Analysis and simulation of a MMCC-SSBC converter

End of Master Degree Project Master in Energy Engineering Analysis and simulation of a MMCC-SSBC converter Author: Sander Claeys Director: Dr. Francisco Diaz Gonzalez Escola Tècnica Superior d’Enginyeria Industrial de Barcelona Abstract Battery energy storage systems (BESS) are the most versatile type of energy storage. With an increasing share of renewable energy, they could prove to be essential to provide the much needed flexibility. The MMCC-SSBC might be the most suitable converter for modern BESS. It is modular, and allows for an individualized treatment of the connected battery modules. The main objective of this thesis is to develop a tool which simulates the behavior of a MMCC-SSBC converter. This objective is fulfilled by the core deliverable of the thesis: a Matlab implementation of a dynamic model of the converter. As a secondary objective, this thesis aims to demonstrate the usefulness of this tool. It applies the tool to a specific use case, and analyzes three key characteristics based on the simulations: efficiency, power quality and reliability. This leads to some concise design guidelines, and continued operation under a battery short-circuit fault. 1 Contents 1 Introduction 4 1.1 Context and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Objectivesandscope............................ 5 1.3 Outline.................................... 6 2 Literature review 7 2.1 Currentcontrol ............................... 8 2.2 Modulation ................................. 9 2.3 Reliability.................................. 10 2.4 Lossmodeling................................ 11 3 Methodology 14 4 Modeling and control 18 4.1 Interfaceandcontrol ............................ 18 4.1.1 Electrical equations governing the system . . . . . . . . . . . . . 19 4.1.2 Control loop regulating the power flow . . . . . . . . . . . . . . 21 4.2 Converteroperation............................. 25 4.2.1 Converter hardware . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.2 Modulation ............................. 27 4.3 Lossmodel ................................. 32 4.3.1 Modeling the semi-conductor losses . . . . . . . . . . . . . . . . 33 4.3.2 Event-based model . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 Case study 37 5.1 Data..................................... 37 5.1.1 Converterratings.......................... 37 5.1.2 Semiconductor components . . . . . . . . . . . . . . . . . . . . 39 2 Analysis and simulation of a SSBC-MMCC converter 3 5.2 Simulationtool ............................... 42 5.2.1 Simulation framework . . . . . . . . . . . . . . . . . . . . . . . 42 5.2.2 Basecase .............................. 44 5.2.3 Initialresults ............................ 45 5.3 Powerlossanalysis ............................. 48 5.3.1 Analyticformulas.......................... 48 5.3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3.3 Efficiency .............................. 53 5.4 Powerquality ................................ 55 5.4.1 Fourieranalysis........................... 55 5.4.2 Regulation and standards . . . . . . . . . . . . . . . . . . . . . 56 5.4.3 Analysis of base case . . . . . . . . . . . . . . . . . . . . . . . . 57 5.4.4 Variation under key parameters . . . . . . . . . . . . . . . . . . 58 5.5 Reliability.................................. 64 5.5.1 Requirements for continued operation . . . . . . . . . . . . . . . 64 5.5.2 Simulation of a battery short-circuit . . . . . . . . . . . . . . . 66 6 Conclusion 68 7 Acknowledgement 71 A Datasheets 77 A.1 IGBT module FS100R17KE3 . . . . . . . . . . . . . . . . . . . . . . . 77 B Matlab script 84 C Simulink model 87 D dq0 transform 95 Chapter 1 Introduction 1.1 Context and motivation Nowadays, climate change and it’s human cause is an established scientific fact. In the words of the intergovernmental panel on climate change: The SYR confirms that human influence on the climate system is clear and growing, with impacts observed across all continents and oceans. Many of the observed changes since the 1950s are unprecedented over decades to millennia. The IPCC is now 95 percent certain that humans are the main cause of current global warming. [1] Emissions of greenhouse gases have reached an unprecedented height [1]. This led to several international initiatives to tackle greenhouse gas emissions. In 2007, the European Union (EU) agreed on a climate and energy package, setting specific targets for 2020. The three key targets are a 20% cut in greenhouse gas emissions, a 20% of EU energy from renewables and a 20% improvement in energy efficiency. [2]. By enacting this package, the EU strives to fight climate change, create green jobs and secure energy supplies. Additionally, the EU launched the Horizon 2020 programme. It is the biggest EU Research and Innovation programme, containing €80 billion of funding. With this, the EU seeks to drive economic growth and create jobs. [3] More flexibility is essential to accommodate renewable generation in the energy mix. Flexibility is not new; flexible generators already allow the system to balance the supply and demand [4]. Traditionally, flexibility is provided by the supply side [5]. But recently, more and more renewables entered the energy mix. The EU for example committed to increasing the share of renewables to 20% by 2020 [2]. Variable 4 Analysis and simulation of a SSBC-MMCC converter 5 renewable energy sources (RES) cannot provide supply side flexibility due to their lack of controllability [5]. On the contrary, they heighten the need for flexibility due to their volatility. Signs of an inflexible power system include difficulty balancing supply and demand, renewable energy curtailments, price volatility and negative market prices [4]. One promising source of flexibility is energy storage. Generic energy storage could provide several services. They can be classified in two broad categories according to duration. Short-duration services include frequency control and system stability services, whilst long-duration services include energy management and power reserves. Which functions are more prominent for a given storage type, depend on its key characteristics: response time, capacity both in terms of power and energy etc. Battery energy storage systems (BESS) are the most versatile type of energy storage. They can do peak shaving, defer investments in transmission and distribution, regulate voltage, replace spinning reserve and many more [6,7]. A common value proposition is using BESS in conjunction with a renewable energy source. For example, Prompinit and Khomfoi use a BESS to reduce the volatility of PV generation in a microgrid [8]. Battery modules and a power converter are the main components of a BESS. The power converter forms the interface between the battery modules and the electrical grid. Traditional BESS use a multipulse converter together with a complicated zigzag transformer [9]. This transformer has several disadvantages: it is expensive, bulky and likely to fail [10]. Instead, modern BESS use a multilevel converter. Maharjan et al. developed a BESS which uses a MMCC-SSBC converter [10]. They further claim that the MMCC family of topologies might be the most suitables ones for modern BESS. These topologies are modular, and allow for individualized treatment of the connected battery modules. These are desirable characteristics. 1.2 Objectives and scope The main objective of this thesis is to develop a tool which simulates a MMCC-SSBC converter. Such a tool is crucial for the design and analysis of a BESS based on this type of converter. Therefore, as a secondary objective, this thesis attempts to demonstrate the potential of this tool. Based on simulations carried out by the tool, the thesis will analyze key characteristics of the converter and provide design recommendations. A functional converter is a complex device, with several levels of control. For each of these levels of control, there are multiple methods available in the literature. Therefore, Analysis and simulation of a SSBC-MMCC converter 6 the scope for each of these individual levels is narrow. This thesis chooses the most conventional and straightforward control methods rather than the very latest experimental ones. Specifically, rotating frame PI-control and sub-harmonic unipolar PWM are implemented. The merit of this thesis will not be to advance one of those levels of control, but to bring all those levels together in one simulation tool. Furthermore, this thesis considers only the converter itself. It models the grid as an infinite bus and the batteries as ideal voltage sources. 1.3 Outline This thesis contains 7 chapters and several appendices. The current chapter, the introduction, starts of by showing the reader the motivation for this research. It continues with defining the research objective and the scope of it. Finally, this section clarifies the structure of the document. The upcoming chapters follow roughly the flow of the actual research. First, chapter 2 reviews the literature and identifies the state-of-art, relevant to the construction of the simulation tool. It covers a wide range of literature. Next, Chapter 3 clarifies the methodology. It shows in detail all the steps of the research, and how they are connected. Chapter 4 and 5 contain the core of the work. Based on the literature, chapter 4 develops in-depth a model of the SSBC-MMCC converter. Chapter 5 describes the developed tool and applies it to a case study. The analysis discusses three key characteristics: efficiency, power quality and reliability. Finally, chapter 6 concludes the thesis with a summary of the obtained results and suggestions for further research. Chapter 7 expresses the gratitude of the author of this thesis towards the people who enabled this work. Chapter 2 Literature review In the past, researchers have proposed several names for the MMCC-SSBC converter. These names reflect some of the key characteristics. Akagi studied the history of the converter, and proposed a naming convention and classification based upon it [11]. The converter is both modular and cascaded. The converter wires together several H-bridge or chopper cells, which are the building blocks of the converter. Cells are connected in series to create the multilevel output [12]. Hence, the converter consists of groups of cascaded modules, hereafter referred to as clusters. This justifies the first half of the name: modular multilevel cascade converter (MMCC). Two things remain undefined: the topology and cell type. Firstly, Fig. 2.1 shows four variations of the topology. Secondly, the cell type is uniform across the converter, and is either a chopper cell (CC) or an H-bridge cell (BC). This leads to the second half of the name: single star bridge cell (SSBC). Ota et al. recommend the MMCC-SSBC converter for use in a BESS. It provides a cost-effective, practical and flexible solution [13]. Maharjan et al. researched this setup extensively, validating the results with an experimental verification [10, 14–16]. These studies provide a high-level discussion, assuming a background knowledge on several topics such as current control, pulse width modulation (PWM) techniques and many more. Knowledge on all of these topics is required to build a fully functional simulation tool. Therefore, the remainder of this chapter will discuss the state-of-art regarding several of these topics. 