load, equivalent load impedance and voltage control effects are used. An improved model that takes into account laboratory results, circuit laws and voltage stability theory is proposed. Using IEEE test systems, different results are shown for the conventional and proposed models.
1
Introduction
Electrical machines have long been represented by simple combinations of resistance and reactance [1]. A piece of electrical equipment, however, is clearly not an electrical circuit, as a circuit merely represents an approximation of the electrical, magnetic and even mechanical behaviour (should the machine be rotating) of the equipment. A given model for a piece of electrical equipment may not necessarily be the only or most convenient model for every purpose. Hence, the choice of model must depend on which model serves a particular purpose best [2]. However, the extent to which researchers are familiar with one or more methods is a determining factor in this choice and explains the slow uptake at the beginning of the past century of excellent tools such as the symmetrical components method [2]. The modelling of tap-changing transformers is fundamental in voltage stability analysis, both in terms of the information provided to the operator about voltage stability margins and the effects of voltage control actions. The objective of this paper is to describe problems that arise with the conventional model for a tap-changing transformer and to propose an improved model that can be used in any steady-state study with tap-changing transformers.
2 2.1
Basic concepts Voltage stability
Fig. 1 shows a typical constant φ (power factor angle) curve, which is also known as a ‘nose curve’. This curve was plotted for a generic bus k, where Si is the apparent power injected in the bus for a particular operating point. In this situation, there is a solution in the normal region (point A, with a high-voltage Vk and low current) and another in the abnormal region (point B, with a low-voltage Vk and high IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
current). Point C corresponds to the maximum apparent injected power in the bus (Sm). 2.1.1 Equivalent impedance at the maximum load point: The graph in Fig. 1 refers to a simple 2-bus system with the transmission impedance (Zt∟αt) between the generation (V0 |u0 ) and the load buses (V1 |u1 ), where the load (Z1 |f1 ) is modelled as a constant impedance. The curve can be plotted by varying the load from Z1 = ∞, where V1 = V0 to Z1 = 0, where V1 = 0. According to [3], the magnitude of the load impedance at the maximum load point is Zt. In other words, when the magnitude of the load impedance Z1 is equal to the magnitude of the transmission impedance Zt, the power transmitted to the load is a maximum (Sm). On the other hand, if a generic representation of the 2-bus circuit is used, with elements in parallel with the generator and the load, as shown in Fig. 2a, the Thévenin equivalent circuit seen from the load is shown in Fig. 2b. Thus, the same reasoning can be used to obtain the load equivalent impedance at the maximum loading. Analogously to the simple 2-bus system, the magnitude of the load impedance at the maximum load point will be Zth. In other words, when the magnitude of the load impedance Z1 is equal to the magnitude of the equivalent Thévenin impedance seen from the load terminal, the transmitted power is a maximum (Sm). 2.1.2 Actions to increase the maximum transmission capacity and control the voltage: If a capacitor is introduced in the load bus in the simple 2-bus system, a new constant φ curve in the SV plane (where S is the apparent power injected in the bus and V is the voltage magnitude) is obtained. In Fig. 3, it can be observed that the maximum load point increases because of the introduction of the capacitor. It can also be seen that if the system operates at point A (normal region) and a capacitor is introduced to increase the voltage, this control action has 1289
& The Institution of Engineering and Technology 2013
www.ietdl.org where 2 N ′ Z˙ k (V) = Z˙ k (V) m Nk
Fig. 1 Constant φ curve in the SV plane
