Preliminary Design of Power Controller Devices Using ... - IEEE Xplore

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012

Preliminary Design of Power Controller Devices Using the Power-Flow Control and the Ideal Phase-Shifter Methods Nicklas Johansson, Member, IEEE, Lennart Ängquist, Member, IEEE, and Hans-Peter Nee, Senior Member, IEEE

Abstract—This paper introduces a new method for the preliminary design of power controllers (PCs) in the electric power grid. The method, which is denoted the ideal phase-shifter (IPS) method, utilizes the concept of the power controller plane where the active power of the PC line is plotted versus the difference in voltage angle between the PC terminals. The power controller plane makes it possible to graphically visualize the working area of a PC in a power grid and thus determine the grid situations which are dimensioning for the PC. The IPS method offers the possibility of plotting the grid characteristics in the power controller plane which are unbiased with respect to the reactive properties of the PC. This makes the method suitable for comparison and preliminary design of PCs of different types and with different characteristics by simple geometrical considerations. In this process, the IPS method uses the power-flow control method for deriving the PC characteristics. This paper includes an application example of the method where it is used for dimensioning of two different PCs in a 26-bus test system. Index Terms—Flexible AC transmission systems (FACTS), phase-shifting transformer (PST), power controller plane, power-flow control, thyristor-switched series capacitor (TSSC).

I. INTRODUCTION LEXIBLE ac transmission systems (FACTS) devices have received a large amount of attention recently. These devices, which can be used to improve the stability, control power flows, and improve voltage characteristics in a power system are important components on the road to smarter power grids. This paper focuses on power controllers (PCs) which can be used to resolve congestion and reduce wheeling–and loop flows in power grids to improve the transfer capabilities of critical transmission lines. These issues are becoming more common due to the deregulation of the national power grids and the increased amount of renewables used for generation. The phaseshifting transformer (PST) is the most common apparatus in the group of PCs which, among others, also include the thyristorswitched series capacitor (TSSC) [1], the dynamic power-flow

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Manuscript received May 23, 2011; revised October 14, 2011; accepted April 14, 2012. Date of publication June 01, 2012; date of current version June 20, 2012. This work was supported by the Elektra Program at Elforsk AB, Stockholm, Sweden. Paper no. TPWRD-00432-2011. N. Johansson is with ABB Corporate Research, Power Technologies, Västerås 721 78, Sweden (e-mail: [email protected]). L. Ängquist and H.-P. Nee are with the School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm 100 44, Sweden (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRD.2012.2196449

controller (DPFC) [2], [3], or hybrid flow controller (HFC) [4], the unified power-flow controller (UPFC), the interphase power controller (IPC) [5], and the static synchronous series compensator (SSSC). In addition, a number of distributed devices for power-flow control, which are attached directly to high-voltage conductors, have recently been investigated [6], [7]. The most common PCs are described in some detail in [8]. To determine suitable steady-state settings of a PC to optimize the steady-state security, relieve overloads, and minimize power losses, optimal power-flow (OPF) control [9]–[13] can be used. Sensitivity analysis [14]–[16] can be used to investigate how a certain FACTS controller impacts the power flows and voltages in a power system. This approach may be used, for example, for the optimal siting of new PC devices. Several PSTs have recently been installed in Europe to control active power flows and, thus, improve cross-border transfer capabilities between the Netherlands and Germany [17], and cope with transit power flows in Belgium [18]. The design of a PC to meet specific targets for the power flow is a process which normally requires power-flow calculations at many grid operating points for several considered PC candidates. This process is simplified by the use of the power controller plane, which was described in [19] and used for design of an interphase power controller in [20]. The method described in [19], which is here denoted as the power-flow control (PFC) method, describes the working area of the PC graphically in one plot showing all possible steady-state operating points of the device for all possible grid configurations and load cases. This approach makes it easy to find the dimensioning cases for the PC which can then be further analyzed to determine the required PC design to solve the power-flow problem. In [19], it is shown that for PCs installed in certain small generic power grids, the grid and PC characteristics can be approximately decoupled. This decoupling is very attractive and provides a comprehensive insight into the capabilities of different PCs in the grid and how they are affected by the selected PC characteristics and the operating point of the grid. With this method, the grid characteristics describing the sensitivity of the power flow in the PC line, relative to an inserted shift in voltage phase angle, are analytically derived and plotted in the power controller plane. Since these characteristics are shown to be close to independent to the PC characteristics, the operating point of a certain PC can be found by graphical comparison of the PC- and grid characteristics in the power controller plane. The mentioned method is in [19] used to provide an analytical