7 Analysis and simulation of a SSBC-MMCC converter 8 (a) SS (b) SD (c) DS (d) DD Figure 2.1: The topology is characterized by two letters. The first letter is either S or D, which stands for single and double. This refers to the number of clusters per phase. The second letter is also either S or D, but stands for star and delta instead. This refers to how the clusters are interconnected. 2.1 Current control Bahrani et al. discuss the methods which have been developed for current control [17]. Over the years, several controllers emerged. They can be classified in two main groups: linear and non-linear controllers. The non-linear techniques add additional complexity, but without a significant improvement in performance. Therefore, linear controllers are the most popular. A further distinction is based on the used reference frame. The linear controllers can be applied either directly or after transforming the signals to the rotating frame. Bahrani et al. state that the most well-known control method is a PI-controller applied in the rotating frame [17]. Schauder et al. proposed this method more than two decades ago [18]. This method is easy to implement and provides a satisfactory performance. This thesis will adopt the conventional rotating frame PI-controller. Bahrani et al. recognize its satisfactory performance, but also mention some disadvantages [17]. In the rotating frame, the current has two components, direct and quadrature. The conventional approach decouples these two components and applies a separate PIcontroller to each component, ignoring the cross-coupling. In a real system however, some cross-coupling remains. The decoupling is based on the system parameters, so deviations in these parameters affect the cross-coupling. Furthermore, disturbances in one component also affect the other control loop. For these reasons, Bahrani et al. propose two modified controllers to tackle these Analysis and simulation of a SSBC-MMCC converter 15 The full model covers many topics in the literature. In order to reduce the complexity, this thesis separates the model in three interacting layers. These three layers are: interface and control, converter operation and loss model. Each layer is developed on its own. Within a given layer, the operation of the layer below is idealized with a set of assumptions. The layer itself is designed in order to achieve as closely as possible the assumptions of the layer above. By doing so, the model can temporarily ignore the underlying complexity and focus on the complexity in a specific layer. This layered approach facilitates significantly the Simulink implementation. The Simulink implementation is interwoven with the model development. The layered structure of the model transfers to the Simulink implementation. First, the model of the top layer is developed and directly implemented in Simulink. Directly implementing the model deepens the understanding of it and points out potential shortcomings. If the underlying layer is needed in order to test the implementation, it can be replaced with a simplified version. For example, when implementing the interface and control layer, the converter was replaced by a controllable voltage source. This approach facilitates the debugging of the implementation, as the complexity of each implementation round is significantly reduced. This approach is repeated for each subsequent layer. A base case offers a starting point for the simulation. Most of the parameters depend on the application. These parameters include the rated power, battery module voltage and cascade number. Furthermore, short circuit requirements are provided by the operator of the grid to which the converter is connected. All these are set to typical values, common for converters used in power grids. The main parameters to be sized, are the specifics of the filter. In the base case, the short circuit requirements lead to a minimal sizing of the filter. The effect on the harmonics is neglected in the base case. With all parameters set to specific values, the tool simulates the operation of the converter. Running the simulation leads to a set of raw output data. In order to draw conclusions, several Matlab scripts convert the raw data to a more presentable form. Most importantly, the scripts convert the waveforms to the frequency domain, exposing the harmonic components. The base case does not consider the power quality of the output waveform in the initial sizing of the filter. Therefore, it is crucial to verify whether the harmonics stay within reasonable limits. Conceptually, the impact of the filter on the harmonics is assessed. Based on reasoning, the necessary sizing of the filter is predicted. Fig. 3.1 indicates this check with the purple box. If the power quality is Analysis and simulation of a SSBC-MMCC converter 16 unacceptable, the chart flows back to the sizing step. This is iterated until the power quality falls within the required limits. The steps described by Fig. 3.1 will fulfill the objectives of this thesis. The first three steps, literature review, model development and Simulink implementation, lead to a tool with and underlying model, based on relevant literature. The fourth step, case study, shows the usefulness of this tool. The case study uses the tool to size the filter in order to keep the harmonics within acceptable limits. Analysis and simulation of a SSBC-MMCC converter 17 Case study Sizing Simulation Post-processing Short circuit requirements Input data Literature review START STOP Power quality? 1. Interface and control 2. Converter operation 3. Loss model Model development 1. Interface and control 2. Converter operation 3. Loss model Simulink implementation Resize filter Figure 3.1: The thesis goes through 4 steps in order to achieve the objectives. The arrows indicate the interaction and flow between the steps. Chapter 4 Modeling and control With only three cascade levels, the SSBC-MMCC converter already contains 9 batteries, 36 IGBTs and 36 diodes. Proper grouping and simplification of sub-modules is necessary to properly understand the converter. This thesis divides the model in three layers. Each layer uses an idealized representation of the layer below and strives to approximate the assumptions made by the level above. Fig. 4.1 shows a schematic overview of the layers and their interaction. Some topics, such as harmonics reduction, are related to several layers. But for the purpose of conceptual understanding, this subdivision is useful. For example, the top layer, interface and control, assumes the converter can instantly achieve an arbitrary voltage at its output. Based on this assumption, the top layer then implements a control loop which achieves the desired power output. The control loop generates a reference voltage for the converter, which is passed on to the middle layer (converter operation). The middle layer operates the converter in such a way that it approximately achieves the reference voltage. This chapter is structured according to these three layers. Each of the following sections discusses a layer in detail, and mentions how it fits in the bigger picture. The order is top-to-bottom, in order to keep the reader engaged. The layer above states the purpose for the layer below through the assumptions it makes about it. 4.1 Interface and control The top layer of the model regulates the power flow to the grid. It assumes that the converter can be represented by an ideal voltage source. Taking into account the parameters of the filter, the top layer implements a negative feedback control loop. The 18 Analysis and simulation of a SSBC-MMCC converter 19 Interface and control Converter operation Assumes: converter is a VSC Sets: voltage setpoint Loss model Assumes: ideal switches Sets: switch states Calculates: switching and conduction losses Figure 4.1: The thesis divides the model in three layers: interface and control, converter operation and real switches. Each layer assumes ideal operation of the layer below, and set the operation point for it. control system adjusts the voltage of the VSC to obtain a specified power output. The power output is measured, and fed back through a control loop to adjust the voltage setting. Fig. 4.2 shows a diagram of the top layer system. This section is organized in two subsections. The first subsection will derive the dynamic equations governing the electrical circuit. These equations are transformed to a more convenient form thereafter. The second subsection discusses the design of a control system to achieve the desired power output. 4.1.1 Electrical equations governing the system Fig. 4.3 shows the electrical diagram of the system. Applying Kirchhoff’s voltage law and the transfer characteristics of an inductor and a resistor, this leads to the set of equations shown by (4.1).      