Dragan Jovcic
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Electrical power converters have become indispensable units in modern power systems, with an increasing number of installations, application fields, and ratings. They are used for power conditioning; for controlling network, regulating generation, or load power; for integrating storage systems; and importantly for integrating most renewable energy plants. As a typical example, a modern wind generator will have two ac–dc converters back-to-back connected, facilitating optimal energy extraction from variable speed wind turbines and ensuring that the power plant meets network connection requirements.
In most cases converters are controllable, incorporating a complex control system that facilitates regulation of some variables and ensures system stability under disturbances. The inherent high controllability and increasing penetration of converters bring challenges to system designers and operators because of the impact that converters have on system stability and more generally on security of power exchange.
By its nature, converters will facilitate electrical power exchange between two different systems, and a typical example will be power conditioning between dc and three-phase ac circuits. These two electrical systems have fundamentally different electrical variables (oscillating versus nonoscillating), causing complications in joint system design and operation.
Converters are built using semiconductor power switches, typically from transistor or thyristor families. In power electronic applications, these semiconductors are exclusively used as switches, i.e., they are in either in an ON (conducting) or OFF (nonconducting) state. The switching between states occurs at high frequencies (many times per second) and can be controlled. However, the current flowthrough of a semiconductor while in the ON state cannot be controlled. The current control through a converter (at the system level) is only indirectly controlled by modulating turn ON and OFF instants. The changes in switch states are abrupt and lead to highly nonlinear waveshapes of converter variables.
The above aspects bring a range of challenges in converter modeling. They need substantially different models compared with traditional electromechanical power system elements, like transformers or rotating generators.
Power systems engineers require converter models for numerous system studies, including component dimensioning, power flow, stability, protection, or analysis of contingencies during all stages of system design and operation.
Thanks to the rapid development of computing power and software tools in the past 30 to 40 years, engineers are nowadays able to perform quite detailed and accurate converter modeling and simulation. Converter simulation in the time domain is often employed using detailed modeling on electromagnetic transient (EMT) platforms with several commercial suppliers established worldwide. These platforms perform calculations of all system variables in each step and simulation progresses step-by-step in the time domain using small steps of a few microseconds. Mathematically, the simulation methodology is simple, and it achieves good convergence and robustness. Their major advantage is in flexibility, since users can enter any circuit, but changes in topology are also allowed while the simulation is running (like switching of semiconductors). The models are accurate, flexible, with good graphical interface, and they are widely adopted in the professional community. The EMT modeling and most converter integration studies can be done by practicing engineers without advanced skills or research.
However, EMT simulation has some limitations and challenges, including:
Scientists and engineers have always strived to develop models that capture the essence, key properties, and dependencies of the system under examination, while being simple and convenient for the studies. Some ingenious scientific principles and engineering modeling methods have been recorded in history, and a few that have made notable impact in electrical power engineering will be mentioned.
Park’s transformation, introduced in late 1920s by Robert H. Park, simplified analyses of three-phase ac circuits, and in particular helped understanding ac rotating machines. This method introduces a rotating coordinate frame that eliminates time-varying parameters and translates oscillating variables into dc variables, which are more convenient for analysis. In turn, Park’s transformation builds on the more general earlier coordinate frame transformation theories, like the theory of moving observer used by Einstein in his theory of relativity.
Another very important principle is Fourier series expansion, introduced by Joseph Fourier in the 18th century, that enables representation of any signal as a sum of simple oscillating signals at a range of frequencies. This method is widely used in engineering for studying periodic signals, and is especially important with ac power circuits, where all variables are oscillating at the operating frequency (typically 50 or 60 Hz). Thanks to the Fourier series, each ac variable can be expressed as a simple sine signal at fundamental frequency and numerous other sine signals at harmonic frequencies.
The third essential method is Taylor series expansion, introduced by Brook Taylor in the 18th century. It facilitates representing a nonlinear function/signal as a sum of simple linear terms dependent only on the input variable and its derivatives. Considering the highly nonlinear nature of converter behavior, linearization plays a key role in converter studies.
This article describes analytical modeling principles for power converters. Development of a converter analytical model is generally much more challenging than EMT modeling, since it requires in-depth converter knowledge and the use of advanced mathematical concepts, like coordinate frame transformation and Fourier or Taylor series expansion. It is not routinely used in industry, although most operators and manufacturers will use some analytical models at least at some stages of substantial studies. The effort in developing an analytical model may be justified when new topology is evaluated, repeated (control) issues are observed in practice, or a highly complex system is brought for feasibility evaluation.