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JOHANSSON et al.: PRELIMINARY DESIGN OF PC DEVICES

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Fig. 1. One-machine infinite bus system.

background to the more exact method for final design of PCs provided in [19] which uses power-flow calculations to plot the exact working area of the PC in the power controller plane. This paper introduces a new screening method, which is here denoted as the IPS method, for preliminary design of PCs. The proposed method makes it possible to decouple the PC and grid characteristics in complex power grids. In the proposed method, the grid characteristics are found by power-flow calculations and plotted in the power controller plane only once, in the first phase of the design process. Following this, the capabilities of PC candidates of different types and with different characteristics can be found by plotting the PC curves determined by analytical expressions in the same graph where the grid curves are plotted. The PC curves are specified according to the PFC method introduced in [19]. The grid characteristics are found by introducing an ideal phase shifter in the grid model and varying the phase shift of the line voltage at the intended PC location. The resulting grid characteristics are nonbiased with regard to the reactive properties of the PC and can thus be used to compare PC candidates of different types. This paper is organized as follows. In Section II, the concept of the PFC method is introduced. Section III describes the IPS method and Section IV describes the used test system. The results and discussion are found in Section V, and Section VI concludes the study. II. PFC METHOD The PFC method for the design of PCs was introduced in [19]. In this pioneering work, it is shown that the characteristics of the grid and PC can be conveniently plotted in the plane, showing the power transmitted on the PC line versus the voltage-angle difference between the terminals of the PC. This plane is also denoted as the “power controller plane.” In [19], it is shown that for a number of small, generic power grids, the characteristics of the grid and the PC can be decoupled under the assumptions that the voltage-angle differences are small and that the terminal voltages of the PC are close to their nominal values. In such power grids, the grid characteristics can be found analytically and plotted in the power controller plane. To illustrate the principle of decoupling in a small power grid, a test system from [19], shown in Fig. 1, is used. Here, it is 1 p.u., 1 p.u., and that the angles are assumed that small so that and . By expressing and in the angle and reactance variables and eliminating , the PC line power in this system can be expressed as (1) with and . Equation (1), which describes how the simple grid responds to an ideal angle shift

Fig. 2. Typical working area of a PST A in the P

0

plane.

inserted by a PC, corresponds to a straight line in the plane. If it is assumed that can assume values between and , corresponding to all relevant grid configurations including contingencies, a set of lines describing the system can be plotted for each value of . It is now possible to consider the PC characteristics separately. For example, the approximations (2) (3) can be used for the steady-state characteristics of a PST with an angle shift at zero current having a leakage reactance of and a TSSC with a reactance (which is always negative since the device is capacitive). Each of these equations represents a family of lines which can be plotted in the plane. For a PC in the grid of Fig. 1, both the device equation given by (2) for a PST or (3) for a TSSC and the grid equation given by (1) must be fulfilled. If the lines described by the equations are plotted in the same graph for a certain loading condition, line configuration, and PC setting, the crossing point of the lines yields the predicted power flow on the PC line and the voltage-angle difference between the PC terminals. By considering the PC lines corresponding to the extreme values of the device settings and the grid lines corresponding to the maximum and minimum transmitted power for all possible values of , the working area of the PC can be plotted. This area in the plane defines the operating limits of the PC, and all operating points of the device in any grid condition fall within it. An example of a working area of a PST installed in the system of Fig. 1 is shown in Fig. 2 as area . It has been assumed that and the phase shift of the PST is in the range that the total active power transmitted across the corridor is and . Note that when the power on the PST between transmission line is zero, the voltage-angle difference across the . Thus, the PST lines cross zero power at device will be the angles and as indicated in Fig. 2. For the same grid, the working area of a TSSC with a degree (0%) and is illustrated in of compensation between Fig. 3. The working area, which is the aggregate of the areas