vCu vCv vCw     −     vSu vSv vSw      =Ld dt      iu iv iw      +R     iu iv iw      (4.1) Alternative method for reverse recovery losses if not given directly in the datasheet, but in terms of reverse recovery charge and current slope Analysis and simulation of a SSBC-MMCC converter 20 V V V A A A + - + - + - RL Grid Ammeter Filter VSC CONTROL Voltmeter Figure 4.2: The control system measures both the voltage and current at the point of common coupling, at the grid tie. By doing so, the control system calculates the immediate power flow to the grid and sets the VSC accordingly. + - vSu vCu iuLR Figure 4.3: This diagram shows the electrical variables of the u-phase of the complete system displayed by Fig. 4.2. The other phases have an identical layout, the variable names are only changed according to the phase. Fig. 4.2 shows clearly that there are two nodes in the system: the neutral point of the grid and the neutral point of the VSC. Applying Kirchoff’s current law gives only one linearly independent equation, shown by (4.2). The sum of all phase currents is equal to zero, since there is no return wire. iu+iv+iw= 0 (4.2) Equations (4.1) and (4.2) can be combined in an elegant way by transforming equation (4.1) to the dq0 reference frame. The full derivation is left for Appendix D. The set of equations (4.3) show the final result of the transformation. Analysis and simulation of a SSBC-MMCC converter 21      vCd −vSd vCq −vSq vC0−vS0      =     L     0−ω0 ω0 0 0 0 1      +R          id iq i0      +Ld dt      id iq i0      (4.3) ⇒ vCd −vSd vCq −vSq  = L 0−ω0 ω0 0 +R  id iq +Ld dt  id iq (4.4) Since the sum of the phase currents is equal to zero, this implies that the 0component current is also zero. Applying this to the set of equations (4.3), eliminates the last equation. This means that the 4 equations shown by (4.1) and (4.2) have now been reduced to only the 2 equations (4.4), containing the same information. This will significantly simplify the design of the controller. 4.1.2 Control loop regulating the power flow The goal of the control loop is to govern the power flow to the grid. The instantaneous active power flow depends algebraically on the line-to-line voltage and current at the grid tie. The line-to-line voltage at the grid tie is external to the control system. Therefore, controlling the power injected in the grid is equivalent to controlling the current injected into it. In the dq0 reference frame, the active and reactive power are defined as shown by (4.5). The classical definitions for active and reactive power are only defined for symmetrical, perfectly sinusoidal waveforms. A power invariant dq0 transform gives the same value as the classical definitions under these specific conditions. If the transform is not invariant for power, then the values will differ by a constant factor. Inverting the equation (4.5) gives an expression for the current as a function of power and voltage (4.6). [39]  p q = vduq uq−ud  id iq (4.5) ⇔ id iq =1 v2 d+v2 q vduq uq−ud  p q (4.6) Through (4.6), a set point for the power corresponds to a value for idand iq. The Analysis and simulation of a SSBC-MMCC converter 22 current is related to the voltage at the output of the converter, as shown by equations (4.4). The current control loop will adapt the converter output voltage vCin order to obtain the current set point. There are several control techniques available to achieve this. The studies conducted by Bahrani et al. [17] [19] describe state-of-the-art control techniques. These control techniques provide superior axis-decoupling and are less sensitive to deviations in the model parameters. Studying these is beyond the scope of this thesis. Maharjan et al. develop a control system for an eneergy storage specifically, and suffice with a classical PI-controller [14] [10]. This thesis adopts the same high-level control strategy. Equations (4.7) and (4.8) show the equivalent of equations (4.4) in the Laplace domain. The first term on the left-hand side introduces cross-coupling between the two equations. VCd −VSd +ωLIq= (pL +R)Id(4.7) VCq −VSq −ωLId | {z } coupling = (pL +R)Iq(4.8) Intermediate variables udand uqare introduced in order to decouple the equations. By setting these variables equal to the left-hand side of equations (4.7) and (4.8), both equations are decoupled in terms of uand i. equations (4.9) show the definition of u. Fig. 4.4 shows how the inside of the control block in Fig. 4.2 is implemented. The decoupling block shows the implementation of equations (4.9). ud=vCd −vSd +ωLiq uq=vCq −vSq −ωLid⇔vCd =ud+vSd −ωLiq vCq =uq+vSq +ωLid (4.9) From the perspective of the PI-controller, this greatly simplifies the complexity. The transfer function from udto idis now simply given by equation (4.10). The same applies for uq, with the appropriate variables. Fig. 4.5 shows the block diagram of the control system. The electrical system together with the decoupling block are now contained in the Gsblock. (4.11) shows the two standard forms of the transfer function of the PI-controller. Id Ud = 1 R 1 + pL R (4.10) Analysis and simulation of a SSBC-MMCC converter 23 current reference PI PI decoupling Figure 4.4: This figure shows the implementation of the inside of the control block in Fig. 4.2. The blue signals on the left are measurements taken in the electrical system, and the blue signals on the right go to actuators. The green signal is set as desired by a higher-level controller. 1 + pTn pTi =Kp+Kp 1 p Kp=Tn Ti Ki=1 Ti (4.11) The zero of the PI-controller is set equal to the pole of the transfer function of the system. This eliminates the pole in the open-loop transfer function as shown by (4.12). Go(p) = GP I (p)·Gs(p) = 1 pTiRTn=L R(4.12) The closed-loop transfer function is given by (4.13). The closed-loop system is a first-order system. The dynamic response of such a system is well-known. A step change in the input will cause the output to follow, where the error will decay exponentially. The exponential decay of the error is characterized by the time constant τ. Analysis and simulation of a SSBC-MMCC converter 24 Figure 4.5: The system block Gsrepresents the dynamics of udto id; it includes the decoupling block and the electrical system. Exactly the same block diagram applies for the q-quantities, with appropriate names. Gc(p) = 1 1 + 1 Go(p) =1 1 + pTiRτ=TiR(4.13) By eliminating the pole of the system, the PI-controller has one remaining degree of freedom, Ti. This parameter directly affects the time constant of the closed-loop system. It can be set to obtain a desired response to a step in the input signal. In this context, it is important to keep in mind that the analysis so far regarded the converter as in deal voltage source. In reality, the converter is a non-linear device which introduces a delay in the loop. This effect becomes significant for small time constants. Analysis and simulation of a SSBC-MMCC converter 31 (a) Output waveform of the first cell (b) Aggregate waveform of the whole cluster, with the reference signal Figure 4.10: The waveform of the entire cluster is the aggregate of the waveforms of the individual cells. This Figure shows the waveforms for a cluster with three cells, N= 3, and a specific ratio for the periods, Tp Tsw = 20. The converter is slightly oversized; the range of the converter is larger than the range of the reference signal. Analysis and simulation of a SSBC-MMCC converter 32 4.3 Loss model Up until this point, there are no losses present in the model. The top and middle layer control the converter, and do not consider the losses. But the losses are a very important aspect, justifying choosing one converter over another. This section describes the bottom layer, the loss model. The losses are caused by the switches. There are two main types of losses: conduction losses and switching losses [38]. When real switches are closed, they cause a small voltage drop across their terminals. By conducting current at the same time, they consume an amount power which leaves the converter as heat. These are the conduction losses. The switching losses are a consequence of the finite switching time of real switches. When the switch changes state, it needs a certain amount of time to realize the change of state. For example, when the switch closes, the voltage across it has to drop and the current through it increases. These two changes happen at the same, and incur a loss, the switching loss. Fig. 4.12 shows the switch models used in this thesis. The first two layers use ideal switches, shown by Fig. 4.12a. In reality, a switch consists of an IGBT in parallel with a freewheeling diode, shown by Fig. 4.12b. The IGBT takes care of the actual switching, whilst the diodes are necessary because of the finite switching time. The dynamics of both semiconductor devices will determine the losses incurred during switching, whilst the conduction losses are a function of the transfer characteristics. Figure 4.11: The transient dynamics of the switches take place in a much smaller timescale Ttr than the switching period Tsw. In order to capture the transient dynamics, the solver has to take substantially smaller time steps as indicated by the dashed arrow. The dynamic behaviour of the IGBT and diode determine the switching losses. The time required to change the state of a switch is typically in the order of 1µs, whilst the switching period is typically three orders of magnitude larger. As shown by Fig. 4.11, the time step should be lowered substantially in order to capture the transient dynamics of the switches. This would result in a substantially slower simulation. This thesis therefore opts for an event-based model of the switches. Furthermore, the datasheets characterizing the devices are not detailed enough to provide an accurate dynamic model. Implementing the computational costly model doesn’t guarantee a more accurate result. Analysis and simulation of a SSBC-MMCC converter 33 This section contains two subsections. The first subsection will describe the dynamics of the semiconductor switch. Understanding the dynamics is essential to develop an accurate event-based model, replacing the direct simulation of the dynamics. The second subsection deals with the event-based model itself. (a) Ideal switch (b) IGBT and diode (c) IGBT model Figure 4.12: The first two layers of the model use an ideal switch. In reality, the switch consists of an IGBT and a diode. The conduction losses are included in the dynamic model finally as shown by Fig. 4.12c. 4.3.1 Modeling the semi-conductor losses The first two layers consider the IGBT to act as an ideal switch. But in reality, the IBGT is a three-terminal device which requires a finite time to realize a change in state [41]. There are several physics-based models of IGBTs, aiming to