Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00004549/00001
## Material Information- Title:
- Open circuit fault detection and localization through state observer in modular multilevel inverter
- Creator:
- Sen, Mustafa (
*author*) - Language:
- English
- Physical Description:
- 1 electronic file (64 pages). : ;
## Subjects- Subjects / Keywords:
- Automatic control ( lcsh )
- Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- Modular multilevel converters (MMC) have become attractive since being pro- posed by Marquardt in 2001 [1] by means of being available for medium and high voltage/power applications in the way of such properties as modularity, eciency, high output voltage and power quality apart from being alternative approach for conventional power converters. According to proposed multilevel converter topologies, MMCs are comprised of many submodules (SMs) which are made of semiconductors such as isolated gate bipolar transistors (IGBTs) and diodes. However, because of its design reliability is still problem to be handled. One of the biggest problems that threatens sustainability of MMC system operation is open circuit faults in IGBTs in SMs. In this thesis state observer which is used in small scale "5 level" MMC inverter system to observe, detect and locate the fault related with SM capacitor voltages. Phase shifted pulse width modulation (PS{PWM), voltage balancing (sorting) algorithm which is similar with reduced switching frequency (RSF) method and circulating current suppression controller (CCSC) [2] are applied to get result in open loop condition. The method is implemented via detailed Matlab/Simulink computer simulations.
- Thesis:
- Thesis (M.S.)--University of Colorado Denver.
- Bibliography:
- Includes bibliographic references
- System Details:
- System requirements: Adobe Reader.
- General Note:
- Department of Electrical Engineering
- Statement of Responsibility:
- by Mustafa Sen.
## Record Information- Source Institution:
- University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 952058749 ( OCLC )
ocn952058749
## Auraria Membership |

Downloads |

## This item has the following downloads: |

Full Text |

OPEN CIRCUIT FAULT DETECTION AND LOCALIZATION THROUGH
STATE OBSERVER IN MODULAR MULTILEVEL INVERTER by MUSTAFA SEN Bachelor of Science, Yildiz Technical University, 2011 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2016 2016 MUSTAFA SEN ALL RIGHTS RESERVED This thesis for the Master of Science degree by Mustafa Sen has been approved for the Department of Electrical Engineering by Jaedo Park, Chair Jan Bialasiewicz Dan Connors March 11, 2016 iii Sen, Mustafa (M.S., Electrical Engineering) Open Circuit Fault Detection and Localization Through State Observer In Modular Multilevel Inverter Thesis directed by Assistant Professor Jaedo Park ABSTRACT Modular multilevel converters (MMC) have become attractive since being pro- posed by Marquardt in 2001 [1] by means of being available for medium and high voltage/power applications in the way of such properties as modularity, efficiency, high output voltage and power quality apart from being alternative approach for con- ventional power converters. According to proposed multilevel converter topologies, MMCs are comprised of many submodules (SMs) which are made of semiconductors such as isolated gate bipolar transistors (IGBTs) and diodes. However, because of its design reliability is still problem to be handled. One of the biggest problems that threatens sustainability of MMC system operation is open circuit faults in IGBTs in SMs. In this thesis state observer which is used in small scale 5 level MMC inverter system to observe, detect and locate the fault related with SM capacitor voltages. Phase shifted pulse width modulation (PS-PWM), voltage balancing (sort- ing) algorithm which is similar with reduced switching frequency (RSF) method and circulating current suppression controller (CCSC) [2] are applied to get result in open loop condition. The method is implemented via detailed Matlab/Simulink computer simulations. The form and content of this abstract are approved. I recommend its publication. Approved: Jaedo Park IV ACKNOWLEDGEMENT At first, I would like to thank to my advisor, Dr. Jaedo Park, for his guidance, valuable advice and support during the whole period of the study. Also, I greatly thank and acknowledge the invaluable supports and guidance of Turkish Petroleum Corporation. I express my gratitude to my friends Muhannad Alaraj and Ersin Canak their advice and help on my research. I greatly appreciate Dr. Jan Bialasiewicz and Dr. Dan Connors for forming my dissertation defense committee, their valuable discussions. Finally, and most importantly, I would like to thank my family; they were always there cheering me up and stood by me through the good and bad times. v DEDICATION The thesis is dedicated to my family and entourage who have encouraged me to go further during my life. vi TABLE OF CONTENTS Chapter 1. Introduction................................................................ 1 2. System Modeling and Principles of Operation ............................... 10 2.1 Structure of MMC.................................................... 10 2.2 Operation Principles................................................ 12 3. MMC Control............................................................. 19 3.1 Modulation Methods ................................................. 19 3.1.1 Phase Shifted PWM (PS-PWM) ................................... 20 3.1.2 Phase Disposition PWM (PD-PWM)........................ 22 3.2 Voltage Balancing Algorithm......................................... 27 3.3 Circulating Current Suppression Controller (CCSC)................... 29 4. Fault Detection and Localization .................................... 33 4.1 Fault Types and Characteristics..................................... 33 4.2 Proposed Fault Detection Method..................................... 35 4.3 Proposed Fault Localization Method.................................. 35 4.3.1 Design of State Observer...................................... 36 5. Design Guideline and Simulation .................................... 39 5.1 Circuit Parameters ................................................. 39 5.2 Circulating Current PI Controller Gain Selection.................... 39 5.3 Simulation Results.................................................. 39 5.3.1 Ke=50......................................................... 40 5.3.2 Ke=0.1........................................................ 43 5.4 Discussion.......................................................... 45 6. Conclusion................................................................. 46 7. Future Work................................................................ 47 References................................................................. 48 vii Appendix A. Park Transformation.................................................. 52 viii LIST OF TABLES Table 1.1 Fault Types................................................................. 7 2.1 Operation of SM............................................................ 15 4.1 Failure Characteristics of Type 1......................................... 34 4.2 Failure Characteristics of Type 2......................................... 34 4.3 Failure Characteristics of Type 3......................................... 34 5.1 Simulation Parameters...................................................... 39 IX LIST OF FIGURES Figure 1.1 Two Level Three Phase VSC Topology............................................. 2 1.2 Three Level NPC Converter...................................................... 4 1.3 Flying Capacitor Multilevel Converter.......................................... 5 1.4 Cascaded Half-Bridge Multilevel Converter...................................... 6 1.5 MMC Inverter Topology.......................................................... 7 2.1 Voltage Waveform of 2 Level Converter......................................... 10 2.2 Single Module of MMC ......................................................... 12 2.3 MMC Output Voltage Synthesis.................................................. 13 2.4 Overall System Structure ..................................................... 14 2.5 Switching States of SM ....................................................... 15 3.1 Sinusoidal PWM and Output Waveforms For 2 Level VSC...................... 19 3.2 PS-PWM Reference vs Carriers.................................................. 21 3.3 Activated Number of SMs....................................................... 21 3.4 PD-PWM Reference vs Carriers.................................................. 22 3.5 Reference for Number of ON State SMs.......................................... 23 3.6 POD-PWM Reference vs Carriers................................................. 23 3.7 Reference for Number of ON State SMs.......................................... 24 3.8 APOD-PWM Reference vs Carriers................................................ 24 3.9 Reference for Number of ON State SMs.......................................... 25 3.10 Phase C Arm and Inner Difference Currents for PS-PWM.......................... 26 3.11 3 Phase Circulating Currents for PS-PWM....................................... 26 3.12 Phase C Arm and Inner Difference Currents for PD-PWM.......................... 27 3.13 3 Phase Circulating Currents for PD-PWM....................................... 27 3.14 Reduced Switching Frequency Voltage Balancing Algorithm............... 28 3.15 Transfer Function of the Circulating Currents in dq Reference Frame........... 30 x 3.16 Closed Loop d-q Axis Circulating Current Controllers............................... 31 3.17 Overall Circulating Current Suppression Controller................................. 32 4.1 Fault Types....................................................................... 33 4.2 Fault Detection Flowchart......................................................... 36 4.3 Fault Localization Flowchart...................................................... 37 5.1 Phase C Upper-Lower Arm Capacitor Voltages......................................... 40 5.2 3 Phase Circulating Currents ...................................................... 41 5.3 Phase C Vuci and Vuciest.............................................. 41 5.4 Phase C Vuc2 and Vuc2est.............................................. 42 5.5 Phase C Vuc3 and Vuc3est.............................................. 42 5.6 Phase C VUC4 and VUC4est.............................................. 43 5.7 Fault Status of SMs............................................................... 43 5.8 Phase C VMCi and Vuciest for Ke=0.1.............................................. 44 5.9 Fault Status of SMs for Ke=0.1..................................................... 44 A.l The stationary abc reference frame and the rotating dq reference frame ... 