the expected effect both with a constant power load (the new operating point will be A′) and a constant impedance load (the new operating point will be A″). However if, as shown in Fig. 4, the system operates at point B (abnormal region) and a capacitor is introduced to increase the voltage, this control action has the expected effect only if the load is a constant impedance load (the new operating point will be B″). If it is a constant power load, the control action has an unexpected effect (the new operating point will be B′), that is, the voltage decreases when the capacitor is introduced. The insertion of a capacitor is merely an example of an action to control the voltage and increase the maximum transmission capacity. Other actions would have the same qualitative results and include increasing V0, inserting another transmission line in parallel with the existing one, inserting series capacitive compensation and changing transformer tap, as will be addressed in this paper. 2.2
Steady-state transformer model
Fig. 5 shows a steady-state model for a transformer, where the impedance composed of the magnetisation reactance and resistance of the iron is ignored because of its high value compared with the impedance of the windings Z˙ [4]. This impedance, referred to a particular side of the transformer, can be obtained using the short-circuit test [4] and is given, in Ω, by ′ ˙ Z(V) = Z˙ k (V) + Z˙ m (V)
(1)
(2)
In Fig. 5, the impedance is referred to side m, without any loss of generality. In per-unit (p.u.), the impedance values referred to side k or m will have equal values as long as the base voltage values chosen for each side of the transformer follow the turns ratio (Nm/Nk). In addition, the base power must be the same on each side of the transformer and throughout the whole system [5]. The transformer can therefore be completely represented by its p.u. impedance without the ideal transformer, since no transformation is needed. The current will have the same value on both sides. This is an advantage of representing the transformer in p.u. values. The problem is that it is common when operating electrical power systems to change transformers taps in order to control the voltage profile. In this situation, the base voltage and impedance on the system side, where the tap was changed would have to be changed, so that the advantages associated with the use of p.u. values can be observed. Thus, voltages and transmission line and transformer impedances would have to be changed each time a tap was changed, a procedure that is clearly impracticable.
3 Conventional model for a tap-changing transformer 3.1
Conventional model representation
To solve the problem outlined above, the ideal transformer is reintroduced in the representation of the tap-changing transformer, as shown in Fig. 6a. This model is used worldwide. In addition to the impedance, the tap is also represented in p.u. in Fig. 6a, where the base tap is equal to the nominal tap, which is equal to the turns ratio (Nm/Nk) under nominal conditions. Thus, a = 1 when the ratio is nominal, with the circuit in Fig. 6a without the transformer. The base voltages remain constant and follow the nominal tap rate, irrespective of the tap value. As mentioned, the impedance Z˙ is measured in a short-circuit test, so its value is dependent on the tap position used for the test [6]. However, the impedance value used corresponds to the nominal tap; this approach is justified by the small tap variation of around 1 p.u..
Fig. 2 Generic 2-bus system a Generic 2-bus system b Thévenin equivalent circuit for the generic 2-bus system 1290 & The Institution of Engineering and Technology 2013
IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
www.ietdl.org [1, 2, 7, 8] and [9]. This circuit was first described in [1] and [8]. 3.2 Conventional model for a tap-changing transformer in voltage stability studies
Fig. 3 Voltage control analysis with an operating point in the normal region
Fig. 4 Voltage control analysis with an operating point in the abnormal region
Fig. 5 Steady-state transformer model
It can be seen from the conventional model for a tap-changing transformer in Fig. 6 that if the generator is connected to bus k and the load to bus m, the maximum power transfer occurs when Zm = Zth = Z (see Section 2.1.1), where Zm is the equivalent load impedance, Zth is the Thévenin impedance seen by the bus m and Z is the total transformer impedance (on side m). All measurements are in p.u. values. In this situation, the impedance at the maximum load point does not change when the tap is changed. However, when the power flow is inverted, that is, with the generator connected to bus m and the load to bus k, the maximum power transfer occurs when Zk = Zth = Z/a 2, where Zk is the equivalent load impedance and Z/a 2 is the total transformer impedance referred to side k. In this case, unlike in the previous situation, the impedance at the maximum load point varies when the tap is changed. Considering, once again, the conventional