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before the PC type has been specified. In order to plot the decoupled grid curves, it is necessary to determine the dependency of the power, on the line where PC installation is considered, on an inserted shift in voltage phase angle (with a unity voltage ratio between the terminals) at the planned PC location for each relevant grid condition. Without these decoupled grid curves, a comparison of different PC candidates by simple geometrical considerations in the same power controller plane is not straightforward. III. IDEAL PHASE-SHIFTER METHOD

Fig. 3. Typical working area of a TSSC, A and A in the P

0

plane.

and in the figure, is bounded by the limiting grid lines and the TSSC lines for minimum and maximum degree of compensation. The TSSC lines turn counterclockwise in the plane (the angle increases) when the level of compensation is increased. From the working area of a PC in the power controller plane, the maximum line power, which can be reached, is seen together with the maximum angle across the PC in any working condition. These two values yield the maximum current through and maximum voltage across the device from which the rating of the PC can be calculated. For large power grids, the authors of [19] propose two different methods for plotting the working area of a PC as follows. 1) A simplified, less computationally demanding method uses the methodology from the decoupling method described before. However, as the grid characteristics in the plane cannot be found from inspection of the grid parameters, the method uses a limited amount of power-flow data calculated with the specified PC inserted in the grid model, to find the required parameters to plot the grid characteristics. This approximative method is used to illustrate the capabilities of the PC in the grid in a pedagogical manner. 2) The exact method plots the working area of the specified plane by power-flow calculations for PC in the all possible device settings and relevant grid topologies and loading situations. The two methods may be combined so that the first method is first used to identify the dimensioning cases for the PC. These cases can then be further studied by the second approach to yield the exact working area of the PC. The examples of decoupling in the small test system above suggest an attractive possibility for preliminary design of PCs in complex power grids: If the grid and PC characteristics can be decoupled, it is possible, once the grid characteristics have been plotted, to compare possible choices of PCs with different characteristics in the same plot by simple geometrical considerations without additional power-flow studies. This could potentially simplify the design process, especially in cases where a high complexity of the power grid makes it difficult to determine the type of PC which is to be used. The complication in this approach is how to determine the plane grid curves of a complex power grid in the

In this paper, a method to determine the decoupled grid curves of a power grid is proposed and studied. This method makes it possible to design and compare PCs of different types and characteristics in an arbitrary power system by comparing the analytically specified PC characteristics and the grid characteristics obtained by power-flow calculations geometrically in the plane. The IPS method is not biased in any direction with respect to the reactive power properties of the PC and it is not restricted to small angles. The IPS method is thus suitable at an early stage in the design process when the type and characteristics of the PC are still unspecified. The proposed method plots the grid curves of an arbitrary power system by inserting an ideal symmetric PST (without leakage reactance and with a unity voltage ratio) in the power system model at the position where the PC is planned to be lois varied to cover cated. The angle shift of the ideal PST the entire relevant region of angle shifts, and the power flows in the system at each angle are calculated. This process is repeated for the undisturbed system and for all contingency configurations of interest for the relevant load-generation situations of the grid. Commonly, it is sufficient to study typical load situations which result in maximum and minimum load of the power corridor where the PC is to be installed. The resulting active power flow of the PC line is plotted as a function of the angle shift for each case. This yields grid curves similar to the lines described by (1). In this paper, the power flows were calculated by the method of Newton–Raphson [21] and the ideal PST was modeled according to [21]. In order for the algorithm to be numerically stable, it was necessary to assign a very small positive value to the reactance of the ideal PST. Now, to determine the working area of a PC in this power system, the PC characteristics, which are analytically specified according to the PFC method, are plotted in the same graph as the grid curves. In this process, no small angle assumptions are made. However, the terminal voltages of the PC, which are unknown, are approximated to 1 p.u. This means that a PST is, for example, modeled by the first part of (2), assuming that its and are equal to 1 p.u. If the terminal terminal voltages voltages would show large variations (e.g., larger than 0.02–0.04 p.u.) in a real case, additional reactive compensation would be required. Thus, the IPS method can be said to yield the PC capability assuming that the terminal voltages of the PC are kept close to the nominal values. In Fig. 4, the principle of dimensioning PCs by the IPS method is illustrated. The grid curves of two grid scenarios: Case #1 and Case #2, are plotted in the graph. Each scenario corresponds to a transmission-line configuration and a certain