52 xi 1. Introduction Developments for the new technologies in the last century increased the atten- tion in electric power systems. Studies on electric power generation, conversion and transmission devices have become more and more important day by day. Also, to protect and preserve the environment from the pollution which is caused by nuclear and fossil energy sources like oil, coal and natural gas the interest in electrical power generation from green energy (renewable) sources such as wind power and solar sys- tems has been enhanced and they are supposed to play important role in world-wide energy production in the following years. Not only industrial applications even the electrical network requirements display the importance of energy supply and control in the recent researches. As a consequence, power conversion and control in power transmission process has to be reliable, safe and available in order to accomplish all the requirements. In [3] it is stated that in a power transmission system the most important process is to control active and reactive power flow to keep the system voltage stable. This goal can be accomplished by use of power converters through its ability to convert energy from DC to AC or vice versa. So far there has been two types of configurations as regards three phase power converters which high voltage direct current (HVDC) transmission system can utilize. These are conventional voltage source converters (VSCs) which is commonly used in the area and shown in Figure 1.1 and current source converters (CSCs). Main characteristic of VSCs is that as it is shown in Figure 1.1 it is a composition of three identical half-bridge converters and it operates with specified vector control strategy which can perform active and reactive power control separately [4], This makes it convenient for connection to weak AC networks without local voltage sources. Also, during the power reversal the DC voltage polarity remains the same for VSCs based transmission system and the power transfer depends on only the direction of the DC current.On the other hand, in a Current Source Converter the 1 DC current is fixed with a small ripple using a large inductor, thus forming a current source on the DC side. The direction of power flow through a CSCs is determined by the polarity of the DC voltage while the direction of current flow remains the same. Figure 1.1: Two Level Three Phase VSC Topology Although these power converters have lately been used in high voltage direct current (HVDC) power transmission applications, However, according to [5] there are some drawbacks for both. CSCs require large reactive power quantities during the process and it is inconvenient to control active and reactive power independently. Also, because of generating low frequency harmonics it causes losses and expensive Liters needed. Additionally, conventional 2 level VSCs produce large high frequency harmonics which result in bigger losses to compare with CSCs due to higher switching frequencies. Moreover, its design need a large number of switches that are connected in series.Thus, this situation may cause multiple failure points. To overcome these problems several different multilevel topologies have been pro- posed and the main reason for the interest on multilevel converters instead conven- 2 tional two-level converters lies in the improved quality of their output waveform, possibility to achieve higher power levels and higher efficiency [6]. Some of the most common multilevel topologies are: * Neutral-Point Clamped (NPC) * Flying Capacitor (FC) * Cascaded H-Bridge (CHB) * Modular Multilevel Converters (MMC) The NPC multilevel converter was initially proposed as a three-level inverter as shown in Figure 1.2. The midpoint of the switches is connected to the neutral point of the converter over the clamping diodes, so it enables zero voltage level generation. Thanks to this, for the same DC-link voltage, the voltage which the devices in the converter have to tolerate is reduced to half in comparison with the two-level topol- ogy. On the contrary, this topology has several disadvantages such as under certain operating conditions the NPC converter may experience unbalanced capacitor volt- ages, creating a potential between the neutral point and ground and causing distorted output waveforms. In FC converter topology which is shown in Figure 1.3, each capacitor in the phase is charged to a different voltage level, therefore by changing the states of the switches, various output voltage levels can be obtained [7]. This topology can have phase redundant switching states that can be used for capacitor voltage regulation and it brings sort of advantage compared to the NPC converter topology. In the way of the energy storage in the capacitors, the converter can ride through short duration outages. As a disadvantage, before the start-up, capacitors have to be pre-charged which is known as initialisation as requirement. Also this topology presents unequal duty distribution between the switches. Even though the FC converter topology can 3 fa fb Vc Figure 1.2: Three Level NPC Converter be extended to an arbitrary number of cells, the addition of capacitors leads to an increase in cost. Thus the number of level is usually limited with four [6,8]. The CHB topology is based on the series connection of single-phase full-bridge inverter cells with isolated DC supplies. It is demonstrated below in Figure 1.4. The main advantages of the CHB topology over the NPC and FC ones are its modular structure and the possibility to have an independent control over the zero-sequence component in the current. In case of rectifier applications, the need of many isolated DC sources in series limits the number of cells in the leg, keeping this topology unfavourable for bidirectional power applications [4,6]. However, a proposal for CHB approach in HVDC applications using a re-injection circuit can be found in [8]. Since MMCs were proposed to be alternative approach for existing ones, they have become commercially attractive for medium and high power applications and brought many advantages over conventional types of converters [9-12],In Figure 1.5 n + 1 level MMC Inverter topology is shown. These benefits are mainly 4 Figure 1.3: Flying Capacitor Multilevel Converter * modularity and scalability * lower switching frequency of individual levels, so in comparison with tradi- tional VSCs obtaining the same waveform properties with lower switching losses * improved reliability in case some modules have fault as converter can function * magnitude of harmonics is significantly reduced or possibly eliminated, there- fore no need for filter banks * flexible control of the voltage level and simple realization of redundancy if required * no requirement for expensive transformers On the other hand, one of the biggest challenges about MMCs is reliability. MMCs consist of a large number of power switching devices such as IGBTs in SMs depends 5 n Figure 1.4: Cascaded Half-Bridge Multilevel Converter on volume of the application and these switches have to be taken into consideration as potential failure points. According to table 1.1 which shows SM and converter level faults respectively and also is given in [13],in SM level, faults which are related with IGBTs such as open circuit and intermittent gate misfiring faults are shown as one of the most common fault types which may disrupt the operation of MMC or even destroy it due to effect on current and voltage in MMC [14], So it is clear that fault detection and localization in a short time after occurrence are really vital for sustainability of the MMC operation. There are several methods that are proposed on open circuit fault detection in MMC. In [15], Kalman Fitering (iv) is being recommended to detect the fault con- dition by means of comparison between the measured and estimated inner difference 6 Klc Figure 1.5: MMC Inverter Topology Table 1.1: Fault Types Sub-module level Capacitor Fault in capacitor structure Diode Open diode faults Short Circuit Faults IGBT Open circuit faults Intermittent eate misfiring; faults Converter level Capacitor voltage unbalance Circulating current among three phase unit Unbalance between upper and lower arm voltage Control of energy stored in the leg Single phase to ground fault Double phase to ground fault Triple phase to ground fault current based on specified current difference threshold for some certain period. After fault is detected, SM capacitor voltages in the concerned phase which is identified as 7 faulty are compared with minimum capacitor voltage value in the same arm for some threshold period and value to locate faulty SM in the upper\lower arm. Another proposed method is based on sliding mode observer (SMO) which has been presented in [16]. In here, voltage and current relationships between the both sides of SMs are calculated under normal and open circuit fault case in each switch. Through SMO estimated and actual states are compared if difference between them is more than threshold value for certain period which is already specified in terms of systems sen- sitivity, fault is detected in this way. According to [17], it is stated that this SMO method is accurate, but not fast for MMCs with high number of SMs. In this case, it may be encountered with additional damages because of instant increase of faulty SM capacitor voltages. So, it suggests different method which is based on voltage across inductor and arm current observation in the upper\lower arms in any phase instead of using observer to make this process faster. Another presented approach relies on adaptive backstepping observer to take place of sensors and reduce the complexity of implementation which is given in [18]. Apart from these methods, state observers can be designed to detect and locate the open circuit faults in SMs by reducing number of measurement sensors in parallel with total cost by estimating capacitor voltages and output currents [19]. In this thesis study, design of state observer approach for linear systems is benehtted to estimate capacitor voltages in SMs locate the faulty module to be alternative for existing approaches. To obtain proper results in open loop condition PS-PWM technique and voltage balancing algorithm which is similar with the way expressed in [2] are employed in 5 Level MMC inverter. In [20], PS-PWM is presented and technical aspects are discussed. The thesis is organized in following way; in chapter 2, structure modelling and operational principles of MMC are explained. Afterwards, chapter 3 is about control approach related with the preferred PWM technique, the capacitor voltage balancing 8 algorithm and circulating current suppression control (CCSC). In chapter 4, fault types, characteristics on SMs with proposed fault detection and localization meth- ods are explained. Chapter 5 is including design guideline and simulation results. Conclusion is added in chapter 6. 9 2. System Modeling and Principles of Operation 2.1 Structure of MMC It can be stated that MMCs are sort of VSC and the related principles of operation for VSCs can be applied to it. Because if only 2 SMs were used in MMC (we could say that one for positive DC connection and the other one is negative), synthesized output voltage waveform would be in Figure 2.1 which is like 2 level voltage source converter. So, operation of this MMC would be equivalent to VSC. Also, It can be used as rectifier and inverter in back to back HVDC power transmission systems that Siemens has a plan of putting this converter into practical applications with the trade name HVDC-plus. The system configuration of the HVDC-plus has a power of 400 MVA, a dc link voltage of 200 kV, and each arm composed of 200 SMs [21]. Fn ___________ _____________________ -tv 2 --------------------------- Figure 2.1: Voltage Waveform of 2 Level Converter Additionally, The MMC configuration also offers more advantages over the tra- ditional VSCs [5]. These are * Not all the switches in a leg are opened or closed at the same time as they would in a two-level VSC, but they are operated at different time instants to follow the sinusoidal reference command more closely, the switching frequency of each switch can thus be low, while still generating a large apparent switching frequency. * The voltage blocking requirements of an individual switch is limited to the voltage across the modules capacitor. 