model in Fig. 6, the constant φ curves in the SV plane can be plotted for two different transformer tap values. The equivalent load impedance curve passing through the maximum load points can also be plotted. It can be seen from Fig. 7 that if the power flows from bus k to bus m, the tap change does not modify the equivalent load impedance, as already mentioned. However, the maximum load point changes. If, as shown in Fig. 8, the power flows from bus m to bus k, the opposite is observed, that is, the maximum load point remains constant and the equivalent impedance at the maximum load point changes as the tap changes. Curves similar to those in Fig. 8 are described in [10] and [11]. In [12] also, the authors report that the maximum load point remains constant when the tap changes. This result is qualitatively inconsistent with the theory presented in Section 2.1.2. The finding that a change in the direction of the power flow resulted in the differences described above was unexpected, as both the maximum load point and equivalent load impedance at this point had been expected to change when the tap changed, regardless of the direction of the power flow. This intuitively expected result had been confirmed in laboratory tests with a real tap-changing transformer [13]. Thus, the model for tap-changing transformers used
To eliminate the ideal transformer in the conventional model shown in Fig. 6a, so that the admittance matrix can be easily assembled, the electrical circuit is represented by a π equivalent circuit, as shown in Fig. 6b and described in
Fig. 6 Conventional models a Conventional model based on an ideal transformer in series with an impedance b Conventional model represented by the Π equivalent circuit IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
Fig. 7 Constant φ and Z curves for two tap values with the conventional model and with power flow from bus k to bus m 1291
& The Institution of Engineering and Technology 2013
www.ietdl.org It can be easily verified that the relationship between the impedances Z˙ k (V) and Z˙ m (V) is the quadratic turns ratio, as is the relationship between the base impedances on sides k and m. Thus, the impedances on each side of the transformer have the same p.u. values ˙ Sbase Z(V) Z˙ k (p.u.) = Z˙ m (p.u.) = 2 2 Vbasem
(7)
Z˙ Z˙ k (p.u.) = Z˙ m (p.u.) = 2
(8)
or
Fig. 8 Constant φ and Z curves for two tap values with the conventional model and with power flow from bus m to bus k
worldwide yields qualitatively incorrect results in voltage stability studies.
4 Proposed model for a tap-changing transformer 4.1
Proposed model representation
Z˙ k (V) and Z˙ m (V), which are given by (1) and (2), are the impedances in ohms referred to the windings on each side of the transformer. These impedances cannot be obtained separately even if separate short-circuit tests are carried out on each side of the transformer, because of the linear dependence between the two equations generated. However, according to [4, 14] and [15] ˙ Z(V) ′ Z˙ k (V) = Z˙ m (V) = 2
(3)
2 ˙ Nk Z(V) Z˙ k (V) = Nm 2
(4)
Thus
˙ where Z(V), as previously mentioned, is obtained in a short-circuit test with a nominal tap. The base impedances are given by 2 Vbasem Zbasem = (5) Sbase Zbasek =
Vbasem 2 N 2 k
Sbase
Nm
(6)
The proposed model is shown in Fig. 9a, from which it can be seen that no impedance is reflected on either side. The proposed model shown above takes into account the impedances in each transformer winding. If the impedance from side k in the circuit in Fig. 9a is reflected to side m based on a tap ratio 1:a, the total impedance will vary when the tap changes, as it is shown in Fig. 9b. This contrasts with the conventional model and leads to qualitative differences between the results obtained with the two models, as will be shown. It can also be observed that the total impedance on side m is equal to Z˙ only when the tap rate is nominal (a = 1). In this case, the proposed model coincides with the conventional one. The π equivalent circuit for the proposed model is similar to Fig. 6b. The difference is that the impedance values are multiplied by (a 2 + 1)/2. 4.2 Proposed model for a tap-changing transformer in voltage stability studies With the proposed model for a tap-changing transformer shown in Fig. 9, if the generator is connected to bus k and the load to bus m, the maximum power transfer occurs when Zm = Zth = Z(a 2 + 1)/2, where Zm is the equivalent load impedance and Z(a 2 + 1)/2 is the transformer impedance on side m plus the impedance on side k referred to side m. In this case, the impedance at the maximum load point changes when the tap changes, unlike in the conventional model. By inverting the direction of the power flow, that is, with the generator connected to bus m and the load to bus k, the maximum power transfer occurs when Zk = Zth = Z(a 2 + 1)/(2a 2), where Zk is the equivalent load impedance and Z (a 2 + 1)/(2a 2) is the transformer impedance on side k plus the impedance on side m referred to side k. Once again, the impedance at maximum load changes when the tap changes. Still considering the circuits in Fig. 9, the constant φ curves can be plotted in the SV plane for two different tap values, as