accurately represent the dynamic behaviour [30] [36] [32] [42]. Not only are these models computationally expensive, the parameters of these models are hard to extract from the data sheets alone. Lauritzen et al. [31] aim to offer an easy parameter extraction for their model. But in doing so, they make some additional assumptions. These models tend to be quite complicated, and using them in practice requires additional simplifications because of limited available data. Therefore, this thesis adopts another approach. Another approach is what this thesis will refer to as the algebraic model. The switching waveforms of the IGBT mostly depend on the voltage and current levels, which are dictated by the H-bridge operation. Qualitative knowledge of the dynamics is essential to justify the assumptions made in the algebraic approach. IGBTs need a certain amount of time to turn on or off, ton and toff . Therefore, switching the IGBTs in a single leg cannot be done at the same time; this would create a short-circuit in the battery. Fig. 4.14a for example shows the commutation of the left leg. The upper switch turns off first at t1. But because of the inductive load, the Analysis and simulation of a SSBC-MMCC converter 34 current cannot change. Therefore, a freewheeling diode in that leg will take over the current, depending on the current direction. Both IGBTs in the left leg are now turned off. Next, the other IGBT turns on at t2, taking over the current from the freewheeling diode. Subsection 4.3.2 discusses these sequences in greater detail. The key point is that an IGBT always takes over the current from a freewheeling diode when it turns on. The diode itself turns on almost instantly, but the turn-off causes some significant losses compared to the turn-on. These losses are known as the reverse recovery losses. The diode consumes negative current in the final stage of turning-off, to remove the remaining charge in its junction. This reverse recovery current will affect the transient waveforms of the IGBT when turning on. Based on these type of operating considerations, Rajapakse et al. [29] develop formulas which give the switching losses of both diode and IGBT. A similar approach is used by other authors [43] [38] [33]. The formulas developed by [29] depend on datasheet parameters and operating conditions of the H-bridge, the battery voltage and the load current at time of switching. These formulas contain many terms, due to the piece-wise nature of the approximate waveforms. When applied, it turned out that not all parameters were available in the datasheets of the devices. Instead, this thesis adopts a more straight-forward approach. The datasheets of the IGBT and diode contain graphs of the switching losses versus the load current at a specified voltage Vref . Dieckerhoff et al. [34] use equation (4.16) to model the current and voltage dependency of the switching loss. The parameters a,band care the result of a quadratic fit of the datasheet graphs. The same equation is suggested by the IGBT manual of ABB, a major IGBT producer [35]. This thesis uses this equation to model the depency of the turn-on, turn-off and reverse recovery losses. Esw =V Vref a·I2+b·I+c(4.16) 4.3.2 Event-based model The event-based model offers an alternative to simulating the full dynamics of the switches. The same approach is adopted by Kouro et al. [33], besides that this thesis uses equation (4.16) to model the voltage and current dependency of the switching losses. Fig. 4.13 shows the switching signals of the U-PWM, augmented with a blanking Analysis and simulation of a SSBC-MMCC converter 35 time tblank. This blanking time is necessary to prevent short-circuits; it guarantees that the closing switch has enough time to completely close, before the other IGBT starts to open. The direction of the load current determines which freewheeling diodes will conduct during the leg commutation. Table 4.1 gives a full oversight of which devices are changing state at which moment. Take for example time instance t1. The upper IGBT in the left leg switches off. But due to the inductive load, the current finds a new path. Depending on the current direction, either the upper or lower diode provides a path for the current. If the upper diode switches on, then nearly no voltage buildup occurs across the IGBT. But if the lower diode switches on, a voltage builds up across the IGBT and induces a significant switching loss. Fig. 4.14 shows a graphical representation of the events for a positive load current. Whenever a switching event occurs due to the converter operation, Table 4.1 indicates which devices change state as a consequence of this. By measuring the voltage and current at that time, equation (4.16) allows for the calculating of the switching loss in each device. This is purely algebraic, and computes significantly faster than a dynamic model of the switches. So far, the conduction losses were not discussed. An ideal switch has no voltage across its terminals when it is closed. When it is open, it blocks all current. A real switches does both of these. But in terms of power loss, only the closed state is of importance. The conduction loss then simply follows from the transfer characteristic. The data sheet typically contains a graph of this characteristic. This thesis approximates the graph with an electrical circuit containing a constant voltage drop and a linear resistance. Section 5.1 explains the parameter extraction in detail. The conduction loss of the diode is insignificant compared to the conduction loss of the IGBT. The diodes only conducts during the blanking interval, whilst the IGBTs conduct during the remaining timeslots in the switching period. Since the blanking interval is typically much smaller than the switching period, the diode conduction losses are neglected. By including the circuit shown by 4.12c, the conduction losses are included in the dynamic simulation of the converter as a whole. The switching losses are calculated separately, adding to the total switching losses whenever a switching event occurs. The effect of voltage and current at time of switching is taken into account. Analysis and simulation of a SSBC-MMCC converter 36 Figure 4.13: The blanking time tblank guarantees that the closing switch has enough time to completely close, before the other IGBT starts to open. A short-circuit would occur if this was not in place. (a) t1→t4 (b) t5→t8 Figure 4.14: The load current finds a path through the converter at all times. During the blanking interval, when both IGBTs are eventually turned off, a freewheeling diode provides a path in the commutating leg. This Figure shows only the commutation for a positive load current, corresponding to Table 4.1a. Chapter 5 Case study This chapter will apply the developed model to a specific use case. This is an important step towards reaching the objective of the thesis. In addition to developing a conceptual model of the converter, an actual tool is implemented based on it. This tool is then used to simulate a specific use case. The results of this simulation then enable further analysis, demonstrating the potential of the tool. The first section presents the input data. This includes things such as nominal ratings of the converter, semiconductor components etc. The second section presents the tool and a base case. This base case forms a starting point from where the design can be explored. The three last chapters extensively use the tool to analyze key performance indices (KPI) such as harmonics, losses and reliability. In doing so, they change the base case to examine how certain parameters affect these KPI. 5.1 Data 5.1.1 Converter ratings The specifications of the converter are set to typical values for converters applied in the power network, fitting the research interests of CITCEA. The nominal power of the converter is 100kW. 9 battery modules are available, at a nominal voltage of 350V each. The short circuit power absorbed by the converter is allowed to be 1% of the nominal power, and the short circuit impedance should be not lower than 10% of the nominal one. Table 5.1 summarizes these parameters. The AC phase voltage follows from the modulation strategy and number of batteries connected in series in a phase. Each module is controlled by unipolar PWM. Since there 37 Analysis and simulation of a SSBC-MMCC converter 38 Figure 4.15: A real H-bridge has 4 diodes and IGBTs. Table 4.1: These tables indicate the switching events for all semiconductor devices under U-PWM as shown by Fig. 4.13. Fig. 4.15 shows the location of the devices within the H-bridge. The diode turn-on power loss is negligible compared to the other events, and is indicated in light gray. Furthermore, only the events indicated in boldface cause significant losses. t1t2t3t4t5t6t7t8 T1OFF ON D1 T2ON OFF D2ON OFF ON OFF T3ON OFF D3ON OFF ON OFF T4OFF ON D4 (a) Positive current t1t2t3t4t5t6t7t8 T1OFF ON D1ON OFF ON OFF T2ON OFF D2 T3ON OFF D3 T4OFF ON D4ON OFF ON OFF (b) Negative current Analysis and simulation of a SSBC-MMCC converter 39 Table 5.1: The ratings of the converter are set to typical values for converters applied in the power network. Parameter Value Srated 100kW VDC 350V PSC 0.01Prated ZSC 10% are 9 battery modules in total, 3 battery modules are connected in series in a single phase. The modulation index is lower than 1 to prevent saturation of the controller when there is some ripple in the reference voltage. With a modulation index maof 0.9, the resulting AC phase voltage is VAC,f =3maVDC √2= 668V(5.1) The current rating follows from the power and voltage rating IAC =Srated 3VAC,f = 50A(5.2) 5.1.2 Semiconductor components Each H-bridge consists of 4 IGBT modules, consisting in turn of an IGBT and a freewheeling diode. The IGBT module FS100R17KE3 produced by infineon provides the required voltage and current handling capabilities. The full datasheet is attached in Appendix A.1. This subsection will focus on how the relevant model parameters were extracted from this datasheet. The specifications of the IGBT module is the determining factor for the switching and conduction losses. Therefore, a good approximation is essential for the correct representation of the losses. are set to typical values for converters applied in the power network Transfer characteristics The transfer characteristics