10 * If enough modules are used, the voltage across each switch can be low enough not to require series connected switches. In a 2-level VSC, each arm (valve) must be able to block several hundreds of kVs while open. An IGBT is usually rated for a few kVs only, so multiple switches must be connected in series to achieve the desired blocking capabilities. During switching, all these devices must operate together. If one IGBT operation is delayed, it is exposed to a high voltage, which could damage it. In the case of the MMC, the high number of modules, each only blocking the voltage across its internal capacitor eliminates this drawback and improves the failure rate. * The modular concept allows operation even if some modules have failed. Failed modules can simply be bypassed, and kept bypassed until it is possible to replace them, and operation can continue. The DC bus voltage is then divided among the remaining N-l modules and normal operation can continue. * It is possible to design a MMC converter with spare modules which can be used if one module were to fails. If spare modules are available, they can be inserted in the arm as needed. As it can be seen from the Figure 2.4 that demonstrates the overall system, SM is the basic building cell for MMC. There may be hundreds or thousands of SMs in MMC in terms of required power level depends on application [22], SMs consist of two IGBTs (Si,S2), two diodes (Di,D2) which are connected in parallel with them and a submodule capacitor (C^m). Also these diodes are used due to protection of IGBT switches. Figure 2.2 shows SM diagram. In a three-phase MMC system, for each phase there are phase legs which are composed of upper and lower arms. These arms are formed by series connected n number of SMs. There is also an inductor in each arm, Larm, in order to smooth the voltage difference (to reduce circulating currents between phase units) that is 11 [ produced when a SM is connected or disconnected [23]. 2.2 Operation Principles The general concept of multilevel converter is the synthesis of a sinusoidal voltage by several levels of voltages. In case of MMC, these voltage levels are obtained from the capacitor voltages, Vc, of each SM. At any instant, a number of SMs that are switched on equals n totally which is including upper and lower arm in any phase, so that the voltage at the converter terminals equals the instantaneous value of the voltage to be synthesized [24] as it is shown in in Figure 2.3 which are output voltages of 3, 5 and 9 level MMCs varies between Vdc/2 respectively. As it is seen if number of voltage level is increased, resolution of sinusoidal waveform for the output voltage gets better. Besides, average Vc is denoted as Vc = (2.1) n As regards gate signal of related IGBT switch SM has two states in normal operation. These are ON and OFF states. Figure 2.5 indicates the concerned states respectively. *ON State Si is switched on, S2 is switched off As upper IGBT is in the conducting mode in any SM in any phase, Vc equals to 12 i VdÂ£ \ 1 1 1 t Vdc 2 ' VdÂ£ 2 t Figure 2.3: MMC Output Voltage Synthesis the SM output voltage (Vc = Tdm) regardless of current direction. If the current is positive it freewheels through anti-parallel diode D1 and Vc increases or when current polarity is negative it flows through the Si and discharges the C^m. *Off State Si is switched off, S2 is switched on In here, the Td,m doesnt change. Because related SM is bypassed. Arm current either flows through the S2 or the D2 depending on its direction. Thereby, Td,m is zero. Table 2.1 shows the operational status of the SM in terms of complementary switching states of IGBTs. Also, the relation between V^m and Vc can be formulated in the following way: 13 MMC VSI Figure 2.4: Overall System Structure SM SWITCHED ON SM SWITCHED OFF Figure 2.5: Switching States of SM bsm,i.jk SijkVc,ijk (2/2) Table 2.1: Operation of SM SM State S Si s2 ^sm Arm Current n wsm vc ON 1 ON OFF uc Positive Charge Increased Negative Discharge Decreased OFF 0 OFF ON 0 Positive Bypass Unchanged Negative The arm voltage can be computed considering the status of the SMs switches, Syk, as follows [25]: Vn = k= 1 SVjkl/o)ijk (2.3) 15 where i = u,Â£ represents the upper and lower arm, respectively; j = a,b,c is the phase; and k = 1,2,..., n denotes the SM. The arm currents can be determined as luj = + Idiffj (2-4) hi = -y- + biff, (2.5) According to [26] Idiffj has two components. These are one third of dc source current (/<&/3) and the other part is circulating current (ICirc,j) So the previous equation can be written in the following way. 4uj = + -^ + 4circj, (2.6) Aj = + icircj (2.7) where IcirCj is the circulating current for phase j, and /CirCa + /CirCb + /CirCc = 0. These circulating currents have no effect on the ac side or dc side voltages. It can be proved based on arm current equations as it is demonstrated below [5]. On the other hand, they have significant impact on SMs capacitor voltages and the rating values of the MMC components [25]. So Lima becomes IQ, I dc r ' g ~T~ -^circ,a g \ -^circja Iua I la. lie j Ala Ila 3~ + ua 2 Adc '~3~ Alia, + Ii la (2,8) (2,9) Icirc,b and ICirc,c can also be defined as in the following. 16 Iarct=-If+1^^- (2.10) (2.11) By summing all three circulating currents it is obtained that Icirc,a I Icirc,b I Icirc,c IirrM I hzrrj Idc Va ~L I (.a, ~3~ + 2 Vc Vb "h lib ~3~ + 2 Vc Vc "h I(,c ~3~ + 2 Idc Vc 0 Via + Vib + Vic + Va + lib + Vc Vdc "h Vdc (2.12) Simultaneously, correlation between ac side and dc side voltages of the MMC is ex- tracted through Kirchoffs Voltage and Current Laws (KYL-KCL). Vic 2 Vdc V vdiffj Ki + L iIai Vj + La dt dlf.\ V -RarmVij I Vj ^ + Raimiij Vj d dt Kj + Vj+ harm (+ ~7rl + -Rarm(Vj + Vj) dt dt 1 dlj ^(Vj Kj harm Varm/j) d/ciiflf; + i?arm/diffj dt 1 i(W Kj vy (2.13) where V^ represents inner voltage difference between phase units in MMC and i?arm is the equivalent resistance of the SMs in an arm. It is related to number of the SMs in an arm via the resistance of one of SMs IGBT switch, Rsm, 17 -Rarm = nRsm. (2.14) Equivalent converter arm capacitance (C^m) when n modules are inserted in the related converter arm equals to the total capacitance of the n series connected sub- module capacitors as it is shown in equation 2.15. n v~/arm Csm/^ (2.15) 18 3. MMC Control 3.1 Modulation Methods If two level VSC which is in the form of half bridge circuit as it is shown in Figure 1.1 is considered, conventional pulse width modulation which is using one modulating signal and one carrier waveforms to generate gate signals for complimentary IGBT switches in the same phase [4]. Figure 3.1 pictures an example for sinusoidal pulse width modulation and the result. As it is seen from the Figure 3.1 modulating signal is compared with triangular carrier wave signal. As a result of it, in case modulating signal is higher than carrier, this means upper IGBT in VSC is on, otherwise if it is lower than carrier, lower one is on. This PWM method is repeated with proper phase shifts in terms of other phases when 3 phase two level VSC is used. For modular multilevel converters, there are several of these half-bridge circuits (SMs) that all of them has to be individually controlled. The solution for this is to use multi-carrier PWM methods which means that one carrier waveform for each SM in MMC. Time (sec) -0 5 -------1------------1----------i-----------I---------1---------- 0.3 0 0 3 05 0.37 0.375 0.38 0.38 5 0.39 Time (sec) Figure 3.1: Sinusoidal PWM and Output Waveforms For 2 Level VSC In [10] and [27] details about multilevel converter PWM methods are given. There are two specified methods in terms of MMC multi-carrier modulation techniques. These are phase-shifted and phase disposition PWM methods. For both methods 19 there is one carrier waveform for each SM in the upper\lower arm of the MMC phase leg. In both methods, multiple carrier waveforms for number of SMs in the related arm are compared with single arm voltage reference signal (modulating) and this comparison dictates how many submodules need to be switched on or bypassed. Apart from these methods there are other multilevel modulation techniques such as Nearest Level Modulation (NLM) and Space Vector Modulation (SVM). However, these methods have some drawbacks [28]. For instance, NLM technique is suitable with the converters which have large number of SMs due to small voltage steps. This means if it is used with small scale MMC which means low number of switchings this may cause larger voltage fluctuation in the capacitor voltages and SVM brings implementation complexity. 3.1.1 Phase Shifted PWM (PS-PWM) Phase-shifted PWM for modular multilevel converters use one carrier for each submodule in the MMC and these carrier waveforms have the same amplitude and frequency which are shifted by A0 which depends on number of SMs in the arm. This angle A6 is calculated using the equation 3.1. A0 360 (3.1) n n indicates number of SMs in an arm. Two key parameters are the amplitude and frequency modulation index in this method [20]. The equations to calculate these parameters are shown in equations 3.2 and 3.3. The frequency modulation index (nrif) relates the frequency of the carrier wave (/c) to the frequency of the reference sine wave (/r). m, = Â£ (3.2) Jr The amplitude modulation index is the ratio of the amplitude of the carrier waveform and reference waveform. 