Fig. 9 Proposed model a Proposed model b Proposed model with impedance reflection 1292 & The Institution of Engineering and Technology 2013
IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
www.ietdl.org
Fig. 10 Constant φ and Z curves for two tap values with the proposed model and with flow from bus k to bus m
Fig. 11 Constant φ and Z curves for two tap values with the proposed model and with flow from bus m to bus k
well as the curves of the equivalent load impedances that passes through the maximum load points. It can be seen from Fig. 10 that if the power flows from bus k to bus m, both the maximum load point and the equivalent load impedance at this point change when the tap changes, as expected. When the conventional model is used, the equivalent load impedance remains constant when the tap changes, as shown in Fig. 7. As shown in Fig. 11, if the power flows from bus m to bus k, once again both the maximum load point and equivalent load impedance at this point change when the ‘tap’ is changed, as expected. When the conventional model is used, the maximum load point remains constant when the tap changes, as shown in Fig. 8. It is important to mention that the results obtained with the proposed model are qualitatively coherent with the theory presented in Section 2.1.2. Therefore it can be seen that when the proposed model is used, both the maximum load point and equivalent load impedance at this point change when the tap is changed, regardless of the direction of the power flow. This agrees with what would intuitively be expected and was proved in laboratory tests [13].
5 Effects of voltage control actions: a comparison of the conventional and proposed models
Fig. 12 Comparison of the conventional and proposed models for power flow from bus m to bus k
the tap, which is on side m, must be set to a lower value. The operating point is also assumed to be in the abnormal region, as shown in Fig. 12. When the conventional model is used, the voltage control action (change to a lower tap value) causes an increase in the voltage for a constant power load. With a constant impedance, however, the voltage decreases, as described in [10, 11, 12]. The model proposed here yields results that are the opposite of those obtained using the conventional model. When a lower tap is selected, the voltage decreases for a constant power load and increases for a constant impedance load. It should also be noted that the results for the proposed model when it is used to analyse the effects of voltage control actions are independent of the direction of the power flow and are coherent with the theory presented in Section 2.1.2; unlike those obtained with the conventional model.
6 6.1
IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
Laboratorial simulation
The objective is to draw the constant φ (power factor angle) curve with two different values of transformer tap. Moreover, the two power flow directions through the transformer are considered. Nominal transformer values are: 1 kVA, 220:220 V, with taps at 110 and 190 V on both sides. Load growth is the parallel connection of resistors. Initially transformer operation is with 110:220 V. Second, the tap on the high-voltage side was changed to 190 V. In both tap configurations, that is, 110:220 and 110:190 V, power flowed from the low to high voltage side in one test and from the high to the low voltage side in another test. The main results are outlined in Fig. 13. As can be seen, the maximum loading point changes when the transformer tap is changed, regardless of the power flow direction, as expected. From current measurements made at the point of maximum loading, it is easy to calculate that the equivalent load impedance at this point also varies with the tap, regardless of the direction of power flow [13], as expected. Thus, the curve shapes indicate that (i) the conventional model is not adequate and (ii) the proposed model is appropriate.