determine the conduction losses of the components. Fig. 5.1a and 5.1b show the transfer characteristic of respectively the diode and the IGBT. The developed model does not include a thermal model. Therefore, the characteristics at 125◦Cwere used, to obtain the worst-case losses. The effect of the gate voltage is not considered. Both the IGBT and the diode have an exponential transfer characteristic. Analysis and simulation of a SSBC-MMCC converter 40 This can be approximated electrically by an ideal diode causing a constant voltage drop in series with a linear resistance. This representation is often used for real diodes. Fig. 5.1a and 5.1b show the resulting characteristic of the approximation with a red line. Table 5.1c gives the numerical values corresponding to these fits. Switching losses The current dependency of the switching losses is shown by Fig. 5.2a and 5.2b. The relation between both is approximately quadratic, and was therefore approximated by a quadratic fit. Table 5.2c shows the coefficients obtained by a least-square quadratic fit of several points along the curve. The resulting approximate curves are shown in color in Fig. 5.2a and 5.2b. Analysis and simulation of a SSBC-MMCC converter 47 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 -100 -50 0 50 100 (a) u-phase current 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 -100 -50 0 50 100 (b) v-phase current 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 -100 -50 0 50 100 (c) w-phase current Figure 5.5: The current loop controller regulates the current by changing the converter voltage. The harmonics present in the converter voltage cause a ripple current, modulated on the fundamental sinusoidal current. Analysis and simulation of a SSBC-MMCC converter 48 5.3 Power loss analysis The losses are one of the key performance indicators which could motivate the choice for a specific converter. This analysis looks at the losses in the converter itself, due to imperfect switches and parasitic resistances. There is another important aspect when it comes to losses: harmonics. The harmonics enter the grid and induce losses in other devices. This harmonic loss is not included in this section. According to Kou et al., this is exactly one of the benefits of multilevel inverters; the reduced harmonics leads to lower losses [45]. This section examines the implications of the loss model discussed in section 4.3. First, approximate formulas are derived for the steady-state operation of the converter. The predictions obtained from these formulas are then compared with the simulated operation of the converter. 5.3.1 Analytic formulas The losses in the converter are caused by the imperfect switches and parasitic resistance. These losses can be grouped in two classes: conduction losses and switching losses. This subsection will derive approximate formulas for both loss components during steady-state operation. Steady-state in this context means that the current waveform is periodic i(t) = ˆ Isin (ωt) = √2Isin (ωt)(5.9) If the converter is not generating any reactive power, then the current is simply related to the active power by I=P 3VAC (5.10) Conduction losses The conduction losses are caused by the IGBTs and the filter. The voltage across the terminals of an IGBT increases with the current. The relationship between both is called the transfer characteristic. Since the voltage is non-zero, energy leaves the IGBT as heat. The transfer characteristic was approximated by vIGBT (t) = VCES +RCE ·|i(t)|(5.11) Analysis and simulation of a SSBC-MMCC converter 49 where VCES is the collector-emitter on-state voltage drop and Rce is a linear approximation of the current dependency of the total voltage drop. For steady-state operation, time-averaging the instantaneous power over one period gives a formula for the average power loss due to conduction hPcond,IGBT i=1 TZT 0vIGBT (t)·|i(t)|dt =1 T·ZT 0VCES ·|i(t)|+RCE ·i2(t)dt =2 TZT 2 0VCES ·ˆ Isin (ωt) + RCE ·ˆ I2sin2(ωt)dt =2 TZT 2 0VCES ·ˆ Isin (ωt) + RCE ·ˆ I2"1−cos (2ωt) 2#dt =2 πVCES ·ˆ I+1 2RCE ·ˆ I2 ⇒ hPcond,IGBT i=2√2 πVCES ·I+RCE ·I2(5.12) In each cell at any time, two IGBTs are conducting the current. During the blanking time, the diodes conduct the current, but since the blanking time is much smaller than the switching period, these losses are negligible. Each cell in a cluster conducts the same current, because they are connected in series. The current waveform is phaseshifted across the phases. This means that the instantaneous current is different, but the time-averaged value is the same. These three arguments justify that the total conduction loss is simply obtained by scaling (5.12) with the total number of IGBTs hPcondi= (2 ·Nc·3)Pcond,IGBT = 6Nc"RCE ·I2+2√2 πVCES ·I#(5.13) where Ncis the cascade number, indicating the number of modules connected in series in each phase. The filter contains an inductance, which also has a parasitic resistance. Other losses, such as the iron and eddy current losses in the core, are neglected for simplicity. The power consumed by the filter is therefore given by Pfilter =R·I2(5.14) Analysis and simulation of a SSBC-MMCC converter 50 Switching losses In the event-based model, the switch transitions occur instantly, and induce a power loss according to the voltage and current at that given time. The dependency is given by equation 4.16. As explained later, for all non-negligible switching events, the voltage is constant and equal to the battery module voltage VDC . Time-averaging 4.16 over one period of the current gives an expression for the average energy loss. For example, for the turn-on energy this gives hEoni=1 TZT 0 V Vref haon ·i2(t) + bon ·|i(t)|+conidt =1 TZT 0 V Vref haon ·i2(t) + bon ·|i(t)|+conidt =2 TZT 2 0 V Vref haon ·ˆ I2sin2(ωt) + bon ·ˆ Isin (ωt) + conidt ⇒ hEoni=V Vref "aon ·I2+2√2 πbon ·I+con#(5.15) As indicated by Table 4.1, 12 switching events occur per switching period. The diode turn-on event is negligible in power loss compared to the turn-off. Furthermore, when the current switches between two devices connected in parallel (IGBTs and diode), the voltage stays nearly zero. This means that according to equation (4.16), the incurred loss is nearly zero as well. Ingoring these two type of events, leaves only 6 significant switching events per period. These events are evenly divided across the three categories: 2 IGBT turn-on events, 2 IGBT turn-off events and 2 diode turn-off events. These time-averaged values are then multiplied with the number of times they occur during a period of the current hPswi=fsw ·3Nc·[2hEoni+ 2hEoff i+ 2hEreci] = 6Ncfsw VDC Vref "at·I2+2√2 πbt·I+ct#(5.16) where at=aon +aoff +arec bt=bon +boff +brec ct=con +coff +crec (5.17) Analysis and simulation of a SSBC-MMCC converter 51 Impact of Nc Overall, the cascade number Ncincreases the power losses. The same amount of current has to go through a higher number of switching devices, which increases the conduction losses. The switching losses are not affected; this is because the losses scale with the battery module voltage. To make the comparison fair, the total amount of voltage remains the same N0 c˙ V0 DC =Nc˙ VDC =VDC,total (5.18) This is also achievable in reality for a given system, because battery modules often consist of several smaller modules. By removing modules from all existing units, a new unit is created. Since the number of modules in use stays the same, so does the total available voltage. Then, according to equation (5.16), the switching losses remain unaffected. 5.3.2 Simulation results Fig. 5.6 shows are comparison of the simulation results and the predictions of the analytic formulas. Overall the formulas offer a good approximation, reaching an accuracy of around 5% at nominal power. There are two differences between the formulas and the simulation. First and foremost, the formulas neglect the harmonics in the steady-state current. When the power output increases, the fundamental component of the current increases; the harmonics stay the same. Therefore, the relative share of the harmonics in the current becomes less significant. This also reduces the relative error in the analytic formulas. Fig. 5.6b clearly demonstrates this. As the power output goes to zero, the relative error becomes larger and larger. Secondly, the analytic formula makes a continuous approximation of the discrete switching events. This also causes a deviation between the formula and the simulation results. Analysis and simulation of a SSBC-MMCC converter 52 0 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 (a) The switching and conduction losses increase with the load. The dots show the simulation results, whilst the full line indicates the prediction of the analytic formulas. 0 1 2 3 4 5 6 7 8 9 10 -5 0 5 10 15 20 (b) Relative error Figure 5.6: The analytic formulas offer a good approximation, especially at higher load. The harmonics are not included in the analytic formulas, and are the main cause of the difference with the simulation results. The harmonics don’t change with the load. At higher load, the harmonic are less significant compared to the fundamental, and the approximation improves. Analysis and simulation of a SSBC-MMCC converter 53 5.3.3 Efficiency The efficiency of the converter is the ratio of the output power to the input power. The difference between both is the power loss η=Pout Pin =Pin −Ploss Pin = 1 −Ploss Pin (5.19) For highly efficient systems, the efficiency is approximately equal to η≈1−Ploss Pout (5.20) What the input and output is, depends on whether the converter is charging or discharging. For example, when the batteries are charging, power flows from the batteries, through the converter, to the grid. Time-averaged values of the power flows are used. This means that only the fundamental current has an impact on the output power; the harmonics produce no average power flow when connected to an ideal voltage source. The efficiency is then equal to η= 1 −Pcond +Psw + 3Pfilter 3V I (5.21) Using the analytic formulas, the loss components can be expressed as a function of the current. This leads to η= 1 −A·I+B+C·1 I(5.22) where A,Band Crepresent compactly a product of several parameters. This clearly shows the current dependency. When the current and hence the output power is close to zero, the C-term dominates. As the current increases, the