20 (3-3) Ar 771(1 = X 2 Figure 3.2 shows an example of PSC-PWM with 4 carrier and 1 reference wave- forms which dictates how many submodules need to be inserted in the arm in order to achieve the desired voltage level as it is reflected in Figure 3.3. Also there is 27t/4 phase shift between each carrier waveform which satisfies equation 3.1. .Q 2 I____I________I________I_______I________I_______I________I_______I________I_______I 0.33 0.332 0.334 0.336 0.338 0.34 0.342 0.344 0.346 0.348 0.35 Time (sec) Figure 3.2: PS-PWM Reference vs Carriers Figure 3.3: Activated Number of SMs 21 3.1.2 Phase Disposition PWM (PD-PWM) Phase Disposed PWM is similar to the previously described methods in that again the number of carrier waveforms is equal to the number of submodules in each arm. Also the same amplitude. The differences are that there is no phase shift between the carriers and they are displaced with respect to zero axis. Also amplitude of them (Ac) depends on the equation shown in equation 3.4 Ac = - (3.4) n In [29] it is stated that there are various types of phase disposed PWM techniques which depend on whether or not the carrier waveforms are 0 degrees out of phase or 180 degrees out of phase. The first is phase-disposition PWM. Figure 3.4 shows an example of phase-disposition PWM by showing 4 carrier waveforms superimposed with a single sinusoidal reference waveform. Figure 3.5 shows the reference signal using PD-PWM. Here, each carrier waveform has the same frequency and phase but amplitude of carriers is 0.25 which is calculated by using the equation 3.4. Time (sec) Figure 3.4: PD-PWM Reference vs Carriers 22 Figure 3.5: Reference for Number of ON State SMs The next level-shifted technique discussed is phase opposite disposition PWM which is identical to phase-disposition PWM except that the lower half carrier wave- forms are 180 degrees out of phase. Figure 3.6 shows the carrier waveforms for phase opposite disposition PWM and we can see that the bottom half carrier waveforms are 180 degrees out of phase. Figure 3.7 then shows the resultant waveform. Time (sec) Figure 3.6: POD-PWM Reference vs Carriers 23 3.5 - CZJ CO Â£2 o 0>1 'i Â£ 0.33 0.332 0.334 0.336 0.338 0.34 0.342 0.344 0.346 0.348 0.35 Time (sec) Figure 3.7: Reference for Number of ON State SMs The final level-shifted technique to be investigated is alternating phase opposite disposition PWM in which every other carrier waveform is phase shifted 180 degrees out of phase. Figure 3.8 shows the carrier waveforms for APOD-PWM and Figure 3.9 shows the resultant waveform. Time (sec) Figure 3.8: APOD-PWM Reference vs Carriers In this thesis, PS-PWM method is chosen based on the assumptions given in the [30] which says concisely PS-PWM can automatically suppress low order harmonics 24 3.5 - CO Pi 2 O CL 'i 0.33 0.332 0.334 0.336 0.338 0.34 0.342 0.344 0.346 0.348 0.35 Time (sec) Figure 3.9: Reference for Number of ON State SMs for MMCs. Also when the simulation which is used in the theis study is run with PS-PWM and PD-PWM seperately, Figures 3.10 and 3.11 show that phase C arm currents, inner difference current which is including circulating current as explained in chapter 2 have less oscillation in comparison with the Figures 3.12 and 3.13 obtained by use of PD-PWM method. Also modulating signals (MUjref = VUjref, Mgjref = Vejref)which means voltage references for the upper and lower arms as input for PWM block in any phase of MMC is calculated in the following way [31]; VUjref ~ ~^ ^diffJref ~ ^Jre/ (^-b) = Vdiff,jref + Vjrf,f (3-6) Vjref represents the desired output phase voltage of MMC [32] and can be expressed as V V -^(mcos(ut +
ref 2
reference inner unbalance voltage as explained in chapter 2 and it is obtained from |

Full Text |

PAGE 1 OPENCIRCUITFAULTDETECTIONANDLOCALIZATIONTHROUGH STATEOBSERVERINMODULARMULTILEVELINVERTER by MUSTAFASEN BachelorofScience,YildizTechnicalUniversity,2011 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience ElectricalEngineering 2016 PAGE 2 c 2016 MUSTAFASEN ALLRIGHTSRESERVED PAGE 3 ThisthesisfortheMasterofSciencedegreeby MustafaSen hasbeenapprovedforthe DepartmentofElectricalEngineering by JaedoPark,Chair JanBialasiewicz DanConnors March11,2016 iii PAGE 4 Sen,MustafaM.S.,ElectricalEngineering OpenCircuitFaultDetectionandLocalizationThroughStateObserverInModular MultilevelInverter ThesisdirectedbyAssistantProfessorJaedoPark ABSTRACT ModularmultilevelconvertersMMChavebecomeattractivesincebeingproposedbyMarquardtin2001[1]bymeansofbeingavailableformediumandhigh voltage/powerapplicationsinthewayofsuchpropertiesasmodularity,eciency, highoutputvoltageandpowerqualityapartfrombeingalternativeapproachforconventionalpowerconverters.Accordingtoproposedmultilevelconvertertopologies, MMCsarecomprisedofmanysubmodulesSMswhicharemadeofsemiconductors suchasisolatedgatebipolartransistorsIGBTsanddiodes.However,becauseof itsdesignreliabilityisstillproblemtobehandled.Oneofthebiggestproblemsthat threatenssustainabilityofMMCsystemoperationisopencircuitfaultsinIGBTs inSMs.Inthisthesisstateobserverwhichisusedinsmallscale"5level"MMC invertersystemtoobserve,detectandlocatethefaultrelatedwithSMcapacitor voltages.PhaseshiftedpulsewidthmodulationPS{PWM,voltagebalancingsortingalgorithmwhichissimilarwithreducedswitchingfrequencyRSFmethodand circulatingcurrentsuppressioncontrollerCCSC[2]areappliedtogetresultinopen loopcondition.ThemethodisimplementedviadetailedMatlab/Simulinkcomputer simulations. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:JaedoPark iv PAGE 5 ACKNOWLEDGEMENT Atrst,Iwouldliketothanktomyadvisor,Dr.JaedoPark,forhisguidance, valuableadviceandsupportduringthewholeperiodofthestudy. Also,Igreatlythankandacknowledgetheinvaluablesupportsandguidanceof TurkishPetroleumCorporation. IexpressmygratitudetomyfriendsMuhannadAlarajandErsinCanaktheir adviceandhelponmyresearch. IgreatlyappreciateDr.JanBialasiewiczandDr.DanConnorsforformingmy dissertationdefensecommittee,theirvaluablediscussions. Finally,andmostimportantly,Iwouldliketothankmyfamily;theywerealways therecheeringmeupandstoodbymethroughthegoodandbadtimes. v PAGE 6 DEDICATION Thethesisisdedicatedtomyfamilyandentouragewhohaveencouragedmetogo furtherduringmylife. vi PAGE 7 TABLEOFCONTENTS Chapter 1.Introduction...................................1 2.SystemModelingandPrinciplesofOperation................10 2.1StructureofMMC............................10 2.2OperationPrinciples...........................12 3.MMCControl..................................19 3.1ModulationMethods..........................19 3.1.1PhaseShiftedPWMPS-PWM................20 3.1.2PhaseDispositionPWMPD-PWM..............22 3.2VoltageBalancingAlgorithm......................27 3.3CirculatingCurrentSuppressionControllerCCSC.........29 4.FaultDetectionandLocalization.......................33 4.1FaultTypesandCharacteristics....................33 4.2ProposedFaultDetectionMethod...................35 4.3ProposedFaultLocalizationMethod..................35 4.3.1DesignofStateObserver.....................36 5.DesignGuidelineandSimulation.......................39 5.1CircuitParameters...........................39 5.2CirculatingCurrentPIControllerGainSelection...........39 5.3SimulationResults............................39 5.3.1 K e =50...............................40 5.3.2 K e =0.1..............................43 5.4Discussion................................45 6.Conclusion....................................46 7.FutureWork...................................47 References ......................................48 vii PAGE 8 Appendix A.ParkTransformation..............................52 viii PAGE 9 LISTOFTABLES Table 1.1FaultTypes..................................7 2.1OperationofSM...............................15 4.1FailureCharacteristicsofType1......................34 4.2FailureCharacteristicsofType2......................34 4.3FailureCharacteristicsofType3......................34 5.1SimulationParameters............................39 ix PAGE 10 LISTOFFIGURES Figure 1.1 TwoLevelThreePhaseVSCTopology ....................2 1.2 ThreeLevelNPCConverter ..........................4 1.3 FlyingCapacitorMultilevelConverter .....................5 1.4 CascadedHalf-BridgeMultilevelConverter ..................6 1.5 MMCInverterTopology ............................7 2.1 VoltageWaveformof2LevelConverter ....................10 2.2 SingleModuleofMMC ............................12 2.3 MMCOutputVoltageSynthesis ........................13 2.4 OverallSystemStructure ...........................14 2.5 SwitchingStatesofSM ............................15 3.1 SinusoidalPWMandOutputWaveformsFor2LevelVSC ..........19 3.2 PS-PWMReferencevsCarriers ........................21 3.3 ActivatedNumberofSMs ...........................21 3.4 PD-PWMReferencevsCarriers ........................22 3.5 ReferenceforNumberofONStateSMs ....................23 3.6 POD-PWMReferencevsCarriers .......................23 3.7 ReferenceforNumberofONStateSMs ....................24 3.8 APOD-PWMReferencevsCarriers ......................24 3.9 ReferenceforNumberofONStateSMs ....................25 3.10 PhaseCArmandInnerDierenceCurrentsforPS{PWM ..........26 3.11 3PhaseCirculatingCurrentsforPS{PWM ..................26 3.12 PhaseCArmandInnerDierenceCurrentsforPD{PWM ..........27 3.13 3PhaseCirculatingCurrentsforPD{PWM ..................27 3.14 ReducedSwitchingFrequencyVoltageBalancingAlgorithm ..........28 3.15 TransferFunctionoftheCirculatingCurrentsindqReferenceFrame .....30 x PAGE 11 3.16 ClosedLoopd-qAxisCirculatingCurrentControllers .............31 3.17 OverallCirculatingCurrentSuppressionController ..............32 4.1 FaultTypes ..................................33 4.2 FaultDetectionFlowchart ...........................36 4.3 FaultLocalizationFlowchart ..........................37 5.1 PhaseCUpper{LowerArmCapacitorVoltages ................40 5.2 3PhaseCirculatingCurrents .........................41 5.3 PhaseC V uc 1 and V uc 1 est ............................41 5.4 PhaseC V uc 2 and V uc 2 est ............................42 5.5 PhaseC V uc 3 and V uc 3 est ............................42 5.6 PhaseC V uc 4 and V uc 4 est ............................43 5.7 FaultStatusofSMs ..............................43 5.8 PhaseC V uc 1 and V uc 1 est for K e =0.1 ......................44 5.9 FaultStatusofSMsfor K e =0.1 ........................44 A.1 Thestationaryabcreferenceframeandtherotatingdqreferenceframe ...52 xi PAGE 12 1.Introduction Developmentsforthenewtechnologiesinthelastcenturyincreasedtheattentioninelectricpowersystems.Studiesonelectricpowergeneration,conversionand transmissiondeviceshavebecomemoreandmoreimportantdaybyday.Also,to protectandpreservetheenvironmentfromthepollutionwhichiscausedbynuclear andfossilenergysourceslikeoil,coalandnaturalgastheinterestinelectricalpower generationfromgreenenergyrenewablesourcessuchaswindpowerandsolarsystemshasbeenenhancedandtheyaresupposedtoplayimportantroleinworld-wide energyproductioninthefollowingyears.Notonlyindustrialapplicationseventhe electricalnetworkrequirementsdisplaytheimportanceofenergysupplyandcontrol intherecentresearches.Asaconsequence,powerconversionandcontrolinpower transmissionprocesshastobereliable,safeandavailableinordertoaccomplishall therequirements. In[3]itisstatedthatinapowertransmissionsystemthemostimportantprocess istocontrolactiveandreactivepowerowtokeepthesystemvoltagestable.This goalcanbeaccomplishedbyuseofpowerconvertersthroughitsabilitytoconvert energyfromDCtoACorviceversa.Sofartherehasbeentwotypesofcongurations asregardsthreephasepowerconverterswhichhighvoltagedirectcurrentHVDC transmissionsystemcanutilize.Theseareconventionalvoltagesourceconverters VSCswhichiscommonlyusedintheareaandshowninFigure1.1andcurrent sourceconvertersCSCs.MaincharacteristicofVSCsisthatasitisshowninFigure 1.1itisacompositionofthreeidenticalhalf{bridgeconvertersanditoperateswith speciedvectorcontrolstrategywhichcanperformactiveandreactivepowercontrol separately[4].ThismakesitconvenientforconnectiontoweakACnetworkswithout localvoltagesources.Also,duringthepowerreversaltheDCvoltagepolarityremains thesameforVSCsbasedtransmissionsystemandthepowertransferdependsononly thedirectionoftheDCcurrent.Ontheotherhand,inaCurrentSourceConverterthe 1 PAGE 13 DCcurrentisxedwithasmallrippleusingalargeinductor,thusformingacurrent sourceontheDCside.ThedirectionofpowerowthroughaCSCsisdeterminedby thepolarityoftheDCvoltagewhilethedirectionofcurrentowremainsthesame. Figure1.1: TwoLevelThreePhaseVSCTopology Althoughthesepowerconvertershavelatelybeenusedinhighvoltagedirect currentHVDCpowertransmissionapplications,However,accordingto[5]there aresomedrawbacksforboth.CSCsrequirelargereactivepowerquantitiesduring theprocessanditisinconvenienttocontrolactiveandreactivepowerindependently. Also,becauseofgeneratinglowfrequencyharmonicsitcauseslossesandexpensive