6.2 In the same 2-bus system, consider a tap change from a1 to a2 that is intended to increase the voltage on side k, where the load is connected (power flow from bus m to bus k). Then
Results
Computational simulation
In the following simulations, all the system load buses were subjected to additional load while keeping the power factor 1293
& The Institution of Engineering and Technology 2013
www.ietdl.org
Fig. 13 Constant φ curves a Constant φ curves for two tap values with power flow from the low to the high voltage sides b Constant φ curves for two tap values with power flow from the high to the low voltage sides
Fig. 14 Comparison of the conventional and proposed models a bus 17/IEEE 118-bus system b bus 30/IEEE 118-bus system
constant, according to (9) and (10) P = P0 (1 + g)
(9)
Q = Q0 (1 + g)
(10)
where γ is the additional loading, which is supplied only by the voltage angle reference generator. First, results are given for simulations in which the transformer taps are kept at non-nominal values and no voltage control is carried out. In subsequent simulations, the transformers control the bus voltages by changing the tap settings. The simulations were carried out using IEEE test systems described in [16]. 6.2.1 IEEE 118-bus system: Fig. 14 shows constant φ curves for buses 17 and 30 obtained using a continuation power flow algorithm [17]. Between these buses, there is one of the tap-changing transformers of the system, whose taps are kept at non-nominal values in the simulation. As these transformers taps are different from the nominal value [16], there are differences between the circuits for the conventional and proposed models even when there is no voltage control. Differences can be observed between the voltages obtained with the conventional and proposed models, especially (i) as 1294 & The Institution of Engineering and Technology 2013
the load system increases and (ii) in the abnormal operating region. The maximum load points are also different. Analysing Fig. 14, if the system is operating in the abnormal region, with a small load shedding from the additional loading of 0.8 p.u., contrary information regarding the voltage behaviour are provided: the voltage
Fig. 15 Comparison of the conventional and proposed models/bus 32/voltage control/IEEE 57-bus system IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
www.ietdl.org increases or decreases according to the transformer model. It is also important to note that the bottom half of the curve is within normal voltage range. 6.2.2 IEEE 57-bus system: In this test system, the transformer situated between buses 32 and 34 controls the voltage on bus 32, whereas the transformer situated between buses 10 and 51 controls the voltage on bus 50. The constant φ curve in Fig. 15 was obtained using a continuation power flow algorithm, and the voltages on terminal or remote buses can be controlled by changing transformer tap settings. The voltage on bus 32 is controlled until the transformer tap reaches its maximum value of 1.15 p.u., after which it is no longer controlled. Differences can be observed between the two models in terms of: (i) the load at which the maximum load tap is reached (for about 0.08 p.u., the voltage is controlled or not according to the transformer model); (ii) the different voltage values for each additional load and (iii) the maximum load point, which is lower for the proposed model. 6.2.3 33-Bus Brazilian test system: The differences between the two models were also noted in the power flow values. In some simulations, these differences were considerable. The worst case was noted in the 33-bus Brazilian test system [18], where the power flow from bus 896 to 897 is equal to − 279.6 MW, using the conventional model, and 23.9 MW, using the proposed model. Besides the substantial quantitative difference, the two flows are in opposite directions. The system contains four generators with off-nominal tap-changing transformers, and nine others at the end of bulk transmission lines. Thus, there are 13 off-nominal tap-changing transformers, out of the total of 40 network branches, with the equivalent circuit depending on the used model. The large amount of tap-changing transformers in the network is responsible for quite different operating points when the two models are compared. Regarding the voltage values, differences can also be noted, specially comparing the angles. The voltage of buses 896 and 897 are equal to 0.982∟4.91 and 0.997∟6.05° p.u., respectively, using the conventional model, and 1.001∟12.40 and 1.004∟12.29° p.u., respectively, using the proposed model. Once the voltage angles have influence in active power flow [6], the substantial quantitative differences on the flow from bus 896 to 897 is explained. It can also be noted that the angular displacement is − 1.1°, using the conventional method, and 0.2°, using the proposed method. The different signals explain the flows in opposite directions. Furthermore, the exact different results can also be manually checked. The impedance value between the lines 896 and 897, to be used in the well known active power flow equation, is (0.0005 + j0.0073) p.u..
IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 11, pp. 1289–1295 doi: 10.1049/iet-gtd.2012.0466
7
Conclusions
This paper has proposed an improved model for tap-changing transformers that more accurately represents this type of equipment and is more suitable for voltage stability studies. Unlike in the conventional model, the impedance referred to a particular side changes with the tap value. Thus, the effect of a tap change on the voltage to be controlled and the power margin can be correctly evaluated up to maximum load. The proposed model can be used in any steady-state analysis with tap-changing transformers. It gives results that are not only more accurate than those obtained with the conventional model, but also, as shown in this paper, qualitatively different.
8
References
1 MacLeod, D.R.: ‘New equivalent circuits for autotransformer and transformer circuits’, Gen. Electr. Rev., 1929, 32, pp. 120–126 2 Boyajian, A.: ‘Inversion currents and voltages in auto-transformers discussion by I.H. Summers’, Artif. Intell. Eng. Trans., 1930, 49, pp. 819 3 Close, C.M.: ‘The analysis of linear circuits’ (Harcourt, Brace & World, New York, 1966) 4 Members of the Staff of the Department of Electrical Engineering Massachusets Institute of Technology: ‘Magnetic circuits and transformers’ (MIT Press, Cambridge, 1965, 15th edn.) 5 Grainger, J.J., Stevenson Jr., W.D.: ‘Power system analysis’ (McGraw-Hill, New York, 1994) 6 Monticelli, A.J., Garcia, A.: ‘Electrical energy systems introduction’ (Unicamp Publisher, Portuguese, 2003) 7 Clarke, E.: ‘Circuit analysis of A–C power systems’ (John Wiley & Sons, New York, 1950), vol 2 8 Boyajian, A.: ‘New theory of transformer and autotransformer circuits’, Gen. Electr. Rev., 1929, 32, pp. 110–119 9 Boyajian, A.: ‘Inversion currents and voltages in auto-transformers’, Artif. Intell. Eng. Trans., 1930, 49, pp. 810–818 10 Baracho, F.R.A., Café, F.F.: ‘Voltage instability analysis in electrical systems’, presented at the XI SNPTEE, (in Portuguese), Rio de Janeiro, 1991) 11 El-Sadek, M.Z., Mahmoud, G.A., Dessouky, M.M., Rashed, W.I.: ‘Tap changing transformer role in voltage stability enhancement’, Electr. Power Syst. Res., 1999, 50, pp. 115–118 12 IEEE Power System Stability Committee, Voltage Stability Assessment: Concepts, Practices and Tools, Cap 2: Basic Theoretical Concepts, August 2002 13 Ferreira, C.A., Prada, R.B., da Costa, V.M.: ‘Changeable tap transformer usual model evaluation’, presented at the VIII CLAGTEE, (in Portuguese), Ubatuba-SP, 2009 14 Central Station Engineers of the Westinghouse Electric Corporation: ‘Electrical transmission and distribution reference book’ (East Pittsburg, Westinghouse, 1964, 4th edn.) 15 Del Toro, V.: ‘Basic electric machines’ (Prentice-Hall, New Jersey, 1990) 16 PSTCA – Power System Test Case Archive. Available at http://www.ee. washington.edu/research/pstca, Accessed on 8th January 2011 17 Ajjarapu e, V., Christy, C.: ‘The continuation power flow: a tool for steady state voltage stability analysis’, IEEE Trans. Power Syst., 1992, 7, (1), pp. 416–423 18 Brazilian Test Systems for Educational Use and Computational Electrical Power System Analysis. Available at: http://www.sistemasteste.com.br, Accessed on 8th January 2011
1295
& The Institution of Engineering and Technology 2013