efficiency increases as the C-term decreases. This term is due to fixed component of the switching losses. As the current increases further, the A-term starts to dominate. The efficiency reaches a maximum, where after the disproportional increase of the quadratic components leads to a decrease in efficiency. As Fig. 5.6a shows, mostly the conduction losses contribute to the quadratic components. Fig. 5.7 shows the efficiency as predicted by the analytic formulas, and the results of the simulation. The simulated efficiency is lower and reaches its maximum later. This is due to the presence of harmonics in the simulation. The harmonics induce additional switching and conduction losses. These losses behave roughly as a C-term. Analysis and simulation of a SSBC-MMCC converter 54 0 1 2 3 4 5 6 7 8 9 10 92 93 94 95 96 97 98 99 Figure 5.7: The efficiency of the simulation (dots) is lower than the efficiency predicted by the formulas (line). The extra switching and conduction losses due to harmonics are not included in the formulas, and are especially significant at low power output. This discrepancy between simulation and formulas therefore diminishes as the power output increases. Analysis and simulation of a SSBC-MMCC converter 55 5.4 Power quality 5.4.1 Fourier analysis Analyzing the waveforms in the frequency spectrum leads to several key insights and deeper understanding of the converter. In general, the Fourier transformation can transform almost any signal from the time domain to the frequency domain. After the initial transient caused by the change in set point, the current controller quickly moves the converter to steady-state. In steady-state, the waveforms are periodic. All electrical variables repeat periodically with the grid frequency, fn. In steady-state, the frequency spectrum is discretized; only multiples of fnare present. If this was not the case, the waveform would not be periodic. For periodic signals, the general Fourier transform reduces to the Fourier series decomposition. There are a few equivalent definitions of the Fourier series decomposition. The most useful one for this application is xP(t) = ˆ X0 2+∞ X i=0 ˆ Xisin (i·wnt+φi)(5.23) From this definition, it is immediately clear that the periodic signal xPis decomposed into a series of sines at multiples of the grid frequency, each with a unique amplitude and phase shift. These higher frequency components can be represented by a phasor, with the rms value Xias its amplitude [46]. The component of the series at the grid frequency is called the fundamental component. All components at a higher frequency are the harmonic components, or shortly, harmonics. Harmonics are undesirable as they cause losses and disturbances [47]. The objective of this section is to analyze the harmonics present in the current injected into the grid. In this context, the total harmonic distortion (THD) is a metric indicating the presence of harmonics. This metric is useful to compactly display information about the harmonic content. The THD is given by THD =qP∞ i=2 Xi X1 (5.24) =qX2 RMS −X2 1 X1 (5.25) Analysis and simulation of a SSBC-MMCC converter 56 [46].The first equation shows the idea behind the metric: a Euclidean norm on all harmonic components, compared to the fundamental component. The second equation is equivalent but more useful in practical calculations. The equivalence follows from the fact that all components of the Fourier series are orthogonal functions and by assuming that there is no DC-component. 5.4.2 Regulation and standards Harmonics have a wide range of negative effects, reducing the power quality of the system. They induce losses in motors, generators and capacitors and lead to misoperation in electronics, switchgear and relaying [47]. Because of the diverse range of negative effects, it is hard to specify which harmonics are more harmful than others. Standard IEEE Std 519-1992 provides a reference point to assess exactly this. The Institute of Electrical and Electronics Engineers (IEEE) released a standard with best practices to deal with harmonics, referred to as IEEE Std 519-1992. This was motivated by the increased popularity of non-linear loads. Many devices include power converters, which provide many benefits: energy savings, improved process control, higher reliability... But the big drawback is that they turn the devices into non-linear loads from the system perspective, generating harmonics. [48] There are two main parties: the user and the system operator. The user connects to the system at the point of common coupling (PCC). The standard assigns responsibilities to both parties. It is the duty of the system operator to maintain a clean voltage at the PCC. But at the same time, the consumer has to inject a clean current into the system. [47] What makes a current waveform clean or not, is exactly the subject of the remainder of this section. A harmonic emission norm specifically for generators, is given by IEC61000-4-15. The voltage generated by the converter leads to a current injection into the source. IEEE Std 519-1992 defines upper limits for the harmonics present in the injected current. There are two types of restrictions. A first type of restriction defines a limit of each harmonic separately, with lower limits for higher order harmonics. A second type of restriction is defined in terms of total demand distortion (TDD), a metric similar to the THD. Table 5.2 gives an oversight of these limits. Different limits apply depending on the ratio of the maximum load current ILto the short-circuit current ISC. Higher harmonics are tolerated for a higher ILand a lower ISC . The limits on the individual harmonics Analysis and simulation of a SSBC-MMCC converter 63 50 100 150 200 250 300 0 2 4 50 100 150 200 250 300 0 2 4 50 100 150 200 250 300 0 2 4 50 100 150 200 250 300 0 2 4 50 100 150 200 250 300 0 2 4 Figure 5.11: The odd harmonics change with the number of cells Nc. The cummulative voltage of all batteries together is kept constant; all other parameters of the design stay unchanged. Increasing Ncshifts the effective switching frequency to the right according to equation (5.34). Higher frequencies are filtered more, so the harmonic performance increases. Note that an even Ncleads to a significantly worse performance. At Nc= 7, all harmonics are lowered below the allowed limits. Analysis and simulation of a SSBC-MMCC converter 64 5.5 Reliability Most modern inverter designs are based on multilevel topologies. These designs improve the power quality by combining multiple small voltage steps. Additionally, this results in lower losses and improved electromagnetic current. There is one big drawback: the increased number of switches [45]. For the SSBC-MMCC converter, the number of switches is 12Nc. For example, a simple three-level inverter has 36 switches. This is already 6 times more than a classical six-pulse inverter. This leads to concerns regarding the reliability. The likelihood that a single switch fails increases as the total number of switches increases. This is especially troubling in applications where shutting the converter is very costly or even dangerous; downtime in industrial processes and Navy propulsion systems are such applications [45]. Therefore, several authors address this concern by developing techniques allowing the converter to continue operating, even when a fault occurs. [16,25–28,50] This thesis will adopt a mechanism similar to the one proposed by Lezana et al. [27] and Rodriquez et al. [26]. When a cell becomes faulty, the control system bypasses the cell. To compensate for this loss, the control system adjusts the phase shift of the trigger and the reference voltage of the remaining cells. The first subsection derives the changes required by this mechanism. The second subsection will simulate a battery short-circuit to demonstrate this mechanism. 5.5.1 Requirements for continued operation When a battery module becomes unavailable, the other battery modules can compensate and keep the converter operational. There might be several causes for this: a short-circuit in the bridge cell, taking the battery offline for maintenance etc. Temporarily running the converter with one battery module less offers more flexibility and reliability. This subsection will examine what the requirements are for this to be possible. VDC is the available voltage for the modulation of an entire cluster, the sum of the voltages of all battery modules in that cluster. The average output voltage of the cluster depends on the modulation, and ranges from −VDC to VDC. The minimum requirement is VDC >√2VAC (5.35) Analysis and simulation of a SSBC-MMCC converter 65 This guarantees that the converter can achieve the peak of the AC voltage of the grid it is connected to, and dictates a lower limit for the amount of connected battery cells for a given VAC . Therefore, extra battery cells are added, so that when a module goes offline, enough battery cells remain to meet this minimum VDC. There is no physical distinction between normal and backup cells; all cells participate equally in the modulation. These extra cells are even useful: though not necessary to meet the minimal operation requirements, they increase the energy storage capacity of the BESS. The modulation index is the ratio between VDC and the maximum value of the reference signal ma=ˆvref VDC (5.36) When a battery becomes unavailable, the remaining batteries have to increase their share of the total required voltage. They can only compensate for the unavailable battery if the modulation index before was low enough √2VAC < m0 aVDC (Nc−1) ⇔√2maVDC Nc √2< m0 aVDC (Nc−1) ⇔ma< m0 a Nc−1 Nc (5.37) The modulation index was only 90% in the base case. This margin of 10% prevents saturation during steady-state. The ripple present in the current, enters the feedback loop and causes some ripple in the reference voltage. If mawas exactly one, the reference voltage would repeatedly saturate when the reference voltage nears its peak. In order to guarantee that m0 ais still 90% in the base case, mashould be lower than ma<90%3−1 3= 60% (5.38) if the cascade number Ncremains unchanged. This can be achieved by increasing the initial available VDC to V0 DC =VDC Nc Nc−1= 350V3 2= 525V(5.39) Analysis and simulation of a SSBC-MMCC converter 66 5.5.2 Simulation of a battery short-circuit