ltersneeded.Additionally,conventional2levelVSCsproducelargehighfrequency harmonicswhichresultinbiggerlossestocomparewithCSCsduetohigherswitching frequencies.Moreover,itsdesignneedalargenumberofswitchesthatareconnected inseries.Thus,thissituationmaycausemultiplefailurepoints. Toovercometheseproblemsseveraldierentmultileveltopologieshavebeenproposedandthemainreasonfortheinterestonmultilevelconvertersinsteadconven2 PAGE 14 tionaltwo-levelconvertersliesintheimprovedqualityoftheiroutputwaveform, possibilitytoachievehigherpowerlevelsandhighereciency[6].Someofthemost commonmultileveltopologiesare: Neutral-PointClampedNPC FlyingCapacitorFC CascadedH-BridgeCHB ModularMultilevelConvertersMMC TheNPCmultilevelconverterwasinitiallyproposedasathree-levelinverteras showninFigure1.2.Themidpointoftheswitchesisconnectedtotheneutralpoint oftheconverterovertheclampingdiodes,soitenableszerovoltagelevelgeneration. Thankstothis,forthesameDC-linkvoltage,thevoltagewhichthedevicesinthe converterhavetotolerateisreducedtohalfincomparisonwiththetwo-leveltopology.Onthecontrary,thistopologyhasseveraldisadvantagessuchasundercertain operatingconditionstheNPCconvertermayexperienceunbalancedcapacitorvoltages,creatingapotentialbetweentheneutralpointandgroundandcausingdistorted outputwaveforms. InFCconvertertopologywhichisshowninFigure1.3,eachcapacitorinthe phaseischargedtoadierentvoltagelevel,thereforebychangingthestatesofthe switches,variousoutputvoltagelevelscanbeobtained[7].Thistopologycanhave phaseredundantswitchingstatesthatcanbeusedforcapacitorvoltageregulation anditbringssortofadvantagecomparedtotheNPCconvertertopology.Intheway oftheenergystorageinthecapacitors,theconvertercanridethroughshortduration outages.Asadisadvantage,beforethestart-up,capacitorshavetobepre-charged whichisknownasinitialisationasrequirement.Alsothistopologypresentsunequal dutydistributionbetweentheswitches.EventhoughtheFCconvertertopologycan 3 PAGE 15 Figure1.2: ThreeLevelNPCConverter beextendedtoanarbitrarynumberofcells,theadditionofcapacitorsleadstoan increaseincost.Thusthenumberoflevelisusuallylimitedwithfour[6,8]. TheCHBtopologyisbasedontheseriesconnectionofsingle-phasefull-bridge invertercellswithisolatedDCsupplies.ItisdemonstratedbelowinFigure1.4.The mainadvantagesoftheCHBtopologyovertheNPCandFConesareitsmodular structureandthepossibilitytohaveanindependentcontroloverthezero-sequence componentinthecurrent.Incaseofrectierapplications,theneedofmanyisolated DCsourcesinserieslimitsthenumberofcellsintheleg,keepingthistopology unfavourableforbidirectionalpowerapplications[4,6].However,aproposalforCHB approachinHVDCapplicationsusingare-injectioncircuitcanbefoundin[8]. SinceMMCswereproposedtobealternativeapproachforexistingones,they havebecomecommerciallyattractiveformediumandhighpowerapplicationsand broughtmanyadvantagesoverconventionaltypesofconverters[9{12].InFigure1.5 n +1"levelMMCInvertertopologyisshown.Thesebenetsaremainly 4 PAGE 16 Figure1.3: FlyingCapacitorMultilevelConverter modularityandscalability lowerswitchingfrequencyofindividuallevels,soincomparisonwithtraditionalVSCsobtainingthesamewaveformpropertieswithlowerswitchinglosses improvedreliabilityincasesomemoduleshavefaultasconvertercanfunction magnitudeofharmonicsissignicantlyreducedorpossiblyeliminated,thereforenoneedforlterbanks exiblecontrolofthevoltagelevelandsimplerealizationofredundancyif required norequirementforexpensivetransformers Ontheotherhand,oneofthebiggestchallengesaboutMMCsisreliability.MMCs consistofalargenumberofpowerswitchingdevicessuchasIGBTsinSMsdepends 5 PAGE 17 Figure1.4: CascadedHalf-BridgeMultilevelConverter onvolumeoftheapplicationandtheseswitcheshavetobetakenintoconsideration aspotentialfailurepoints.Accordingtotable1.1whichshowsSMandconverter levelfaultsrespectivelyandalsoisgivenin[13],inSMlevel,faultswhicharerelated withIGBTssuchasopencircuitandintermittentgatemisringfaultsareshownas oneofthemostcommonfaulttypeswhichmaydisrupttheoperationofMMCor evendestroyitduetoeectoncurrentandvoltageinMMC[14].Soitisclearthat faultdetectionandlocalizationinashorttimeafteroccurrencearereallyvitalfor sustainabilityoftheMMCoperation. Thereareseveralmethodsthatareproposedonopencircuitfaultdetectionin MMC.In[15],KalmanFitering iv isbeingrecommendedtodetectthefaultconditionbymeansofcomparisonbetweenthemeasuredandestimatedinnerdierence 6 PAGE 18 Figure1.5: MMCInverterTopology Table1.1:FaultTypes Sub-modulelevel Capacitor Faultincapacitorstructure Diode Opendiodefaults ShortCircuitFaults IGBT Opencircuitfaults Intermittentgatemisringfaults Converterlevel Capacitorvoltageunbalance Circulatingcurrentamongthreephaseunit Unbalancebetweenupperandlowerarmvoltage Controlofenergystoredintheleg Singlephasetogroundfault Doublephasetogroundfault Triplephasetogroundfault currentbasedonspeciedcurrentdierencethresholdforsomecertainperiod.After faultisdetected,SMcapacitorvoltagesintheconcernedphasewhichisidentiedas 7 PAGE 19 faultyarecomparedwithminimumcapacitorvoltagevalueinthesamearmforsome thresholdperiodandvaluetolocatefaultySMintheupper n lowerarm.Another proposedmethodisbasedonslidingmodeobserverSMOwhichhasbeenpresented in[16].Inhere,voltageandcurrentrelationshipsbetweenthebothsidesofSMsare calculatedundernormalandopencircuitfaultcaseineachswitch.ThroughSMO estimatedandactualstatesarecomparedifdierencebetweenthemismorethan thresholdvalueforcertainperiodwhichisalreadyspeciedintermsofsystemssensitivity,faultisdetectedinthisway.Accordingto[17],itisstatedthatthisSMO methodisaccurate,butnotfastforMMCswithhighnumberofSMs.Inthiscase,it maybeencounteredwithadditionaldamagesbecauseofinstantincreaseoffaultySM capacitorvoltages.So,itsuggestsdierentmethodwhichisbasedonvoltageacross inductorandarmcurrentobservationintheupper n lowerarmsinanyphaseinstead ofusingobservertomakethisprocessfaster.Anotherpresentedapproachrelieson adaptivebacksteppingobservertotakeplaceofsensorsandreducethecomplexityof implementationwhichisgivenin[18]. Apartfromthesemethods,stateobserverscanbedesignedtodetectandlocate theopencircuitfaultsinSMsbyreducingnumberofmeasurementsensorsinparallel withtotalcostbyestimatingcapacitorvoltagesandoutputcurrents[19].Inthisthesis study,designofstateobserverapproachforlinearsystemsisbenettedtoestimate capacitorvoltagesinSMslocatethefaultymoduletobealternativeforexisting approaches.ToobtainproperresultsinopenloopconditionPS{PWMtechnique andvoltagebalancingalgorithmwhichissimilarwiththewayexpressedin[2]are employedin5LevelMMCinverter.In[20],PS{PWMispresentedandtechnical aspectsarediscussed. Thethesisisorganizedinfollowingway;inchapter2,structuremodellingand operationalprinciplesofMMCareexplained.Afterwards,chapter3isaboutcontrol approachrelatedwiththepreferredPWMtechnique,thecapacitorvoltagebalancing 8 PAGE 20 algorithmandcirculatingcurrentsuppressioncontrolCCSC.Inchapter4,fault types,characteristicsonSMswithproposedfaultdetectionandlocalizationmethodsareexplained.Chapter5isincludingdesignguidelineandsimulationresults. Conclusionisaddedinchapter6. 9 PAGE 21 2.SystemModelingandPrinciplesofOperation 2.1StructureofMMC ItcanbestatedthatMMCsaresortofVSCandtherelatedprinciplesofoperation forVSCscanbeappliedtoit.Becauseifonly2SMswereusedinMMCwecould saythatoneforpositiveDCconnectionandtheotheroneisnegative,synthesized outputvoltagewaveformwouldbeinFigure2.1whichislike2levelvoltagesource converter.So,operationofthisMMCwouldbeequivalenttoVSC.Also,Itcanbe usedasrectierandinverterinbacktobackHVDCpowertransmissionsystemsthat Siemenshasaplanofputtingthisconverterintopracticalapplicationswiththetrade nameHVDC-plus".ThesystemcongurationoftheHVDC-plushasapowerof400 MVA,adclinkvoltageof200kV,andeacharmcomposedof200SMs[21]. Figure2.1: VoltageWaveformof2LevelConverter Additionally,TheMMCcongurationalsooersmoreadvantagesoverthetraditionalVSCs[5].Theseare Notalltheswitchesinalegareopenedorclosedatthesametimeasthey wouldinatwo-levelVSC,buttheyareoperatedatdierenttimeinstantsto followthesinusoidalreferencecommandmoreclosely,theswitchingfrequency ofeachswitchcanthusbelow,whilestillgeneratingalargeapparentswitching frequency. Thevoltageblockingrequirementsofanindividualswitchislimitedtothe voltageacrossthemodule'scapacitor. 10 PAGE 22 Ifenoughmodulesareused,thevoltageacrosseachswitchcanbelowenough nottorequireseriesconnectedswitches.Ina2-levelVSC,eacharmvalve mustbeabletoblockseveralhundredsofkVswhileopen.AnIGBTisusually ratedforafewkVsonly,somultipleswitchesmustbeconnectedinseries toachievethedesiredblockingcapabilities.Duringswitching,allthesedevices mustoperatetogether.IfoneIGBToperationisdelayed,itisexposedtoahigh voltage,whichcoulddamageit.InthecaseoftheMMC,thehighnumberof modules,eachonlyblockingthevoltageacrossitsinternalcapacitoreliminates thisdrawbackandimprovesthefailurerate. Themodularconceptallowsoperationevenifsomemoduleshavefailed.Failed modulescansimplybebypassed,andkeptbypasseduntilitispossibletoreplace them,andoperationcancontinue.TheDCbusvoltageisthendividedamong theremainingN-1modulesandnormaloperationcancontinue. ItispossibletodesignaMMCconverterwithsparemoduleswhichcanbe usedifonemoduleweretofails.Ifsparemodulesareavailable,theycanbe insertedinthearmasneeded. AsitcanbeseenfromtheFigure2.4thatdemonstratestheoverallsystem,SM isthebasicbuildingcellforMMC.TheremaybehundredsorthousandsofSMsin MMCintermsofrequiredpowerleveldependsonapplication[22].SMsconsistof twoIGBTs S 1 ;S 2 ,twodiodes D 1 ;D 2 whichareconnectedinparallelwiththem andasubmodulecapacitor C sm .Alsothesediodesareusedduetoprotectionof IGBTswitches.Figure2.2showsSMdiagram. Inathree{phaseMMCsystem,foreachphasetherearephaselegswhichare composedofupperandlowerarms.Thesearmsareformedbyseriesconnectedn" numberofSMs.Thereisalsoaninductorineacharm, L arm ,inordertosmooth thevoltagedierencetoreducecirculatingcurrentsbetweenphaseunitsthatis 11 PAGE 23 Figure2.2: SingleModuleofMMC producedwhenaSMisconnectedordisconnected[23]. 