Analytical modeling inevitably involves simplifications, which require understanding the tradeoff between accuracy and convenience, and therefore necessitate careful verification of the model accuracy. It may not be easy to understand modeling assumptions, and some practicing engineers will have misgivings about the validity of analytical models and may be concerned about model robustness and flexibility. However, many good analytical models have been reported in the professional community for a range of converter topologies. Numerous research teams have provided invaluable insights into converter design and integration based on analytical models. In particular, analytical models facilitate parametric studies, which lead to qualitative conclusions without repetitive time-domain simulations. One practical example is a system stability study, which can be performed either in time domain by observing the evolution of system variables following a perturbation, or using parametric domain methods from control system theory, like eigenvalue study. Arguably, analytical models bring the most value with multidimensional design tasks, like tuning multivariable controllers or broad robustness studies.
This article will describe essential steps in building analytical models for few typical converter topologies, including averaging, coordinate frame transformation, linearization, and validation.
An illustration of analytical model use for stability analysis on a practical converter test system will be provided in the last section. This case study will include comparison with stability analysis based on detailed EMT modeling for the same test system to highlight differences and limitations of each method.
The first principle with converter analytical modeling is the averaging step. Because of the use of semiconductors, converter variables (voltages and currents) will have “chopped” waveforms, which are described as nonlinear and discrete in mathematical terms. The modeling problem is made worse by the fact that typically both dc and ac chopped signals are involved. Another challenge is that the converter control signal is active (affects the system) only at the switching instants and basically has no impact between turn ON and OFF. However, switchings occur at high frequencies (hundreds of hertz or a few kilohertz) compared to dominant system dynamics, which are typically restricted to a few tens of hertz. This separation of frequency ranges justifies an averaging approach.
The averaging process derives converter variables as continuous and simple signals, which nevertheless should capture well the key converter behaviors. It follows different methodology for ac and dc variables.
Also, averaging differs between converter topologies, and here it will be illustrated through example converters from two common topologies: voltage source converters (VSC) and current source converters (CSC).
Figure 1 shows a typical ac–dc VSC and its key variables. This converter enables power exchange between the dc system and three-phase ac system. A typical application would be converter-enabling interconnection of a battery with an ac grid, or one of two converters in a modern wind generator. It consists of six semiconductor valves (each may contain many semiconductors) S1–S6, an inductor on the ac side L, capacitor on the dc side C, and a controller that sends control signals to switches m1–m6. There are multiple (21 in this example) control pulses per each 50-Hz cycle, and timing of the pulse is achieved according to the desired ac voltage waveform. With VSC topology, dc voltage is constant between switchings, which is facilitated by large capacitor C on the dc side.
Figure 1. VSC with detailed and average variables. (a) VSC topology. (b) Detailed and average variables.
The actual converter ac variables VacA, VacB, VacC, IacA and dc variables Vdc and Idc are shown in blue in Figure 1. The corresponding average variables are shown in red in Figure 1, which are obtained using Fourier series expansion on the ac side and simple averaging on the dc side. The obtained average variables are much more convenient for further analysis and design, and yet they capture essential converter dynamics of interest. A designer is primarily interested in converter variables at fundamental frequency (50 Hz) on the ac side and average variables on the dc side.
The average waveforms on the ac side are obtained using Fourier series expansion and neglecting harmonics. The waveforms on the dc side are commonly obtained by equating powers on the ac and dc sides of the converter, assuming that there is no loss or storage of energy in the converter. The dc voltage can be assumed as a firm, slow-varying variable with all converters from the VSC family.
Table 1 shows the key parameters for the test system converters considered in this article.
Table 1. Key parameters of the test systems.
Figure 2 shows the test CSC and its key variables. The essential differences compared with VSC are:
Figure 2. Line commutated converter with detailed and average variables. (a) Line commutated converter topology. (b) Detailed and average variables.
The actual converter ac variables VacA, VacB, VacC, IacA and dc variables Vdc and Idc are shown in blue in Figure 2. The corresponding average variables are shown in red in Figure 2, which are obtained using Fourier series expansion on the ac side and equating powers for dc side variables.
The dc current can be assumed as a firm, slow-varying variable with all converters from the CSC family.
Oscillating variables are difficult for analysis, as it is awkward to observe and measure changes in such variables. Also, controllers need feedback measurements of system variables, but they have difficulties in processing oscillating signals. The theory of Park’s transformation gives elegant proof that we can use few dc variables to analyze an oscillating signal if our new coordinate frame rotates at the same frequency of the oscillating signal. In practical terms, this means that the observer is positioned to oscillate in synchronism with the system variables at the operating frequency. The key assumptions for the use of Park’s transformation are:
These conditions are usually satisfied with ac transmission/distribution systems, and therefore this transformation is widely applied.