To demonstrate the potential of this mechanism, a battery fault and subsequent control is simulated. At t= 0.04s, a short-circuit occurs in the first bridge cell. The battery management system (BMS) detects an unusual power drain and disconnects the battery. The modulation control is connected to the BMS of each battery, and is notified of the unavailability of the first battery module. Therefore, it switches to two level operation by adjusting the phase shift and reference voltage of the two remaining battery modules. Fig. 5.12 shows the waveforms of the converter for the u-phase, before and after the fault occurs. Initially, Ncis odd. When battery module 1 goes offline, Ncbecomes even. This has a very negative impact on the harmonic performance, causing a significantly bigger ripple in the current. It could be better to further reduce the Ncpost-fault to an odd number. In this specific case that wouldn’t impact the harmonics, because Nc= 1 and Nc= 2 lead to the same harmonic performance. But if Ncwas initially 7, removing two levels would lead to a much smoother current waveform (see Fig. 5.11). The downside of this is that the batteries will have to be oversized even further. Analysis and simulation of a SSBC-MMCC converter 67 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 2 3 -50 0 50 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Figure 5.12: At t= 0.04, battery module 1 goes offline. The cascade number Nc reduces from 3 to 2. The modulation control automatically adjusts the phase shift and the reference voltage of the other battery modules. The converter voltage is maintained. Note that the Nc= 2 operation has a larger ripple current. The ripple current also enters the feedback loop, and causes a ripple in the reference voltage. Chapter 6 Conclusion The main objective of the thesis is to develop a tool which simulates the behavior of a MMCC-SSBC converter. As a secondary objective, the thesis strives to demonstrate the usefulness of this tool. This chapter will discuss how this twofold objective was achieved, and will give suggestions for further research. The main objective is fulfilled by the core deliverable of the thesis: a Matlab implementation of a dynamic model of the converter. Appendices B and C provide a detailed description of the core deliverable. Chapter 4 describes the theoretical underpinning of this implementation. The model consists of three layers: interface and control, converter operation and loss model. For each of these layers, a specific implementation has been developed; both a conceptual description and a Simulink implementation. The Matlab implementation together with the documentation provided by Chapter 4, fulfill the main objective. The thesis also aims to demonstrate the usefulness of the developed tool. This is achieved primarily by the case study described in Chapter 5. The case study chooses specific ratings for a typical converter. With the help of the tool, three key characteristics are discussed: efficiency, power quality and reliability. This leads to some important insights, which will be briefly summarized. Analyzing the power quality led to two simple design rules. Firstly, the cascade number Ncshould be uneven. Secondly, the inductive filter can be scaled according to a simple rule in order to comply with a harmonic emission standard. The efficiency is negatively impacted by harmonic current, even more so for lower load levels. Finally, the thesis demonstrates that the converter can keep on operating when a battery module fails. The tool helped significantly in arriving at and validating these conclusions. Secondarily, the tool is useful due to its layered structure. Each layer’s implemen68 Analysis and simulation of a SSBC-MMCC converter 69 tation can be changed independently. Fig. 6.1a demonstrates this principle. The layers communicate through an interface. As long as a new layer implementation respects the interface connection, the model as a whole remains functional. This allows the user to quickly prototype a new method. For example, assume the user wants to study a phase-disposition modulation method instead. The user only has to change the implementation of the converter operation layer; the other layers remain unaffected. The most critical step for further research is model validation. Due to practical restrictions, it was not possible to build an experimental setup of the SSBC-MMCC converter. The model has only been validated by comparing it to an online simulation tool provided by a semiconductor manufacturer. There was no extensive documentation available for this tool, so it could only be verified that the losses were of a similar magnitude. Therefore, the next step should be a thorough model validation in order to justify any conclusions based on it. Once the model has been validated, there are two main pathways for further research. Firstly, the researcher could further expand on the case study. For example, the cost of the proposed solution has not been considered. Secondly, the layered model provides an excellent basis for comparison studies. Fig. 6.1a illustrates this approach and shows some alternatives. The researcher can easily change a part of the implementation. The tool remains functional under this change. The researcher can then compare the old and new implementation. For example, the researcher could try out a different modulation technique and compare the resulting harmonics. This approach offers a lot of research opportunities. Finally, this thesis modeled the batteries as ideal voltage sources. Further research should extend the model with a realistic battery model, especially when SoC balancing is considered. The thesis fulfilled its main objective. Additionally, the thesis demonstrated the usefulness of the tool in two ways: by applying it to a case study and by pointing out further research based on it. Hopefully, it proves to be instrumental in showing the potential of this topology, and in promoting its adoption in the industry. Analysis and simulation of a SSBC-MMCC converter 70 PI-controller Event-based model SSBC subharmonic U-PWM Interface Interface and control Converter operation Loss model Interface (a) Implementation of layers H2-controller DSBC DSCC phase disposition B-PWM (b) Some alternative implementations Figure 6.1: One pathway for further research is comparing several alternative implementations, and how they affect the converter performance. Chapter 7 Acknowledgement To conclude, I would like to thank the institutions and persons who enabled the creation of this thesis. Firstly, there is the Smart Cities master programme, a collaborative effort between 4 European universities. They allowed me to study abroad at both the Royal Institute of Technology in Stockholm and at the Universitat Politècnica de Catalunya. Furthermore, they provided financial support through KIC InnoEnergy throughout the studies. Secondly, I would like to thank Dr. Gonzalez who supervised the creation of this thesis. He provided guidance when the path forward was unclear, and feedback when I thought the path was clear. I doubt I will ever meet his equal when it comes to answering emails quickly. Finally, I would like to thank Konstantinos Spiliotis. He proofread parts of this thesis as I wrote them. Also, he gave me the chance to collaborate with him on a paper which we presented at the European Energy Market 2016 conference. The experience I gained writing this paper, was very helpful when I wrote this thesis. 71 Bibliography [1] R. K. Pachauri, M. R. Allen, V. Barros, J. Broome, W. Cramer, R. Christ, J. Church, L. Clarke, Q. Dahe, P. Dasgupta et al.,Climate change 2014: synthesis Report. Contribution of working groups I, II and III to the fifth assessment report of the intergovernmental panel on climate change. IPCC, 2014. [2] European parliament and council, “Citizens’ summary: Eu climate and energy package,” http://ec.europa.eu/clima/policies/strategies/2020/docs/climate_package_en. pdf. [3] European commission, “What is horizon 2020?” https://ec.europa.eu/programmes/horizon2020/en/what-horizon-2020. [4] J. Cochran, M. Miller, O. Zinaman, M. Milligan, D. Arent, B. Palmintier, M. O’Maley, S. Mueller, E. Lannoye, A. Tuchy, B. Kujula, M. Sommer, H. Hotlinnen, J. Kiviluoma, and S. Soonee, “Flexibility in 21st century power systems,” NREL, Golden, Tech. Rep., 2014. [5] G. Papaefthymiou, K. Grave, and K. Dragoon, “Flexibility options in electricity systems,” ECOFYS, European Copper Institute, Berlin, Tech. Rep., 2014. [6] A. Joseph and M. Shahidehpour, “Battery storage systems in electric power systems,” in IEEE Power Engineering Society General Meeting, 2006, 2006, pp. 8 pp.–. [7] A. R. Sparacino, G. F. Reed, R. J. Kerestes, B. M. Grainger, and Z. T. Smith, “Survey of battery energy storage systems and modeling techniques,” in 2012 IEEE Power and Energy Society General Meeting, Jul. 2012, pp. 1–8. [8] K. Prompinit and S. Khomfoi, “Ramp rate consideration of a BESS using active power control for PV generation,” in 2015 18th International Conference on Electrical Machines and Systems (ICEMS), Oct. 2015, pp. 1676–1680. [9] L. H. Walker, “10-mw gto converter for battery peaking service,” IEEE Transactions on Industry Applications, vol. 26, no. 1, pp. 63–72, Jan 1990. [10] L. Maharjan, T. Yamagishi, and H. Akagi, “Active-Power Control of Individual Converter Cells for a Battery Energy Storage System Based on a Multilevel Cascade PWM Converter,” IEEE Transactions on Power Electronics, vol. 27, no. 3, pp. 1099–1107, Mar. 2012. 