2.2OperationPrinciples Thegeneralconceptofmultilevelconverteristhesynthesisofasinusoidalvoltage byseverallevelsofvoltages.IncaseofMMC,thesevoltagelevelsareobtainedfrom thecapacitorvoltages, V c ,ofeachSM.Atanyinstant,anumberofSMsthatare switchedonequalsn"totallywhichisincludingupperandlowerarminanyphase, sothatthevoltageattheconverterterminalsequalstheinstantaneousvalueofthe voltagetobesynthesized[24]asitisshownininFigure2.3whichareoutputvoltages of3,5and9levelMMCsvariesbetween V dc = 2respectively.Asitisseenifnumber ofvoltagelevelisincreased,resolutionofsinusoidalwaveformfortheoutputvoltage getsbetter.Besides,average V c isdenotedas V c = V dc n .1 AsregardsgatesignalofrelatedIGBTswitchSMhastwostatesinnormal operation.TheseareON"andOFF"states.Figure2.5indicatestheconcerned statesrespectively. ONState| S 1 isswitchedon, S 2 isswitchedo AsupperIGBTisintheconductingmodeinanySMinanyphase, V c equalsto 12 PAGE 24 Figure2.3: MMCOutputVoltageSynthesis theSMoutputvoltage V c = V sm regardlessofcurrentdirection.Ifthecurrent ispositiveitfreewheelsthroughanti-paralleldiode D 1 and V c increasesorwhen currentpolarityisnegativeitowsthroughthe S 1 anddischargesthe C sm OState| S 1 isswitchedo, S 2 isswitchedon Inhere,the V sm doesn'tchange.BecauserelatedSMisbypassed.Armcurrent eitherowsthroughthe S 2 orthe D 2 dependingonitsdirection.Thereby, V sm iszero. Table2.1showstheoperationalstatusoftheSMintermsofcomplementary switchingstatesofIGBTs.Also,therelationbetween V sm and V c canbeformulated inthefollowingway: 13 PAGE 25 Figure2.4: OverallSystemStructure 14 PAGE 26 Figure2.5: SwitchingStatesofSM V sm ; ijk = S ijk V c ; ijk .2 Table2.1:OperationofSM SMStateS S 1 S 2 V sm ArmCurrent C sm V c ON1ONOFF V c PositiveChargeIncreased NegativeDischargeDecreased OFF0OFFON0PositiveBypassUnchanged Negative ThearmvoltagecanbecomputedconsideringthestatusoftheSM'sswitches, s ijk ,asfollows[25]: V ij = n X k =1 S ijk V c ; ijk .3 15 PAGE 27 where i = u;` representstheupperandlowerarm,respectively; j = a;b;c isthe phase;and k =1 ; 2 ;:::;n denotestheSM.Thearmcurrentscanbedeterminedas I uj = I j 2 + I diff j .4 I `j = )]TJ/F18 11.9552 Tf 10.494 8.088 Td [(I j 2 + I diff j .5 Accordingto[26] I diff j hastwocomponents.Theseareonethirdofdcsource current I dc = 3andtheotherpartiscirculatingcurrent I circ;j .Sotheprevious equationcanbewritteninthefollowingway. I uj = I j 2 + I dc 3 + I circ ; j ; .6 I ` j = )]TJ/F18 11.9552 Tf 10.494 8.088 Td [(I j 2 + I dc 3 + I circ ; j .7 where I circ j isthecirculatingcurrentforphase j ,and I circ a + I circ b + I circ c =0.These circulatingcurrentshavenoeectontheacsideordcsidevoltages.Itcanbeproved basedonarmcurrentequationsasitisdemonstratedbelow[5].Ontheotherhand, theyhavesignicantimpactonSM'scapacitorvoltagesandtheratingvaluesofthe MMCcomponents[25]. I ua = I a 2 + I dc 3 + I circ ; a = I dc 3 + I circ ; a + I ua )]TJ/F18 11.9552 Tf 11.956 0 Td [(I ` a 2 .8 So I circ;a becomes I circ;a = )]TJ/F18 11.9552 Tf 10.494 8.088 Td [(I dc 3 + I ua )]TJ/F18 11.9552 Tf 13.151 8.088 Td [(I ua )]TJ/F18 11.9552 Tf 11.955 0 Td [(I ` a 2 = )]TJ/F18 11.9552 Tf 10.494 8.088 Td [(I dc 3 + I ua + I ` a 2 .9 I circ;b and I circ;c canalsobedenedasinthefollowing. 16 PAGE 28 I circ;b = )]TJ/F18 11.9552 Tf 10.494 8.087 Td [(I dc 3 + I ub + I ` b 2 .10 I circ;c = )]TJ/F18 11.9552 Tf 10.494 8.088 Td [(I dc 3 + I uc + I ` c 2 .11 Bysummingallthreecirculatingcurrentsitisobtainedthat I circ;a + I circ;b + I circ;c = )]TJ/F18 11.9552 Tf 13.15 8.088 Td [(I dc 3 + I ua + I ` a 2 )]TJ/F18 11.9552 Tf 13.15 8.088 Td [(I dc 3 + I ub + I ` b 2 )]TJ/F18 11.9552 Tf 13.15 8.087 Td [(I dc 3 + I uc + I ` c 2 I circ;a + I circ;b + I circ;c = )]TJ/F18 11.9552 Tf 9.299 0 Td [(I dc + I ua + I ub + I uc + I ` a + I ` b + I ` c 2 = )]TJ/F18 11.9552 Tf 9.299 0 Td [(I dc + I dc + I dc 2 =0 .12 Simultaneously,correlationbetweenacsideanddcsidevoltagesoftheMMCisextractedthroughKircho'sVoltageandCurrentLawsKVL-KCL. V dc 2 = V uj + L arm dI uj dt + R arm I uj + V j = V ` j + L arm dI ` j dt + R arm I ` j )]TJ/F18 11.9552 Tf 11.955 0 Td [(V j V dc = V uj + V ` j + L arm dI uj dt + dI ` j dt + R arm I uj + I ` j V j = 1 2 V ` j )]TJ/F18 11.9552 Tf 11.955 0 Td [(V uj )]TJ/F18 11.9552 Tf 11.955 0 Td [(L arm dI j dt )]TJ/F18 11.9552 Tf 11.955 0 Td [(R arm I j V di j = L dI di j dt + R arm I di j = 1 2 V dc )]TJ/F18 11.9552 Tf 11.955 0 Td [(V uj )]TJ/F18 11.9552 Tf 11.955 0 Td [(V ` j .13 where V di j representsinnervoltagedierencebetweenphaseunitsinMMCand R arm istheequivalentresistanceoftheSMsinanarm.ItisrelatedtonumberoftheSMs inanarmviatheresistanceofoneofSM'sIGBTswitch, R sm 17 PAGE 29 R arm = nR sm : .14 Equivalentconverterarmcapacitance C arm whennmodulesareinsertedinthe relatedconverterarmequalstothetotalcapacitanceofthenseriesconnectedsubmodulecapacitorsasitisshowninequation2.15. C arm = C sm =n .15 18 PAGE 30 3.MMCControl 3.1ModulationMethods IftwolevelVSCwhichisintheformofhalfbridgecircuitasitisshowninFigure 1.1isconsidered,conventionalpulsewidthmodulationwhichisusingonemodulating signalandonecarrierwaveformstogenerategatesignalsforcomplimentaryIGBT switchesinthesamephase[4].Figure3.1picturesanexampleforsinusoidalpulse widthmodulationandtheresult.AsitisseenfromtheFigure3.1modulatingsignal iscomparedwithtriangularcarrierwavesignal.Asaresultofit,incasemodulating signalishigherthancarrier,thismeansupperIGBTinVSCison,otherwiseifitis lowerthancarrier,loweroneison.ThisPWMmethodisrepeatedwithproperphase shiftsintermsofotherphaseswhen3phasetwolevelVSCisused. Formodularmultilevelconverters,thereareseveralofthesehalf-bridgecircuits SMsthatallofthemhastobeindividuallycontrolled.Thesolutionforthisisto usemulti-carrierPWMmethodswhichmeansthatonecarrierwaveformforeachSM inMMC. Figure3.1: SinusoidalPWMandOutputWaveformsFor2LevelVSC In[10]and[27]detailsaboutmultilevelconverterPWMmethodsaregiven.There aretwospeciedmethodsintermsofMMCmulti{carriermodulationtechniques. Thesearephase-shiftedandphasedispositionPWMmethods.Forbothmethods 19 PAGE 31 thereisonecarrierwaveformforeachSMintheupper n lowerarmoftheMMC phaseleg.Inbothmethods,multiplecarrierwaveformsfornumberofSMsinthe relatedarmarecomparedwithsinglearmvoltagereferencesignalmodulatingand thiscomparisondictateshowmanysubmodulesneedtobeswitchedonorbypassed. Apartfromthesemethodsthereareothermultilevelmodulationtechniquessuchas NearestLevelModulationNLMandSpaceVectorModulationSVM.However, thesemethodshavesomedrawbacks[28].Forinstance,NLMtechniqueissuitable withtheconverterswhichhavelargenumberofSMsduetosmallvoltagesteps.This meansifitisusedwithsmallscaleMMCwhichmeanslownumberofswitchings thismaycauselargervoltageuctuationinthecapacitorvoltagesandSVMbrings implementationcomplexity. 3.1.1PhaseShiftedPWMPS-PWM Phase-shiftedPWMformodularmultilevelconvertersuseonecarrierforeach submoduleintheMMCandthesecarrierwaveformshavethesameamplitudeand frequencywhichareshiftedby whichdependsonnumberofSMsinthearm.This angle iscalculatedusingtheequation3.1. = 360 n .1 n"indicatesnumberofSMsinanarm.Twokeyparametersaretheamplitudeand frequencymodulationindexinthismethod[20].Theequationstocalculatethese parametersareshowninequations3.2and3.3.Thefrequencymodulationindex m f relatesthefrequencyofthecarrierwave f c tothefrequencyofthereference sinewave f r m f = f c f r .2 Theamplitudemodulationindexistheratiooftheamplitudeofthecarrierwaveform andreferencewaveform. 20 PAGE 32 m a = A r A c 2 .3 Figure3.2showsanexampleofPSC-PWMwith4carrierand1referencewaveformswhichdictateshowmanysubmodulesneedtobeinsertedinthearminorder toachievethedesiredvoltagelevelasitisreectedinFigure3.3.Alsothereis2 = 4 phaseshiftbetweeneachcarrierwaveformwhichsatisesequation3.1. Figure3.2: PS-PWMReferencevsCarriers Figure3.3: ActivatedNumberofSMs 21 PAGE 33 3.1.2PhaseDispositionPWMPD-PWM PhaseDisposedPWMissimilartothepreviouslydescribedmethodsinthatagain thenumberofcarrierwaveformsisequaltothenumberofsubmodulesineacharm. Alsothesameamplitude.Thedierencesarethatthereisnophaseshiftbetween thecarriersandtheyaredisplacedwithrespecttozeroaxis.Alsoamplitudeofthem A c dependsontheequationshowninequation3.4 A c = 1 n .4 In[29]itisstatedthattherearevarioustypesofphasedisposedPWMtechniques whichdependonwhetherornotthecarrierwaveformsare0degreesoutofphaseor 180degreesoutofphase.Therstisphase-dispositionPWM.Figure3.4showsan exampleofphase-dispositionPWMbyshowing4carrierwaveformssuperimposed withasinglesinusoidalreferencewaveform.Figure3.5showsthereferencesignal usingPD-PWM. Here,eachcarrierwaveformhasthesamefrequencyandphasebutamplitudeof carriersis0.25whichiscalculatedbyusingtheequation3.4. Figure3.4: PD-PWMReferencevsCarriers 22 PAGE 34 Figure3.5: ReferenceforNumberofONStateSMs Thenextlevel-shiftedtechniquediscussedisphaseoppositedispositionPWM whichisidenticaltophase-dispositionPWMexceptthatthelowerhalfcarrierwaveformsare180degreesoutofphase.Figure3.6showsthecarrierwaveformsforphase oppositedispositionPWMandwecanseethatthebottomhalfcarrierwaveformsare 180degreesoutofphase.Figure3.7thenshowstheresultantwaveform. Figure3.6: POD-PWMReferencevsCarriers 23 PAGE 35 Figure3.7: ReferenceforNumberofONStateSMs Thenallevel-shiftedtechniquetobeinvestigatedisalternatingphaseopposite dispositionPWMinwhicheveryothercarrierwaveformisphaseshifted180degrees outofphase.Figure3.8showsthecarrierwaveformsforAPOD-PWMandFigure 3.9showstheresultantwaveform. Figure3.8: APOD-PWMReferencevsCarriers Inthisthesis,PS-PWMmethodischosenbasedontheassumptionsgiveninthe [30]whichsaysconciselyPS-PWMcanautomaticallysuppressloworderharmonics 24 PAGE 36 Figure3.9: ReferenceforNumberofONStateSMs forMMCs.Alsowhenthesimulationwhichisusedinthetheisstudyisrunwith PS-PWMandPD{PWMseperately,Figures3.10and3.11showthatphaseCarm currents,innerdierencecurrentwhichisincludingcirculatingcurrentasexplainedin chapter2havelessoscillationincomparisonwiththeFigures3.12and3.13obtained byuseofPD{PWMmethod.Alsomodulatingsignals M uj ref = V uj ref ;M `j ref = V `j ref whichmeansvoltagereferencesfortheupperandlowerarmsasinputforPWM blockinanyphaseofMMCiscalculatedinthefollowingway[31]; V uj ref = V dc 2 )]TJ/F18 11.9552 Tf 11.955 0 Td [(V diff;j ref )]TJ/F18 11.9552 Tf 11.955 0 Td [(V j ref .5 V `j ref = V dc 2 )]TJ/F18 11.9552 Tf 11.955 0 Td [(V diff;j ref + V j ref .6 V j ref representsthedesiredoutputphasevoltageofMMC[32]andcanbeexpressed as V a ref = V dc 2 mcos !t + .7 V b ref = V dc 2 mcos !t + )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(2 3 .8 25 PAGE 37 V c ref = V dc 2 mcos !t + + 2 3 .9 m"demonstratesamplitudeofmodulationindexwhichvariesbetween0and1. and arephasefrequencyandinitialphaseanglerespectively.Also V diff;j ref isthe referenceinnerunbalancevoltageasexplainedinchapter2anditisobtainedfrom CCSCblockasitisshowninFigure3.17. Figure3.10: PhaseCArmandInnerDierenceCurrentsforPS{PWM Figure3.11: 3PhaseCirculatingCurrentsforPS{PWM 26 PAGE 38 Figure3.12: PhaseCArmandInnerDierenceCurrentsforPD{PWM Figure3.13: 3PhaseCirculatingCurrentsforPD{PWM 3.2VoltageBalancingAlgorithm AnotherimportantconcepttobeunderstoodforMMCisthebalancingofthe capacitorvoltages.ImportanceofthevoltagebalancinginMMCisaboutpreventing 27 PAGE 39 SMcapacitorvoltagevariationandasaresultofitphaseunitshavethesamevoltages. Otherwise,itcausescirculatingcurrents I circ;j thatowthroughthesixarmsand distortthesinusoidalarmcurrentwaveforms.Also,becauseofitthermsvalueof thearmcurrentsandtheconverterlossesincrease[25].InMMC,capacitorvoltage ofeachSMshouldbemonitoredandkeptequalforstableoperation.Toachieveit, propervoltagebalancingalgorithmneedstobeused.Inthisstudy,thealgorithm basedonthealgorithmstatedin[2]whichisarmspecicisused.InFigure3.14 owchartofthevoltagebalancingalgorithmisshown. Figure3.14: ReducedSwitchingFrequencyVoltageBalancingAlgorithm Accordingtothisapproach,numberofSMsinthesamearmthatneedstobe switchedonorosignalwhichisdictatedbyPWMblock,SMcapacitorvoltage measurements V c ; ijk andarmcurrent I ij directionsareusedasinputs.If I ij is positiveandextraswitches N> 0needstobeON",thereisnoswitching appliedtoSMswhicharecurrentlyinserted.Thisalgorithmsorts V c ; ijk andchooses SMswiththelowestvoltagesamongtheostateones.If N< 0whichmeanssome 28 PAGE 40 oftheSMswhicharealreadyinonstateneedtobebypassed,SMswiththehighest voltagesamongtheonstateonesarechosenandswitchedo.Incaseofnegativearm currentthisalgorithmworksinthereversewayintermsofchoosingSMsbasedon theirvoltagemeasurement.Additionally,bythetime N =0,sothismeansrecent statusfortheSMsiskept.Oneofthebiggestadvantageousofthisapproachisthatit reducestheaveragedeviceswitchingfrequencyandthetotalMMCswitchinglosses incomparisonwithconventionalmethod.Soitiscalledreducedswitchingfrequency RSFvoltagebalancingalgorithm. 