Fig. 4. Dimensioning of a PC in the P

0

plane by the IPS method.

load-generation characteristic. For the case with no PC, the power on the PC line can, in each case, be found from the crossing point of the grid curve and the -axis as indicated by the ”No PC”-markers in the figure. It is now assumed that the desired power flow on the line (which may be dictated by security or economical constraints) in each case is known and indicated in Fig. 4 by the ”PF target”-markers. Now, in order to find a PC which is capable of controlling the power flow to the target values in both grid scenarios, the PC curves of the considered devices are plotted in the same figure. In this example, two alternative types of PCs are considered, namely the PST and the TSSC. Two different PST designs with different maximum angle and with difshift capabilities ferent leakage reactances (at a maximum advancing angle shift), and are studied. Here, and are considered. Regarding the design of plane, it can be noted that the crossing PSTs in the point of the PST curve with the -axis coincides with the angle shift at zero power . The crossing points of the grid curves and the PC curve of each device indicate the maximum power which can be reached by the PC in the respective grid cases. It is seen that PST1 can reach the target power flows in both grid scenarios while PST2 reaches the target in grid case #2 but fails in grid case #1. Two different TSSC designs are also considered in Fig. 4. The TSSC curves always cross origo and, as the maximum of the TSSC increases, the degree of compensation TSSC curve turns counterclockwise in the plane. Here, , where is the inductive reactance of the line where the PC is planned to be installed is the value of the TSSC reactance when all and capacitive elements are connected in series with the line. In is the case of Fig. 4, it is seen that TSSC 1 with capable of reaching the target in grid case #1 since the crossing point of the TSSC curve and the grid curve in this case indicates a maximum power flow which is higher than the target value. In grid case #2, however, TSSC 1 is not able to reach the target as the crossing point of the TSSC curve and grid curve #2 indicates a power flow which is lower than the target value. TSSC 2, on the other hand, with a degree of compensation

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, is able to reach both targets and can thus be considered to be a candidate for installation in the grid. In some cases, for example, when the PC is to be used to control the power flow on a line other than the PC line, it may to obtain information be helpful to plot the relation versus at the PC location of how an inserted angular difference affects the power flow on this other line. This information can then be used together with the regular working area in the plane to select the appropriate design. Since the IPS method introduces small errors due to the grid-PC decoupling, the method is proposed as a first step in the design process. It is advisable to check and adjust the designs obtained by the IPS method by exact power-flow calculations according to the PFC method to make sure that the requirements are met. Since power-flow calculations are very fast today, the most time-consuming task when plotting the limiting grid curves by the IPS method is probably the selection of the relevant grid scenarios and loading conditions. However, it is important to notice that this task must always be performed regardless of the method which is used to design a PC. If a conventional method for the selection of a PC is used, the operating ranges of different PC candidates in all of the relevant grid topologies and loading conditions must be calculated by power-flow studies. This is commonly an iterative process where the type of PC is first selected followed by a tuning of the necessary PC characteristics. The IPS method has the potential to simplify this process since the selection and rating of a PC, considering PCs of different types, can be accomplished quite accurately geometrically using only the power-flow data of the ideal phase shifter which define the grid curves. The decoupling of the grid and PC characteristics used in the IPS method visualizes the capability of a PC in the particular power grid in a comprehensive manner and facilitates comparison of PCs of different types. Once the grid characteristics have been plotted in the power controller plane, it is very easy to determine the impact which changes in the variables associated with the studied PC (e.g., control variables, leakage reactance, etc.) have on the power-flow control capability. The IPS method yields the performance of a PC which can be expected assuming that the voltages at the PC terminals are close to 1 p.u. If the grid suffers from poor reactive power support at these nodes, there will be differences between the working area of the PC in the power controller plane predicted by the IPS method and the exact working area obtained when the studied PC is included in the admittance matrix of the power grid. Differences seen when comparing the IPS and exact working areas may actually be regarded as an indication of voltage support problems in the grid. The advantage of the IPS method when comparing other design methods such as, for example, OPF is mainly the ability to illustrate and compare the capability of several types of PCs graphically while considering all relevant grid topologies and dispatch settings. Since no complex calculations are required once the grid characteristics are known, the comparison of different PC alternatives is straightforward and fast. The OPF, on the other hand, would have to be recalculated for every PC type and grid topology. One disadvantage of the IPS method is the need to verify the results by power-flow calculations to consider