72 2 TechnischeInformation/TechnicalInformation FS100R17KE3 IGBT-Module IGBT-modules preparedby:MW approvedby:WR dateofpublication:2013-10-03 revision:2.0 VorläufigeDaten PreliminaryData Diode,Wechselrichter/Diode,Inverter HöchstzulässigeWerte/MaximumRatedValues PeriodischeSpitzensperrspannung Repetitivepeakreversevoltage Tvj = 25°C VRRM 1700 V Dauergleichstrom ContinuousDCforwardcurrent IF100 A PeriodischerSpitzenstrom Repetitivepeakforwardcurrent tP = 1 ms IFRM 200 A Grenzlastintegral I²t-value VR = 0 V, tP = 10 ms, Tvj = 125°C I²t 1800 A²s CharakteristischeWerte/CharacteristicValues min. typ. max. Durchlassspannung Forwardvoltage IF = 100 A, VGE = 0 V IF = 100 A, VGE = 0 V VF 1,80 1,90 2,20 V V Tvj = 25°C Tvj = 125°C Rückstromspitze Peakreverserecoverycurrent IF = 100 A, - diF/dt = 2450 A/µs (Tvj=125°C) VR = 900 V VGE = -15 V IRM 155 165 A A Tvj = 25°C Tvj = 125°C Sperrverzögerungsladung Recoveredcharge IF = 100 A, - diF/dt = 2450 A/µs (Tvj=125°C) VR = 900 V VGE = -15 V Qr29,0 48,5 µC µC Tvj = 25°C Tvj = 125°C AbschaltenergieproPuls Reverserecoveryenergy IF = 100 A, - diF/dt = 2450 A/µs (Tvj=125°C) VR = 900 V VGE = -15 V Erec 15,5 27,5 mJ mJ Tvj = 25°C Tvj = 125°C Wärmewiderstand,ChipbisGehäuse Thermalresistance,junctiontocase proDiode/perdiode RthJC   0,39 K/W TemperaturimSchaltbetrieb Temperatureunderswitchingconditions Tvj op -40 125 °C NTC-Widerstand/NTC-Thermistor CharakteristischeWerte/CharacteristicValues min. typ. max. Nennwiderstand Ratedresistance TC = 25°C R25 5,00 kΩ AbweichungvonR100 DeviationofR100 TC = 100°C, R100 = 493 Ω ∆R/R -5 5 % Verlustleistung Powerdissipation TC = 25°C P25   20,0 mW B-Wert B-value R2 = R25 exp [B25/50(1/T2 - 1/(298,15 K))] B25/50 3375 K B-Wert B-value R2 = R25 exp [B25/80(1/T2 - 1/(298,15 K))] B25/80 t.b.d. K B-Wert B-value R2 = R25 exp [B25/100(1/T2 - 1/(298,15 K))] B25/100 t.b.d. K AngabengemäßgültigerApplicationNote. Specificationaccordingtothevalidapplicationnote. 3 TechnischeInformation/TechnicalInformation FS100R17KE3 IGBT-Module IGBT-modules preparedby:MW approvedby:WR dateofpublication:2013-10-03 revision:2.0 VorläufigeDaten PreliminaryData Modul/Module Isolations-Prüfspannung Isolationtestvoltage RMS, f = 50 Hz, t = 1 min. VISOL 3,4 kV MaterialModulgrundplatte Materialofmodulebaseplate    Cu   InnereIsolation Internalisolation Basisisolierung(Schutzklasse1,EN61140) basicinsulation(class1,IEC61140)   Al203  Kriechstrecke Creepagedistance Kontakt-Kühlkörper/terminaltoheatsink Kontakt-Kontakt/terminaltoterminal   10,0 10,0 mm Luftstrecke Clearance Kontakt-Kühlkörper/terminaltoheatsink Kontakt-Kontakt/terminaltoterminal   7,5 7,5 mm VergleichszahlderKriechwegbildung Comperativetrackingindex CTI > 225   min. typ. max. Wärmewiderstand,GehäusebisKühlkörper Thermalresistance,casetoheatsink proModul/permodule λPaste=1W/(m·K)/λgrease=1W/(m·K) RthCH 0,009 K/W Modulstreuinduktivität Strayinductancemodule LsCE 21 nH Modulleitungswiderstand,Anschlüsse- Chip Moduleleadresistance,terminals-chip TC=25°C,proSchalter/perswitch RCC'+EE' 1,80 mΩ Lagertemperatur Storagetemperature Tstg -40 125 °C Anzugsdrehmomentf.Modulmontage Mountingtorqueformodulmounting SchraubeM5-Montagegem.gültigerApplikationsschrift ScrewM5-Mountingaccordingtovalidapplicationnote M 3,00 - 6,00 Nm Gewicht Weight G300 g 4 TechnischeInformation/TechnicalInformation FS100R17KE3 IGBT-Module IGBT-modules preparedby:MW approvedby:WR dateofpublication:2013-10-03 revision:2.0 VorläufigeDaten PreliminaryData AusgangskennlinieIGBT,Wechselrichter(typisch) outputcharacteristicIGBT,Inverter(typical) IC=f(VCE) VGE=15V VCE [V] IC [A] 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 0 20 40 60 80 100 120 140 160 180 200 Tvj = 25°C Tvj = 125°C AusgangskennlinienfeldIGBT,Wechselrichter(typisch) outputcharacteristicIGBT,Inverter(typical) IC=f(VCE) Tvj=125°C VCE [V] IC [A] 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 0 20 40 60 80 100 120 140 160 180 200 VGE = 20V VGE = 15V VGE = 12V VGE = 10V VGE = 9V VGE = 8V ÜbertragungscharakteristikIGBT,Wechselrichter(typisch) transfercharacteristicIGBT,Inverter(typical) IC=f(VGE) VCE=20V VGE [V] IC [A] 5 6 7 8 9 10 11 12 13 0 20 40 60 80 100 120 140 160 180 200 Tvj = 25°C Tvj = 125°C SchaltverlusteIGBT,Wechselrichter(typisch) switchinglossesIGBT,Inverter(typical) Eon=f(IC),Eoff=f(IC) VGE=±15V,RGon=4Ω,RGoff=4Ω,VCE=900V IC [A] E [mJ] 0 25 50 75 100 125 150 175 200 0 10 20 30 40 50 60 70 80 90 100 Eon, Tvj = 125°C Eoff, Tvj = 125°C 5 TechnischeInformation/TechnicalInformation FS100R17KE3 IGBT-Module IGBT-modules preparedby:MW approvedby:WR dateofpublication:2013-10-03 revision:2.0 VorläufigeDaten PreliminaryData SchaltverlusteIGBT,Wechselrichter(typisch) switchinglossesIGBT,Inverter(typical) Eon=f(RG),Eoff=f(RG) VGE=±15V,IC=100A,VCE=900V RG [Ω] E [mJ] 0 5 10 15 20 25 30 35 40 0 20 40 60 80 100 Eon, Tvj = 125°C Eoff, Tvj = 125°C TransienterWärmewiderstandIGBT,Wechselrichter transientthermalimpedanceIGBT,Inverter ZthJC=f(t) t [s] ZthJC [K/W] 0,001 0,01 0,1 1 10 0,001 0,01 0,1 1 ZthJC : IGBT i: ri[K/W]: τi[s]: 1 0,0225 0,01 2 0,0675 0,04 3 0,09 0,06 4 0,045 0,3 SichererRückwärts-ArbeitsbereichIGBT,Wechselrichter (RBSOA) reversebiassafeoperatingareaIGBT,Inverter(RBSOA) IC=f(VCE) VGE=±15V,RGoff=4Ω,Tvj=125°C VCE [V] IC [A] 0 200 400 600 800 1000 1200 1400 1600 1800 0 25 50 75 100 125 150 175 200 225 250 IC, Modul IC, Chip DurchlasskennliniederDiode,Wechselrichter(typisch) forwardcharacteristicofDiode,Inverter(typical) IF=f(VF) VF [V] IF [A] 0,0 0,5 1,0 1,5 2,0 2,5 3,0 0 20 40 60 80 100 120 140 160 180 200 Tvj = 25°C Tvj = 125°C 6 TechnischeInformation/TechnicalInformation FS100R17KE3 IGBT-Module IGBT-modules preparedby:MW approvedby:WR dateofpublication:2013-10-03 revision:2.0 VorläufigeDaten PreliminaryData SchaltverlusteDiode,Wechselrichter(typisch) switchinglossesDiode,Inverter(typical) Erec=f(IF) RGon=4Ω,VCE=900V IF [A] E [mJ] 0 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 30 35 40 Erec, Tvj = 125°C SchaltverlusteDiode,Wechselrichter(typisch) switchinglossesDiode,Inverter(typical) Erec=f(RG) IF=100A,VCE=900V RG [Ω] E [mJ] 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 45 Erec, Tvj = 125°C TransienterWärmewiderstandDiode,Wechselrichter transientthermalimpedanceDiode,Inverter ZthJC=f(t) t [s] ZthJC [K/W] 0,001 0,01 0,1 1 10 0,01 0,1 1 ZthJC : Diode i: ri[K/W]: τi[s]: 1 0,039 0,01 2 0,117 0,04 3 0,156 0,06 4 0,078 0,3 Appendix B Matlab script The full implementation of the Matlab script file is shown below. %clear;clc;close all; %% Sizing %****************************** %−−−−−EDIT−−>>> V_DC = 350*3; m_a = 0.9; f_p = 50; P_rated = 100E3; Nc = 3; % short circuit limits P_sc = 0.01*P_rated; Z_sc = 0.1; %<<<−−−−−−−−−−− levels = Nc; V_bat_unit = V_DC/levels; V_ac = V_bat_unit*m_a*levels/sqrt(2); % minimal sizing of the filter according to short circuit limits R_t = P_sc*(V_ac/P_rated)^2; Z_t = Z_sc*V_ac^2/(P_rated); X_t = sqrt(Z_t^2−R_t^2); L_t = X_t/(2*pi*f_p); %% Time settings 84 Analysis and simulation of a SSBC-MMCC converter 85 %****************************** %time constants %−−−−−EDIT−−>>> % time step of the solver T_s = 0.00001; % switching period T_sw = 0.001; % period of the grid fundamental T_p = 1/f_p; %<<<−−−−−−−−−−− % corresponding frequencies, inverse of the time constants f_s = 1/T_s; f_sw = 1/T_sw; % moving average filter settings FIR_MA_coeff = ones(1,round(1/f_p/T_s))/round(1/f_p/T_s); MA_p = ones(1,round(1/f_p/T_s))/round(1/f_p/T_s); MA_sw = ones(1,round(1/f_sw/T_s))/round(1/f_sw/T_s) %% Control %****************************** tau = T_sw/6; T_n = L_t/R_t; T_i = tau/R_t; PID_P = T_n/T_i; PID_I = 1/T_i; %% Component parameters %****************************** % IGBT Module FS100R17KE3 % CONDUCTION LOSSES %−−−−−EDIT−−>>> %diode (fit of exponential as voltage drop and linear resistance) V_f = 1.083; R_f = 7.638E−3; %IGBT (fit of exponential as voltage drop and linear resistance) V_ces = 0.923; Analysis and simulation of a SSBC-MMCC converter 86 R_ce = 13.01E−3; %<<<−−−−−−−−−−− % when turned off, the ideal diode and switch need to have a finite resistance % the values are set in order to prevent pushing the diode unwanted in % conduction mode G1 = 1E−16; G2 = G1/2*(V_bat_unit/V_f−1)*1.5; % SWITCHING LOSSES % points from datasheet graph %−−−−−EDIT−−>>> % IGBT turn on points I_on_vals = [25.35, 50.69, 76.04, 101.0, 126.4, 151.7, 176.7, 201.7]; E_on_vals = [10.77, 18.31, 25.69, 33.38, 42.62, 54, 68.46, 86.77]*1E−3; V_ref_on = 900; % IGBT turn off points I_off_vals = [25.35, 50.69, 76.04, 101.0, 128.1, 151.7, 176.7, 199.3]; E_off_vals = [10.62, 18.92, 25.69, 32, 38.15, 43.23, 48.62, 53.54]*1E−3; V_ref_off = 900; % diode reverse recovery points I_rec_vals = [20.35, 40.70, 61.05, 81.40, 102.1, 122.5, 142.5, 163.2, 183 .9, 203.2]; E_rec_vals = [11.91, 16.73, 20.86, 24.44, 27.47, 29.94, 31.85, 33.40, 34 .51, 35.19]*1E−3; V_ref_rec = 900; %<<<−−−−−−−−−−− % least−square polynomial fitting (order 2) k_rec = polyfit(I_rec_vals,E_rec_vals,2); k_on = polyfit(I_on_vals,E_on_vals,2); k_off = polyfit(I_off_vals,E_off_vals,2); a_rec = k_rec(1);b_rec = k_rec(2);c_rec = k_rec(3); a_on = k_on(1);b_on = k_on(2);c_on = k_on(3); a_off = k_off(1);b_off = k_off(2);c_off = k_off(3); Appendix C Simulink model The Simulink model is a graphical description of the system. Fig. C.1 shows the highest level diagram. The Simulink tool allows the user to define new blocks, consisting of lower level blocks which generate outputs for given inputs. Fig. C.2 to C.6 show the implementation of important custom blocks. The model descriptions were slightly simplified for clarity, omitting scopes and other non-essential elements. Finally, Fig. C.7 shows an augmented implementation of the block shown in C.3. This altered implementation automatically adapts the PWM when a battery module becomes unavailable. This implementation leads to the result shown in Fig. 5.12. 87 Analysis and simulation of a SSBC-MMCC converter 88 Figure C.1: The highest level diagram implements the top layer of the model: interface and control. Appendix D dq0 transform This appendix shows the full derivation of the transformation of Equation (4.1) to the dq0 reference frame. M−1 dq0     vCd vCq vC0     −M−1 dq0     vSd vSq vS0      =Lac d dt      M−1 dq0     id iq i0           +RM−1 dq0     id iq i0      ⇔     vCd −vSd vCq −vSq vC0−vS0      =Mdq0Lac d dt      MT dq0     id iq i0           +R     id iq i0      = LacMdq0 d dtMT dq0+R!     id iq i0      +LacMdq0MT dq0 d dt      id iq i0      = LacRγ(ωt)MαβγMT αβγ d dtRT γ(ωt) + R!     id iq i0      +Lac d dt      id iq i0      =     Lac      0−ω0 ω0 0 0 0 1      +R          id iq i0      +Lac d dt      id iq i0      The definition of the symbols and some properties are listed below. 95 Analysis and simulation of a SSBC-MMCC converter 96 Mαβγ =s2 3     1−1 2−1 2 0√3 2−√3 2 1 √2 1 √2 1 √2      Mdq0(t) =      cos ωt sin ωt 0 −sin ωt cos ωt 0 0 0 1     ·Mαβγ M−1 dq0=MT dq0MT dq0Mdq0=Mdq0MT dq0=I M−1 αβγ =MT αβγ MT αβγMαβγ =MαβγMT αβγ =I The first derivative with respect to time is given by d dtMdq0(t) = d dtRγ(ωt)·Mαβγ =     −ωsin ωt ω cos ωt 0 −ωcos ωt −ωsin ωt 0 0 0 1     ·Mαβγ d dtMT dq0(t) = MT αβγ ·d dtRT γ(ωt) =MT αβγ ·     −ωsin ωt −ωcos ωt 0 ωcos ωt −ωsin ωt 0 0 0 1     