3.3CirculatingCurrentSuppressionControllerCCSC In[33]itisstatedthatcirculatingcurrents I circ;j whichiscausedbyvoltageunbalancesbetweenthephaseunitsofMMCandcontainssecondharmoniccomponent distortsnotonlyarmcurrents,butalsoincreasetherippleonSMcapacitorvoltages. Itcanbeeliminatedbyaddingparallelcapacitorwhichisresonantlterbetweenthe midpointsoftheuppperandlowerarminductances[34]oneachphaseorusingan activecontroloverACvoltagereference[35].InthisstudyCCSCisbasedonthe approachgivenin[2,33]. Innerdierencecurrent I diff j equationwhichhastwopartsasmentionedinchapter 2isdenedas I diff j = I dc 3 + I circ ; j .10 andaccordingto[36] I circ;j arecomposedofnegativesequencecomponentwithtwice thefundamentalfrequency.So3phase I diff j canberewrittenas I diff a = I dc 3 + I circ ; a cos !t + I diff b = I dc 3 + I circ ; b cos !t + + 2 3 I diff c = I dc 3 + I circ ; c cos !t + )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 .11 29 PAGE 41 ! isthefundamentalfrequencyand istheinitialphaseangle.Innerunbalance V diff j voltagewhichisvoltageacrossthearminductanceandarmresistorisshown inthefollowing V di j = L dI di j dt + R arm I di j .12 andifParktransformationwhichisgivenintheappendixisapplied, dq components ofthreephase V diff j areobtainedasfollows V di d = L arm dI circ d dt + R arm I circ d )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 !L arm I circ q V di q = L arm dI circ q dt + R arm I circ q +2 !L arm I circ d .13 Transferfunctionsofthe I circ;dq isshowninFigure3.15.Alsotodesigncontrol loop,twonewcontrolinputs u d ;u q canbeintroducedtoobtaintwodecoupledrst orderdierentialequationasitissuggestedin[4]. Figure3.15: TransferFunctionoftheCirculatingCurrentsindqReferenceFrame u d = V di d +2 !L arm I circ q = L arm dI circ d dt + R arm I circ d .14 30 PAGE 42 u q = V di q )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 !L arm I circ d = L arm dI circ q dt + R arm I circ q .15 Finally,toeliminate I circ;dq twoidenticalPIcontrollerswithzeroreferencevaluesfor dq componentsseparatelyareappliedasitisshowninFigure3.16.Transferfunction ofthePIcontrolis PI circ;dq s = Kp circ;dq + Ki circ;dq s .16 V diff dq;ref equationsbecome V di d ; ref = )]TJ/F18 11.9552 Tf 11.955 0 Td [(I circ;d Kp + Ki s )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 !L arm I circ q V di q ; ref = )]TJ/F18 11.9552 Tf 11.955 0 Td [(I circ;q Kp + Ki s +2 !L arm I circ d .17 Figure3.16: ClosedLoopd-qAxisCirculatingCurrentControllers Afterobtainingreferencevaluesforinnerunbalancevoltageindqreferenceframe, inverseParktransformationwhichisgiveninappendixisappliedtohaveitinthree phase.Eventually,Thesevaluesareusedtoattainreferencearmvoltagesasitis 31 PAGE 43 showninequations3.5and3.6.Overallcontrolsystemtoobtain V diff;j ref isshown inFigure3.17. Figure3.17: OverallCirculatingCurrentSuppressionController 32 PAGE 44 4.FaultDetectionandLocalization TherearevarioustypesoffaultsinMMCasitisgivenintable1.1.Because ofMMCstructuretherearemanypotentialfailurepointsbasedonsemiconductor devicessuchasIGBTanddiodesinSMsasitshowninFigure2.2anditissovital todetectandlocatethefaultaftertheoccurrencewithinashorttimeinthewayof systemreliability[14].Inthisstudy,onlyIGBTbasedopencircuitfaultisconsidered amongthem. 4.1FaultTypesandCharacteristics InSM,IGBTscanhavethreetypesoffault,whicharecalledType1,2and 3[15].Type1and2faultsarewhen S 1 and S 2 actasopencircuit,respectively. BothswitchesareopencircuitsimultaneouslyinType3fault.Type1faulthappens onlywhenconcernedarmcurrentisnegative I ij < 0andgatesignalof S 1 S ijk is1.Duetothefactthatupperswitchisopencircuit,currentisforcedtocirculate throughcomplementaryswitchdiode D 2 .Thus,SMisbypassedand V sm becomes 0insteadof V c .Type2occurswhenarmcurrentispositive I ij > 0andgatesignal for S 2 S ijk is1.Inthiscasearmcurrentowsthrough D 1 andcharges C sm .Unlike theexpected V sm is0, V sm equalsto V c .OthersituationsSMrunsnormally.What aboutType3isitshowsbothcharacteristicsoftype1and2faults.Faultscenarios aredepictedinFigure4.1respectively.Inthispaper,toillustratethefaultdetection andlocalizationmethodonlysingleType1faultisconsidered. Figure4.1: FaultTypes 33 PAGE 45 Table4.1:FailureCharacteristicsofType1 HealthyOperationType1Fault I arm Gatesignalfor S 1 Directionof I arm V sm Directionof I arm V sm Positive 1 D 1 and C sm V C D 1 and C sm V C 0 S 2 0 S 2 0 Negative 1 C sm and S 1 V C D 2 0 0 D 2 0 D 2 0 Table4.2:FailureCharacteristicsofType2 HealthyOperationType2Fault I arm Gatesignalfor S 1 Directionof I arm V sm Directionof I arm V sm Positive 1 D 1 and C sm V C D 1 and C sm V C 0 S 2 0 D 1 and C sm V C Negative 1 C sm and S 1 V C C sm and S 1 V C 0 D 2 0 D 2 0 Table4.3:FailureCharacteristicsofType3 HealthyOperationType3Fault I arm Gatesignalfor S 1 Directionof I arm V sm Directionof I arm V sm Positive 1 D 1 and C sm V C D 1 and C sm V C 0 S 2 0 D 1 and C sm V C Negative 1 C sm and S 1 V C D 2 0 0 D 2 0 D 2 0 Fromthetables4.1,4.2and4.3thatshowcharacteristicsofSMsinMMCsystem itcanbededucedthatunderalltypesofopencircuitfaultcircumstancesfaultySM capacitorvoltages V sm becomehigherthanthehealthyones. 34 PAGE 46 4.2ProposedFaultDetectionMethod AssoonasType1faulthappensinoneoftheSMsinanyarmoftheMMC, V diff;j betweenthehealthyphaseunitsandfaultyonegetslarger.WhenPWMblock dictatesallSMsinthethesamearmtobeswitchedonafterthefaultoccurrence healthycapacitorvoltagesincreasetorecoverfaultymodule.Inthisperiod,arm currentnegativepolarityissuppressedbecauseofcirculatingcurrentboostedbyfault freephaseunits. Innormaloperationeachcapacitorvoltageis V c = V dc n .1 whichisalreadyshowninequation2.1.Hereby,incaseoffaulthealthySMcapacitor voltagesareexpectedtoincreasetoreachthevoltagewhichcanbeshownasfollows V c threshold = V dc n )]TJ/F18 11.9552 Tf 11.955 0 Td [(n faulty .2 TodetectthefaultinanyphaseunitoftheMMCsystemthisvoltagecanbeused asthreshold V threshold .IfhealthySM V C reachorexceedthis V threshold foracertain periodoftime t min ,sothatarmcanbelabeledasfaultyanditcanbeproceededto locatethefaultymoduleinthearm.Simultaneously,itcanbeextractedfromhere thatifreallifeapplicationsinwhichhundredsofSMsareuseddependsonthevolume oftheapplicationareconsidered,bybenettingequation4.2thresholdvoltagefor faultdetectionbecomesless.Thismightaectthefaultdetectionperiodfavourably. FlowchartofthefaultdetectionprocessisshowninFigure4.2. 4.3ProposedFaultLocalizationMethod Tolocatethefaultymodule,stateobservertoestimatecapacitorvoltagescanbe usedintermsofcomparisonbetweenthedirectmeasurementofitandobservation. SMcapacitorvoltageequationisdenedas 35 PAGE 47 Figure4.2: FaultDetectionFlowchart dV c ; ijk dt = 1 C S ijk I ij .3 Asitcanbeseenfromtheequation4.3itisrstorderdierentialequationandcan beconsideredasstateequationforthecase. V c ; ijk isconsideredasstateand I ij is input.FlowchartofthefaultlocalizationisdemonstratedinFigure4.3. 4.3.1DesignofStateObserver TheStateobservershasbeencommonlyusedinindustriesasaneectivemethod toprovideestimationofarealsystem[37{39].Tobuilduptheobservertherstand mostessentialruleischeckingtheobservabilityofsystem[40].Butinourcasethis isnotnecessarybecauseofhavingtherstordersystem.Ifweconsidertheplant denedby x = Ax + Bu y = Cx .4 observerstateandoutputequationwillbeasfollows; 36 PAGE 48 Figure4.3: FaultLocalizationFlowchart ^ x = A ^ x + Bu + K e y )]TJ/F15 11.9552 Tf 12.747 0 Td [(^ y ^ y = C ^ x .5 K e isobservergainwhichiscorrectiontermbetweensystemoutputandestimated outputasitisshowninequation4.5.Itshouldbechosenproperlyintermsof systemperformancetomakeestimatedstate^ x convergetosystemstatevariablex regardlessofit'sinitialvalue^ x .In[40]itisstatedthatdynamicbehaviourofthe errorvectorwhichisobtainedbysubtractingsystemandestimatedstateequations fromeachotherdeterminedbytheeigenvaluesofmatrice A )]TJ/F18 11.9552 Tf 11.925 0 Td [(K e C asitisshownin equation. e = A )]TJ/F18 11.9552 Tf 11.955 0 Td [(K e C e .6 Ifitisstablematriceerrorvectorwillconvergetozeroforanyinitialerrorvalue e .Ifthesystemisloworderwhichmeanssystemordercanbethirdorderat most,directsubstitutionmethodcanbeappliedtondobservergain.Todothat, 37 PAGE 49 thedesiredcharacteristicpolynomialwhichisextractedfromequation4.5isusedand shownbelow. j sI )]TJ/F18 11.9552 Tf 11.955 0 Td [(A + K e C j = s )]TJ/F18 11.9552 Tf 11.955 0 Td [( 1 .7 1 isthedesiredobserverpolelocationwhichiswithinthelefthalfof j! axisins domain.IfSMcapacitorvoltageequationequation4.3whichisstateequationin hereisconsidered,itisseenthatAwhichisstatematriceis0andCwhichisoutput matriceis1.Asaresultofitfromtheequation4.7itisdeducedthat K e > 0isthe conditionforthestabilityasitisshownbelowinequation4.8.Whatmakesdierence inhereisdierentgainvalueseectingfaultymodulelocalizationtimewhenopen circuitfaultisgenerated. j s + K e j = s )]TJ/F18 11.9552 Tf 11.955 0 Td [( 1 .8 38 PAGE 50 5.DesignGuidelineandSimulation 5.1CircuitParameters TorunMMCinverterwithRLloadinopenloopcondition,circuitparameters aretakenfromthe[15].Theyareshownintable5.1. Table5.1:SimulationParameters V dc n sm C sm L arm R load L load f carrier f load modulationindex V mF mH ohm mH Hz Hz V j ref 3600 4 3.5 3 10 10 1650 50 0.7 5.2CirculatingCurrentPIControllerGainSelection InbothcirculatingcurrentcontrolloopsshowninFigure3.16,KpandKiare derivedasfollows[4]: Kp = L arm = i Ki = R arm = i .1 where i isthetimeconstantofthecirculatingcurrentcontrolloop. i isadesign choice,anditisgenerallyselectedintherangeof0.5-5ms[4].Inourstudyitis chosen0.2mstohavefasterresponse.SoKpwhichisusedhereequals15andKi becomes20. 5.3SimulationResults Mainparametertoimplementfaultlocalizationalgorithmisobservergain K e as mentionedinthepreviouschapter.Toshowhowitaectsthedurationtolocatefaulty moduletwodierentgainsareusedhere.Necessityforobserverstabilityisshownin equation4.8.If K e ischosenhighenough,itisexpectedtoseethatestimationfollows measurementaftergeneratingthefaultandwhencapacitorvoltagesinthesamearm 39 PAGE 51 reach V threshold itstartstodivergeslightly.So,ittakesmuchtimetoexceedV between V c;ijk and V c;ijk est tolocatefaultySM.Tosamplethisscenarioopencircuit faultin S 1 ofrstSMintheupperarmofphaseCisgeneratedat0.8sec.Also, CCSCisactivatedat0.3sectoshowhowtominimizeit. 5.3.1 K e =50 InFigure5.1itisseenthatuntilthefaulthappensbothupper{lowerarmcapacitorvoltagesareslightlyvaryingaround900Vwhichis V dc = 4anditprovesthat balancingalgorithmisabletoequalizetheSMcapacitorvoltages. Figure5.1: PhaseCUpper{LowerArmCapacitorVoltages Figure5.2showsthecirculatingcurrentsin3phaseofMMCsystem.Itisobserved fromthegurethatcontrollersuppressesthecirculatingcurrentswhenitisactivated at0.3sec.Capacitorvoltagevariationbecomessmallerbymeansofit.Assoonas faulthappensbecauseof V diffj itgetshigher. Figures5.3,5.4,5.5and5.6depictthecomparisonbetweentheestimationand measurementofphaseCupperarmcapacitorvoltagesrespectively.Fromthegures itisdeducedthatobservedstatesreachmeasurementfromitsinitialvaluewhichis500 Vinhereandincaseoffaultfaultymodulecapacitorvoltageestimationdivergesfrom 40 PAGE 52 Figure5.2: 3PhaseCirculatingCurrents measurementbecauseoftheerrorafterreaching V threshold asshowninequation4.2. InFigure5.3,rstSMintheupperarmofphaseCiscomparedwithitsestimation. Figure5.3: PhaseC V uc 1 and V uc 1 est Unliketheothersthisisslightlydivergingfrommeasurementanditshowsus thisisfaultyone,butforlocalizationdierencevoltagewhichisspeciedas100V 41 PAGE 53 isn'tenough.Soweunderstandthatbecauseofhighobservationgainitisrecovering theerrorwhichiscausedbyopencircuitfaultinthemodule.Restofthegures estimationfollowsthemeasurementforhealthyonesperfectly. Figure5.4: PhaseC V uc 2 and V uc 2 est Figure5.5: PhaseC V uc 3 and V uc 3 est ItisunderstoodfromtheFigure5.7thatevenallSMcapacitorvoltagesexceed V threshold forfaultdetectionbecauseofhighgainselectionobservedstatedoesn'tshow 42 PAGE 54 Figure5.6: PhaseC V uc 4 and V uc 4 est Figure5.7: FaultStatusofSMs divergentbehaviourtolocatethefaultwithinreasonabletime.Thusfaultstatusof allSMsstays0. 