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voltage variations at the PC terminals. This step would not be necessary with an OPF method. IV. TEST SYSTEM The test system, which is used, is a 26-bus system shown in Fig. 5. The system is a three-phase power grid with a nominal frequency of 60 Hz. The transmission lines in the system are modeled by exact -models and the generators and loads are in the power-flow calculations modeled by either PU, PQ, or slack (swing) buses according to [21]. The line transformers are assumed to be ideal. The system is characterized by its admittance matrix and the power flows in the system are calculated by the Newton–Raphson method [21]. The voltage magnitudes at the slack and PU nodes are assumed to be controlled to 1 p.u. in steady state. In this system, a power-flow bottleneck is identified in the southern part of the grid denoted “SOUTH” in Fig. 5. In this section of the grid, the lines between nodes 1–5 and 4–6 get overloaded in certain load scenarios if certain contingencies occur. To relieve these potential overloads, a PC installation (marked “PC”) is planned as indicated in Fig. 5. The location of the PC is motivated by good controllability of the power flows in the bottleneck and the relatively low-power transfer capability of the PC line compared to that of the parallel lines (which were also candidates for siting of the PC). This means that the potenscenario of an outage of the PC line will be less severe. tial Since it is not possible to consider all possible contingencies and load scenarios of the power system during the PC design, it is necessary to determine a number of contingencies and load-generation patterns which represent the possible states of system operation in an accurate manner. As a starting point, it is known that the voltage at node 3 is well controlled and that the ranges of power consumption/generation injected from the northern part of the grid into node 3 in all anticipated system configurations are known from previous operation of the system. Thus, the design study can focus on the “SOUTH” part of the grid where the power-flow bottleneck is located with the assumption that node 3 is a PU node with a given range of power injection from the “NORTH” part of the grid. The “SOUTH” part of the system in Fig. 5 includes lines on the voltage levels 230, 345, and 500 kV with line parameters 300 km, 300 given in [21]. The line lengths are km, 75 km, 75 km, 37.5 km, and 37.5 km. The 230-kV line between nodes 4 and 6, which has been uprated by reconductoring, has a loadability limit of 600 MW while the 345-kV lines 1–5, 4–5, and 2–3 have loadabilities of 1180 MW. These short lines are all limited by thermal constraints. The 345- and 500-kV lines with a length of 300 km are assumed to have loadabilities of 600 and 1400 MW, respectively. The loadabilities of these lines are determined by voltage requirements at the midpoints, nodes 7 and 8. In some cases, lines 1–5 and 4–6 are overloaded. To correctively relieve these overloads and, thus, increase the transfer capability of the system, the installation of a PC between nodes 1 and 6 in the system is considered. In the power-flow calculations, node 1 is the slack bus, 2–5 are PU buses, and nodes 6–8 are PQ buses. The PCs used in this paper are either thyristor-switched series capacitors/thyristor-switched series reactors (TSSCs/TSSRs), which are modeled as reactances inserted in the admittance

Fig. 5. The twenty-six-bus test system schematic.

matrix of the test system or PSTs which are assumed to be of a two-core symmetric type. The PST is modeled as an ideal transformer with a complex turns ratio in series with a leakage reactance as described in [21]. The leakage reactance of the device is assumed to vary as (4) and are the leakage reactances for the shunt Here, denotes the tap and series transformers, respectively, while , where denotes the maximum reratio with the range tarding angle , and 1 denotes the maximum advancing . The values 20% and angle 10% given in percent of the physical base of each transformer are used in this paper. The considered PSTs are assumed to enable 21 equidistant discrete steps of angle shift. V. RESULTS AND DISCUSSION A. Application Example In order to show the applicability of the IPS method, the test system in Fig. 5 is studied. The aim is to design a PC to achieve specific power-flow control targets in the power grid. In this system, one PST and one TSSC/TSSR are designed by the IPS method to relieve potential overloads by postfault actions. The designs are then compared with designs obtained using exact power-flow data, and the ability of the IPS method to predict the working areas of the PCs in this system is studied. The net loads in the system are at nodes 3 and 5. The load at node 3, corresponding to the power exchange with the ”NORTH” 800 MW and part of the system, is variable between 1500 MW while the load at node 5 is constant at 700 MW. The net generation in the system is at node 2 with 300 MW and at node 4 with 500 MW with the rest of the system load, including the line losses supplied from the slack bus, at node 1. The system is studied in load case and load case 2 for the relevant 1 contingencies which have been identified:


Fig. 6. The 26-bus test system: power controller plane with IPS grid curves P versus and exact power flow data in cases 1C, 1F, 2D, and 2E with no . PC installed. In this system,

=

A B C D E

all transmission lines in service; one line between nodes 1 and 2 is disconnected; one line between nodes 3 and 4 is disconnected; line between nodes 2 and 3 is disconnected; line between nodes 1 and 5 is disconnected;

F

line between nodes 4 and 5 is disconnected.

The possible outage of the PC line is a case which cannot be studied by the IPS method. By regular power-flow calculations, it was found that this scenario does not yield any line overloads with the given conditions of the grid. If the remaining 12 combinations of load- and grid configurations, denoted “1A,” “1B,” “2F,” are studied in the power controller plane, it can be seen that cases 2D and 2E lead to overloads if no PC is used. The PC is required to prevent an overload of the line between 1 and 5 in case 2D and to prevent an overload of the line 4–6 (the PC line) in case 2E. In Fig. 6, the grid curves of the cases where overloads are seen, together with the curves which form the lower boundary of plane where all grid curves are found, the region in the are plotted by the IPS method. The crossing point of each curve with the -axis predicts the power flow from node 6 to node 4 before the PC is installed. In case 2E, it is seen that the line power is 715 MW, which is above the loadability limit of 600 MW. In this case, the PC is intended to reduce the line power to 600 MW. To study the impact of a PC installed at node 6 in the system versus characteristics on the power flow on line 1–5, the of case 2D are plotted in Fig. 7. It can be seen that the power on this line when no PC is installed is 1420 MW which is 240 MW above the loadability limit for this line. In order to relieve to the loadability this overload, the PC is required to lower 1180 MW. From Fig. 7, it can be concluded that the limit PC, in this case, is required to insert a shift in the voltage phase . The target value for the PC line power angle of is found from the grid curve for case 2D in Fig. 6 at which 600 MW. yields With the limiting cases identified, different PC alternatives can be compared by plotting the considered PC device curves

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Fig. 7. The 26-bus test system: power controller plane with the grid curve P versus and exact power-flow data for the designed PCs in case 2D. In this . system,

=

given by the first parts of (2) and (3) with the grid curves shown in Fig. 6 until the designs, which meet the power-flow targets, are found. This iterative procedure yields rough ratings for the PCs. The obtained designs are then inserted in the grid model, and the ratings are adjusted by power-flow calculations so that the required performance is obtained. The IPS method predicts would that a PST with an angle-shifting capability of be capable of reaching both power-flow targets while PF calcu. It is also found by the IPS lations adjusts this value to method that a TSSC/TSSR with a maximum degree of compensation of 55% capacitive and 44% inductive can be used. PF calculations suggest a TSSC/TSSR with 56% capacitive and 41% inductive. These values are given in percent of the PC line reactance. The PC curves of the final designs are plotted together with the limiting grid curves in Fig. 8. It can be seen that the operating points of the PCs at their extreme settings, which are predicted by the crossing points of the grid and PC curves, closely agree with the results of the PF calculations done with the PCs inserted in the grid which are shown by markers in Figs. 7 and 8. The exact working areas of the PCs determined from power-flow calculations according to the PFC method are plotted together with the working areas predicted by the IPS method in Fig. 9 for the PST and in Fig. 10 for the TSSC/TSSR. The exact working areas were calculated by performing power-flow calculations for all device settings in the limiting grid cases forming the upper and lower boundary plane. For all other considered grid cases, the in the power-flow data are calculated for the extreme values of the PC settings. Only minor differences between the results from the IPS method and the exact working areas are seen. The largest deviations for both devices are seen at the highest levels of power flow in the working areas in case 2E. At these operating conditions, corresponding to the largest advancing angle for the PST and the highest level of series compensation of the TSSC, the voltages at the PC terminals deviate the most from the nominal values. The lowest voltage at the PST terminals is 0.967 p.u. in this case and the highest value recorded at the TSSC/TSSR terminals is 1.036 p.u. These values represent the largest deviations from the nominal values of the PC terminal voltages for all studied operating points.

