Nomenclature

A list of the important symbols that used in this book can be found in Tables 1-4. A list of acronyms can be found in Tables 5-7.

Table 1 Superscripts, subscripts, circumflexes, and prefixes

* Complex conjugate ? Reference f,F Filtered value 0 Nominal value + Positive sequence - Negative sequence u,l Upper, lower arm 0 Mean value, zero-sequence component a,b,c Phases a, b, c a,ß Components of the stationary aß reference frame d,q Components of the synchronous dq reference frame Sum ? Difference Mean value Peak value, estimated value Difference ? Ripple quantity, parasitic quantity, difference, increment

Table 2 Variables

h Signed multiple of the fundamental frequency i Submodule index k Phase number m Carrier index n Sample index, sideband index s = d/dt Differential operator (or, where appropriate, complex Laplace variable) t Time z Forward-shift operator (or, where appropriate, complex z-transform variable) d = z - 1 Delta operator iu,l Arm current is = iu - il Output current ic = (iu + il)/2 Circulating current id DC-bus current Output-current control error e´ Modified control error Capacitor voltage in submodule i Sum capacitor voltage per arm Sum capacitor voltage per phase Imbalance sum capacitor voltage Submodule insertion index Insertion index per arm Inserted arm voltage vs = (-vu + v1)/2 Output voltage vc = (vu + v1)/2 Internal voltage va AC-bus voltage vg Grid voltage vPCC PCC voltage vdu,l Pole-to-ground dc-bus voltage vd = vdu + vdl Pole-to-pole dc-bus voltage Imbalance dc-bus voltage vR R-part output Stored energy per arm WS = Wu + Wl Stored energy per phase W? = Wu - Wl Imbalance stored energy Effective stored dc-bus energy P Active output power Pd DC-side input power Q Reactive output power ? Instantaneous angular frequency of the control-system dq frame ? = ? ? dt Angle of the control-system dq frame

Table 3 Parameters and functions

f1 Fundamental frequency ?1 = 2p f1 Fundamental angular frequency fs Sampling frequency fsw Switching frequency Ts = 1/fs Sampling period Tc Computational time delay Td = Tc + 0.5Ts Total time delay K Space-vector scaling constant M Number of phases N Number of submodules per arm C Submodule capacitance Cd Installed dc-bus capacitance Effective dc-bus capacitance L Arm inductance R Parasitic arm resistance RI Insertion resistance Peak value, fundamental component of vs Peak value, fundamental component of is Maximum allowed Maximum allowed ? Phase angle (lagging) of current relative voltage ?h Phase angle of order-h symmetric component da AC-bus-voltage phase angle dg Grid-voltage phase angle ?1 Voltage-reference phase shift ?c Carrier angular frequency ?c Carrier phase shift mf Frequency ratio ma Modulation index Ch Complex Fourier series coefficient Cmn Double complex Fourier series coefficient Jn Bessel function of order n L Sorted list of submodules Re Real part Im Imaginary part sat Saturation function satv Vectorial saturation function

Table 4 Controller parameters and transfer functions

ab PLL low-pass-filter bandwidth ac Output-current control-loop bandwidth ad DC-bus-voltage control-loop bandwidth af Voltage-feedforward-filter bandwidth ah R-part bandwidth aid DC-bus-voltage integrator bandwidth aip PLL integrator bandwidth a1 Power-synchronization control low-pass-filter bandwidth ap PLL bandwidth as Power-synchronization control-loop bandwidth Kh R-part gain Ki I-part gain Kp P-part gain Ks Power-synchronization-control gain Kv Voltage droop gain Ra "Active resistance" for circulating-current control Rs "Active resistance" for power-synchronization control fh Compensation angle for resonant filter Fh R...