5.3.2 K e =0.1 Ifobservergainischosensmallenoughwhichstillneedstobegreaterthanzero, itisseenthatthedivergencebetweenmeasurementandestimationofstatebecomes 43 PAGE 55 moreclearandithelpsustodetectandaddressthefaultfaster.Relatedgures whichare5.8and5.9areshownbelow. Figure5.8: PhaseC V uc 1 and V uc 1 est for K e =0.1 Figure5.9: FaultStatusofSMsfor K e =0.1 44 PAGE 56 5.4Discussion IftheFigures5.3and5.8areconsidered,itisdetectedthatobservergainselection playsanimportantroletolocatethefaultymodulewithinareasonabletime.For smallobservergainerrorbetweenthemeasurementandestimationforeacherrorstep increments.Alsoitisselectivedesigncriteriathatobserverresponseforsmallergain issluggish. AsitisseenfromtheFigure5.9faultislocatedwithinapproximately0.4sec whichcanbeconsideredhighforrealsystemapplications,butasitismentionedin chapter4faultdetectionpart,fortherealMMCsystemswhichcontainhundreds ofMMCsthisfaultdetectionlevel V threshold canbekeptwithinthesmallrange. Forinstance,inthisstudyif10SMsperarmwereusedinstead4SMsatthesame switchingfrequency,faultdetectionvoltagelevelwhichisextractedfromtheequation 4.2andgivenbelowwouldbe400VforeachSMinsteadof1200V.Thus,thatmeans ittakessmallertimetodetectandlocateit. V c threshold = V dc n )]TJ/F18 11.9552 Tf 11.955 0 Td [(n faulty = 3600 10 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 =400 V 45 PAGE 57 6.Conclusion Inthisthesis,"levelthreephaseMMCinverterwithRLloadisdesignedto proposethefaultdetectionandlocalizationalgorithmsbasedonSMcapacitorvoltage measurementandestimation.Thevariouscomponentsinvolvedintherealizationof powercircuitaredescribedandthecontrolapproachisexplainedinparts.Inaddition, thedesignguidelineofthesystemparametersandtuningmethodsforPIcontrollers usedincirculatingcurrentsuppressioncontrollerareexpressed.Theproposedsystem isimplementedviadetailedMatlab/Simulinkcomputersimulations. ItisshownintheMMCcontrolpartthatemployedcapacitorvoltagebalancing algorithmwhichissimilartoreducedswitchingfrequencyapproachgiveninreferencepapersoperatesasdesiredbyequalizingthearmcapacitorvoltages.PS{PWM modulationwhichispreferredmulti{carrierPWMtechniqueinthisstudyprovides lessoscillatorycurrentwaveformsthanPD{PWMmethodincomparison.Additionally,implementedcirculatingcurrentcontrollerminimizescirculatingcurrentsin threephasebynarrowingvoltagevariationofcapacitorvoltagesandprovidesinner unbalancevoltagereferencetobeusedinarmmodulatingsignalreferencecalculation. Todetectthefaultandlocatethefaultymodulestateobserverapproachbased oncomparisonbetweenSMcapacitorvoltagemeasurementandestimationisused. Designofobserverandhowobservergain K e selectioneectslocalizationperiodas acriticalpointisexplained.Moreover,bycomparingthesimulationresultsfortwo dierentgainvaluesitisveried. Ascontributionapartfromtheothermethodswhichisalreadyproposedinthe papers,dierentapproachissuggestedforopencircuitfaultdetectionandlocalization inMMCsystem. 46 PAGE 58 7.FutureWork Thepossiblefutureworkscanbelistedasfollows: Studyontheobservergain K e selectionwhichmattersforfaultdetectionand localizationperiodincaseofhealthyandfaultyconditions. Theobserverapproachforfaultdetectionandlocalizationcanbeappliedfor theothertypesoffaultsinSMorconverterlevel. 47 PAGE 59 REFERENCES [1]R.MarquardtJ.Hildinger,A.Lesnicar.ModularesStromrichterkonzeptfurNetzkupplungsanwendungbeihohenSpannungen,ETG-Fachtagung,BadNeuheim, Germany,2002. [2]Q.Tu,Z.Xu,andL.Xu.Reducedswitching-frequencymodulationandcirculatingcurrentsuppressionformodularmultilevelconverters. IEEETransactions onPowerDelivery ,26:2009{2017,July2011. [3] CurrentSourceConvertersandVoltageSourceConverters http://www. gridtech.eu/project-scope/technologies?id=19 [4]A.YazdaniandR.Iravani. Voltage{SourcedConvertersinPowerSystems .John Wiley&Sons,2010. [5]DominicParadis.RealTimeSimulationofModularMultilevelConverter.Master'sthesis,UniversityofToronto,Canada,2013. [6]L.G.Franquelo,J.Rodriguez,J.I.Leon,S.Kouro,R.Portillo,andM.A.M.Prats. Theageofmultilevelconvertersarrives. IndustrialElectronicsMagazine,IEEE 2:28{39,June2008. [7]JingHuangandK.A.Corzine.Extendedoperationofyingcapacitormultilevel inverters. PowerElectronics,IEEETransactionson ,21:140{147,Jan2006. [8]Y.H.Liu,J.Arrillaga,andN.R.Watson.Cascadedh-bridgevoltagereinjectionpartii:Applicationtohvdctransmission. PowerDelivery,IEEETransactions on ,23:1200{1206,April2008. [9]A.LesnicarandR.Marquardt.AnInnovativeModularMultilevelConverter TopologySuitableforaWidePowerRange.In PowerTechConferenceProceedings,2003IEEEBologna ,volume3,pages6pp.Vol.3{,June2003. [10]M.HagiwaraandH.Akagi.ControlandExperimentofPulsewidthModulatedModularMultilevelConverters. PowerElectronics,IEEETransactions on ,24:1737{1746,July2009. [11]G.P.Adam,O.Anaya-Lara,G.M.Burt,D.Telford,B.W.Williams,andJ.R. McDonald.Modularmultilevelinverter:Pulsewidthmodulationandcapacitor balancingtechnique. PowerElectronics,IET ,3:702{715,September2010. 48 PAGE 60 [12]GumTaeSon,Hee-JinLee,TaeSikNam,Yong-HoChung,Uk-HwaLee,SeungTaekBaek,KyeonHur,andJung-WookPark.Designandcontrolofamodular multilevelhvdcconverterwithredundantpowermodulesfornoninterruptible energytransfer. PowerDelivery,IEEETransactionson ,27:1611{1619,July 2012. [13]HuiLiu,PohChiangLoh,andF.Blaabjerg.Reviewoffaultdiagnosisand fault-tolerantcontrolformodularmultilevelconverterofhvdc.In Industrial ElectronicsSociety,IECON2013-39thAnnualConferenceoftheIEEE ,pages 1242{1247,Nov2013. [14]M.Shahbazi,P.Poure,S.Saadate,andM.R.Zolghadri.Fpga-basedfastdetectionwithreducedsensorcountforafault-tolerantthree-phaseconverter. IndustrialInformatics,IEEETransactionson ,9:1343{1350,Aug2013. [15]FujinDeng,ZheChen,M.R.Khan,andRongwuZhu.Faultdetectionand localizationmethodformodularmultilevelconverters. PowerElectronics,IEEE Transactionson ,30:2721{2732,May2015. [16]S.Shao,P.W.Wheeler,J.C.Clare,andA.J.Watson.Open-circuitfaultdetection andisolationformodularmultilevelconverterbasedonslidingmodeobserver. In PowerElectronicsandApplicationsEPE,201315thEuropeanConference on ,pages1{9,Sept2013. [17]H.Salimian,H.Iman-Eini,andS.Farhangi.Open-circuitfaultdetectionandlocalizationinmodularmultilevelconverter.In PowerElectronics,DrivesSystems TechnologiesConferencePEDSTC,20156th ,pages383{388,Feb2015. [18]V.Najmi,H.Nademi,andR.Burgos.Anadaptivebacksteppingobserverfor modularmultilevelconverter.In EnergyConversionCongressandExposition ECCE,2014IEEE ,pages2115{2120,Sept2014. [19]HuiLiu,KeMa,PohChiangLoh,andF.Blaabjerg.Designofstateobserverfor modularmultilevelconverter.In PowerElectronicsforDistributedGeneration SystemsPEDG,2015IEEE6thInternationalSymposiumon ,pages1{6,June 2015. [20]M.RajanandR.Seyezhai.ComparativeStudyofMulticarrierPWMTechniques foraModularMultilevelInverter. InternationalJournalofEngineeringand Technology ,2013. [21]B.Gemmell,J.Dorn,D.Retzmann,andD.Soerangr.Prospectsofmultilevel vsctechnologiesforpowertransmission.In TransmissionandDistributionConferenceandExposition ,pages1{16,April2008. [22]M.Davies,M.Dommaschk,J.Dorn,J.Lang,D.Retzmann,andD.Soerangr. HVDCPlus{BasicsandPrincipleofOperation. SiemensEnergySector ,3,2008. 49 PAGE 61 [23]J.Peralta,H.Saad,S.Dennetiere,J.Mahseredjian,andS.Nguefeu.Detailed andAveragedModelsfora401{levelMMCHVDCSystem. IEEETransactions onPowerDelivery ,27:1501{1508,Jul.2012. [24]H.Knaak.ModularMultilevelConvertersandHVDC/FACTS:ASuccessStory. In ProceedingsoftheEuropeanConferenceonPowerElectronicsandApplications ,pages1{6,Aug.2011. [25]M.SaeedifardandR.Iravani.Dynamicperformanceofamodularmultilevel back-to-backhvdcsystem. PowerDelivery,IEEETransactionson ,25:2903{ 2912,Oct2010. [26]QiangSong,WenhuaLiu,XiaoqianLi,HongRao,ShukaiXu,andLicheng Li.Asteady-stateanalysismethodforamodularmultilevelconverter. Power Electronics,IEEETransactionson ,28:3702{3713,Aug2013. [27]J.A.AzizandZ.Salam.Apwmstrategyforthemodularstructuredmultilevel invertersuitablefordigitalimplementation.In PowerElectronicsCongress, 2002.TechnicalProceedings.CIEP2002.VIIIIEEEInternational ,pages160{ 164,Oct2002. [28]DanielGamboaArtjomsTimofejevs.ControlofMMCinHVDCApplications. Master'sthesis,AalborgUniversity,Denmark,2013. [29]RyanBlackmon.AnalysisofModulationandVoltageBalancingStrategiesfor ModularMultilevelConverters.Master'sthesis,UniversityofSouthCarolina, USA,2013. [30]L.XuandV.G.Agelidis.Vsctransmissionsystemusingyingcapacitormultilevelconvertersandhybridpwmcontrol. IEEETransactionsonPowerDelivery 22:693{702,Jan2007. [31]HuiLiu,KeMa,PohChiangLoh,andF.Blaabjerg.Asensorlesscontrolmethod forcapacitorvoltagebalanceandcirculatingcurrentsuppressionofmodular multilevelconverter.In EnergyConversionCongressandExpositionECCE, 2015IEEE ,pages6376{6384,Sept2015. [32]BinbinLi,ShaoleiShi,BoWang,GaolinWang,WeiWang,andDianguoXu. Faultdiagnosisandtolerantcontrolofsingleigbtopen-circuitfailureinmodular multilevelconverters. IEEETransactionsonPowerElectronics ,31:3165{ 3176,April2016. [33]JuanA.Martinez-Velasco. TransientAnalysisofPowerSystemsSolutionTechniques,ToolsandApplications .JohnWiley&Sons,2015. [34]W.Nye,D.C.Riley,A.Sangiovanni-Vincentelli,andA.L.Tits.Delight.spice: anoptimization-basedsystemforthedesignofintegratedcircuits. IEEETransactionsonComputer-AidedDesignofIntegratedCircuitsandSystems ,74:501{ 519,Apr1988. 50 PAGE 62 [35]H.Kragh,F.Blaabjerg,andJ.K.Pedersen.Anadvancedtoolforoptimised designofpowerelectroniccircuits.In IndustryApplicationsConference,1998. Thirty-ThirdIASAnnualMeeting.The1998IEEE ,volume2,pages991{998 vol.2,Oct1998. [36]QingruiTu,ZhengXu,H.Huang,andJingZhang.Parameterdesignprincipleof thearminductorinmodularmultilevelconverterbasedhvdc.In PowerSystem TechnologyPOWERCON,2010InternationalConferenceon ,pages1{6,Oct 2010. [37]EllisG. ObserversinControlSystems:PracticalGuide .AcademicPress,2002. [38]P.Peltoniemi,P.Nuutinen,andJ.Pyrhonen.Observer-basedoutputvoltage controlfordcpowerdistributionpurposes. IEEETransactionsonPowerElectronics ,28:1914{1926,April2013. [39]Dong-ChoonLeeandDae-SikLim.Acvoltageandcurrentsensorlesscontrolof three-phasepwmrectiers.In PowerElectronicsSpecialistsConference,2000. PESC00.2000IEEE31stAnnual ,volume2,pages588{593vol.2,2000. [40]OgathaK. ModernControlEngineeringFifthEdition .PrenticeHall,2009. 51 PAGE 63 AppendixA.ParkTransformation Parktransformationtheoryisusedtotransformthree-phaseabcreferenceframe quantitiestothedqreferenceframequantities.Inthistransformtheory,itisassumed thatthedqreferenceframeisrotatingatsynchronousspeedwithrespecttotheabc referenceframewithaphaseangle .ThereferenceframesareshowninFigure A.1.Also,Parktransformationcanbereversed,whichmeansdqquantitiescanbe transformbacktoabcquantities. FigureA.1: Thestationaryabcreferenceframeandtherotatingdqreferenceframe Thedqquantitiesrelatetotheabccounterpartsaccordingto X dq = T dq abc X abc ; X abc = T abc dq X dq ; where T dq abc representstheParktransformationmatrix, T dq abc = 2 3 2 6 4 cos cos )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 cos + 2 3 )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin + 2 3 3 7 5 52 PAGE 64 and T abc dq standsfortheinverseParktransformationmatrix T abc dq = 2 6 6 6 6 4 cos )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin cos )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 cos )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(4 3 )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(4 3 3 7 7 7 7 5 : 53 |