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IPS method. Since these values are calculated from the extreme values of the power-flow and terminal voltage difference, which may occur in a postfault case before the PC has had the time to act, they refer to the short-term ratings of the PCs. Since the maximum power on the line which can be reached by the PCs in case 2E is higher than the thermal rating of the transmission line, it is necessary for the PC to promptly limit the power if this case occurs. VI. CONCLUSION

Fig. 8. The 26-bus test system: power controller plane showing the alternatives of installing a stand-alone PST or a TSSC/TSSR (denoted TSSC/R in the figure) . to solve the power-flow control problem. In this system,

=

Fig. 9. The 26-bus test system: working area of the PST estimated by the IPS method compared to the exact working area given by the PFC method. In this . system,

=

In this paper, a new method for preliminary design of PCs has been presented. The proposed method, which is denoted the IPS method, has the potential to simplify the procedure for design and selection of a PC in a complex power grid. The IPS method relies on the PFC method for plotting the PC characteristics and for verifying the PC designs by plotting their exact working areas in the power controller plane. The main contribution of the IPS method is that it provides a simple and accurate technique to plot the grid characteristics in the power controller plane which is independent of the choice of PC. It is thus possible to decouple the grid and PC characteristics in a complex grid. The advantages of using the IPS method as a first step for PC design are as follows. • The grid characteristics obtained by the method are nonbiased with respect to the reactive properties of the PC. The grid curves will thus only have to be generated once by power-flow calculations. Thereafter, a first design of the PC can be obtained by plotting the analytically specified characteristics of the considered PC candidates, which may be of different types, geometrically in the same graph without further power-flow calculations. • The decoupling of the PC and grid characteristics provides an analytical insight into how the capabilities of different types of PCs are influenced by variations in PC characteristics, grid topologies, and load-generation patterns. This facilitates the selection of the most promising type of PC and characteristics of the PC. REFERENCES

Fig. 10. The 26-bus test system: working area of the TSSC/TSSR by the IPS method compared to the exact working area given by the PFC method. The TSSC is assumed to have three series elements, yielding seven possible equidistant values of capacitive reactance. The TSSR is assumed to have two series elements, yielding three equidistant levels of inductive reactance. In this system, .

=

The power ratings of the designed PCs, predicted by the IPS method, were 344 MVA for the PST and 345 MVA/173 MVA for the TSSC/TSSR. The ratings given by the exact power-flow calculations were 350 MVA for the PST and 360 MVA/163 MVA for the TSSC/TSSR which indicates small errors of the

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New York: McGraw-

Nicklas Johansson (S’05–M’11) was born in Lulea, Sweden, in 1973. He received the M.Sc. degree in engineering physics from Uppsala University, Uppsala, Sweden, in 1998 and the Ph.D. degree in electrical engineering from KTH Royal Institute of Technology, Stockholm, Sweden, in 2011. Currently, he is a Researcher with ABB Corporate Research, Västerås, Sweden. His research is focused on the control of flexible ac transmission systems devices.

Lennart Ängquist (M’09) was born in Växjö, Sweden, in 1946. He received the M.Sc. degree from the Lund Institute of Technology, Lund, Sweden, in 1968, and the Ph.D. degree from KTH Royal Institute of Technology, Stockholm, Sweden, in 2002. He was with ABB (formerly ASEA) in various technical departments. He was working with industrial and traction motor drives in 1974–1987. Since then, he has been working with flexible ac transmission systems applications in electrical power systems. He is an Adjunct Professor with KTH Royal Institute of Technology.

Hans-Peter Nee (S’91–M’96–SM’04) was born in Västerås, Sweden, in 1963. He received the M.Sc., Licentiate, and Ph.D. degrees in electrical engineering from KTH Royal Institute of Technology, Stockholm, Sweden, in 1987, 1992, and 1996, respectively. In 1999, he was appointed Professor of Power Electronics in the Department of Electrical Machines and Power Electronics, KTH Royal Institute of Technology. His interests are power-electronic converters, semiconductor components, and control aspects of utility applications, such as flexible ac transmission systems and HVDC, and variable-speed drives.