An oscillating signal is characterized by three variables—frequency, magnitude, and phase angle—and if frequency is eliminated (it is not changing), then only two variables are adequate. They can be represented in one of the following two ways: using a polar coordinate frame (magnitude and angle) or rectangular coordinate frame (d and q components).
Figure 3 shows the VSC converter phase A ac voltage in both a static ABC frame and in rotating dq coordinate frames. To illustrate correlation between these two coordinate frames, two step changes are applied:
Figure 3. VSC converter phase A voltage in static (a) and rotating (b) coordinate frames.
In the rotating dq frame, the same signal is illustrated using polar (VacAm, FiVacA) and orthogonal components (VacAd, VacAq) for completeness. It is seen that in the dq frame the signal change is readily observed for both excitations. However, the permanently oscillating signal nature clutters the picture and obscures important information in the static ABC frame. The converter controller establishes the position of the rotating coordinate frame by measuring some reference voltage (usually grid connection point), as shown by the signal VacAref in the ABC frame.
The detailed EMT model and the physical system will have all ac variables as oscillating signals. However, Park’s transformation is commonly implemented in practice for online measurement of various ac variables. These dq frame signals are used for monitoring and also as feedback signals in controllers.
The analytical model has no oscillating signals. It represents all internal dynamics and dependencies using dq ac signals and dc signals only. If required, the analytical model can also produce ABC frame oscillating signals by applying an inverse Park’s transformation on the obtained dq frame variables.
Averaged models for most converter topologies will have some nonlinear elements: as an example, multiplication of two signals. Such a nonlinear model can be used for time-domain simulation, for power flow study, component dimensioning, and with some nonlinear controllers.
However, most stability analysis and controller design theories require linear models, and typically they should be represented in state-space format. A linearized model is obtained by applying Taylor series expansion, or similar methods like describing function, to each of the identified nonlinear elements. A very important assumption with linearization is that variables do not deviate much around the steady-state operating point. This is an important factor that has an impact on accuracy and requires careful consideration; the linearized model is also commonly called a small-signal model. The same model may not be valid for large disturbances, like faults.
The model is commonly converted to a specific format, like state-space, depending on the intended use. State-space is the widely adopted model representation in the control engineering field.
The average value model always has a cloud of uncertainty related to accuracy hanging over it. The problems with accuracy may arise because of the overly liberal use of simplifications, contravention of some assumptions with transformations and methods, linearization assumptions, or simply because of errors caused by manual modeling associated with system complexity.
Model accuracy should be examined by comparing responses against a detailed model of known accuracy and for the same excitation. Ideally, a broad range of signals should be observed (dc side, ac side, control signals) to confirm accuracy of all subunits, and testing should be repeated for different excitations.
Figure 4 shows comparisons of dc variables (dc voltage Vdc) and an ac variable (d component of ac current Iacd) between a detailed and average model, for a 5% step increase in power transfer order. We can conclude that the average model has good accuracy based on two observations:
Figure 4. Accuracy verification of average model.
Figure 4 strengthens arguments on the benefits of an average model considering clarity of communicating dynamic phenomena. It is much easier to evaluate dynamic response (like, for example, percentage overshoot) using an average model. The detailed model variables contain lots of noise caused by converter switchings and feedback control amplification.
In electrical power systems there are multiple forms of stability, but preeminent is small signal stability. It describes system response to small perturbations in steady-state operation. In practice, perturbations occur continuously: as an example, small changes in customer circuits, like loading demand or circuit topology (connection/disconnection). A system is stable if all variables remain bounded following a disturbance, while it is more desired that a system is asymptotically stable where all variables are bounded but also converge to a new steady state.
Robust stability is of particular importance but may be challenging in a general case. In practice, a system should be stable when operating under different parameters: as an example, different network characteristics or controller parameters. An optimally tuned controller may turn out to bring disappointment to a system designer when instability is observed under a different but plausible operating scenario.
A system designer would normally first provide a list of all parameters that are expected to vary, and the range of variation for each. Guaranteeing robust stability is difficult, as the number of possible scenarios could be large and dependencies between parameters could be highly nonlinear and unpredictable.
One of the main reasons for using analytical models is multivariable analysis of robustness of small-signal stability. A study of robustness can be tedious with detailed time-domain simulation. Analytical models, however, facilitate parametric stability study, which can be very effective for robustness analysis, as will be elucidated below.
Figure 5 shows a stability analysis case with the detailed EMT model of an HVdc system consisting of two VSC converters, as given in Table 1. The system operates normally at rated power in a stable manner until 1 s. At 1 s controller gain is increased 10 times and it can be observed that the system becomes unstable. Both the dc variable (dc voltage Vdc) and ac variable (d component of ac current Iacd) show oscillatory instability. In practice, such a large variation in variables would be detected by converter self-protection and the converter would be tripped, resulting in loss in capacity. This study enables us to estimate frequency of instability at around 78 Hz, since 7.8 full cycles can be counted in any .1-s interval.
Figure 5. Simulation of instability on detailed model. Controller gain increased 10 times at 1 s.
If a designer is interested in stability at another gain value (say K = 4) they would need to adjust the gain and repeat EMT simulation through the time interval from the beginning until the system settles in the steady state. Unfortunately, each new run demands model preparation by the software platform, which involves building and conditioning model matrices. In the case that new gain value gives stable response, the difference in the time-domain response may not always become readily perceptible.
Figure 6 shows stability analysis for the same system as in the section above, but using an analytical model. The location of system eigenvalues is shown in complex plane with real and imaginary axes. Eigenvalues can be calculated routinely using an analytical state-space model and they provide indication of system stability. To achieve stable operation, all system eigenvalues should be located in the left-half-plane (negative real component). Eigenvalues closest to the imaginary axis (closest to instability boundary) have a dominant impact on the system response. The imaginary component of eigenvalues indicates frequency of the associated oscillating mode in radians per second.
Figure 6. Instability analysis on average linear state-space model. (a) Original system eigenvalues. (b) Eigenvalue location as controller gain increases. Im: imaginary; Re: real.
As seen in Figure 6(a), the original system has all eigenvalues in the stable region, which is consistent with the conclusions of stable operation obtained with the detailed EMT model. There are in total 56 eigenvalues for this system, but only a subset of eigenvalues closest to the imaginary axis is shown. In addition to stability conclusion, a designer engineer can exploit rich control theory to analyze system performance and robustness using eigenvalues. To this end, eigenvalue sensitivity and participation factor analysis would normally be performed.
In Figure 6(b), an illustration of parametric stability analysis is given. The controller multiplying gain K is varied in the range 1 < K < 10 and the position of eigenvalues is recorded. The original position of eigenvalues (K = 1) is indicated with blue circles, the final position (K = 10) with red triangles, and the intermediate positions with black crosses [Figure 6(b)]. It is seen that by increasing gain, the position of most eigenvalues deteriorates (they are moving toward the imaginary axis). Surprisingly however, it is not the dominant eigenvalues that are mostly sensitive to the gain variation, as another pair of eigenvalues first crosses the stability boundary at approximately K = 9. We know that this conclusion is correct, as both the gain value and frequency of instability are consistent with the simulations on the detailed EMT model shown in Figure 5. On the imaginary axis we can read the frequency of the unstable eigenvalues as around 490 rad/s [or 490/(2${\pi}$) = 78 Hz], which corresponds well with the frequency observed using the detailed EMT model.
The parametric study in Figure 6(b) can usually be performed fast, even with complex systems containing numerous eigenvalues and many gain values. This study also provides a full stability picture, showing all eigenvalues, with stability margin and frequency for each eigenvalue. In more advanced design tasks, multiple controller gains can be varied simultaneously to support multidimensional decision making.
Analytical converter modeling complements detailed digital simulation and may expedite system studies. It usually involves multiple development steps, including averaging of all variables, conversion to rotating coordinate frame, linearization, model format conditioning, and verification.
Analytical models facilitate parametric studies, which are not possible with detailed digital models based on time-domain simulation. A good example of model use is the eigenvalue analysis, which provides expedient and comprehensive stability examination.
Work is ongoing on automating the process of developing analytical converter models, and on connecting digital simulation with analytical models.
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D. Jovcic, N. Pahalawaththa, and M. Zavahir, “Novel current controller design for elimination of dominant oscillatory mode on an HVDC line,” IEEE Trans. Power Del., vol. 14, no. 2, pp. 543–548, Apr. 1999, doi: 10.1109/61.754101.
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Dragan Jovcic (d.jovcic@abdn.ac.uk) is professor and director of the Aberdeen HVDC (High Voltage Direct Current) Research Centre, University of Aberdeen, AB24 3FX Aberdeen, U.K.
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