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# A Control Methodology

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### A Control Methodology

4. 4. YAZDANI AND DASH: CONTROL METHODOLOGY AND CHARACTERIZATION OF DYNAMICS 1541 Equation (7) suggests that and, thus, can be controlled by the VSC ac-side current . It should also be noted that (7) represents a nonlinear dynamic system, due to the nonlinear de- pendence of on (see (3) and Fig. 2) as well as the pres- ence of the terms , , and . Fig. 2. Power-voltage characteristic curve of a PV array. The dynamics of the VSC ac-side current are described by the following space-phasor equation: irradiation level, and is a temperature coefﬁcient. Based on (8) (1), the power delivered by the PV array (i.e., ) is expressed as , that is, the VSC ac-side terminal voltage can be controlled as (3) (9) Fig. 2 illustrates the variations of as a function of , for where represents the space-phasor corresponding to the different levels of the solar irradiation. Fig. 2 shows that for a PWM modulating signals which are normalized to the peak given irradiation level, is zero at but increases value of the triangular carrier signal. Substituting for from as is increased. However, this trend continues only up to (9) in (8), one deduces a certain voltage at which reaches a peak value; beyond this voltage, decreases with the increase of . The afore- (10) mentioned behavior suggests that can be controlled/max- imized by the control of . This is referred to as the “max- Equations (7) and (10) constitute a state-space model for the PV imum-power-point tracking” (MPPT) in the technical literature system in which and are the state variables, is the con- [6]. trol input, and and are the exogenous inputs. It should be Dynamics of the dc-link voltage are described, based on the noted that itself is a state variable of the distribution network principle of power balance, as system, as discussed in the next subsection. (4) B. Distribution Network Model If , , , and are chosen as the state variables, the following state-space model can be derived for the distribution where denotes the power delivered to the VSC dc side (see network: Fig. 1). Ignoring the VSC power loss, can be assumed to be equal to , that is, the real component of the VSC ac-side terminal power. , in turn, is the summation of and the real (11) power absorbed by the interface reactor [30]. Thus (12) (13) (5) (14) where denotes the space-phasor representation of a three- phase quantity and is deﬁned as [31] (6) where the PV system ac-side current and the load current act as exogenous inputs to the distribution network subsystem. Substituting for from (5) in (4), one deduces In (14), it is assumed that is a balanced three-phase voltage whose amplitude and phase angle are and , re- spectively. C. Load Model (7) Three types of three-phase, balanced loads—an asyn- chronous machine, a series circuit, and a thyristor-bridge Authorized licensed use limited to: POLITECNICO DI BARI. Downloaded on November 16, 2009 at 03:23 from IEEE Xplore. Restrictions apply.
6. 6. YAZDANI AND DASH: CONTROL METHODOLOGY AND CHARACTERIZATION OF DYNAMICS 1543 Fig. 4. Block diagrams of (a) dq -frame current-control scheme. (b) DC-link voltage-control scheme. Equations (22)–(24) introduce the PLL as a dynamic system of and are determined based on the following control laws: which the inputs are and ; the state variables are , , and ; and the outputs are and . (34) Regulation of at zero also has the effect that the expression for the PV system real-power output (i.e., (35) ) is simpliﬁed to where and are two new control inputs [34]. Substituting (30) and in (32) and (33), respectively, from (34) and (35), one deduces Equation (30) indicates that is proportional to and can be controlled by . As indicated in (7), is controlled to regulate the dc-link voltage and to control/maximize the power extracted (36) from the PV array. Consequently, the control of boils down to the control of , as illustrated in Fig. 1. Similarly, the expression (37) for is simpliﬁed to Equations (36) and (37) represent two, decoupled, linear, ﬁrst-order systems in which and can be controlled by (31) and , respectively. Fig. 4(a) illustrates a block diagram of the -frame current-control scheme [i.e., a realization of (34) and Thus, can be regulated by to adjust the power factor that (35)]. Fig. 4(a) shows that the control signal is the output of a the PV system exhibits to the distribution network. compensator , processing the error signal . B. VSC Current Control Similarly, is the output of another compensator that processes . It should be noted that to produce The need for controlling and was explained in Sec- and , the factor is employed as a feedforward signal tion IV-A. In this section, a current-control scheme is devised to decouple the dynamics of and from those of . The to ensure that and rapidly track their respective reference PWM modulating signals are generated by transformation of commands and . The current-control strategy also and into , , and [not shown in Fig. 4(a)], and enhances protection of the VSC against overload and external the gating pulses for the VSC valves are sent out. faults, provided that and are limited. Since the control plants of (36) and (37) are identical, The current-control scheme is designed based on (32) and and can also be identical. Let (33) which are the -frame equivalents to (10). Thus (38) (32) where and are the proportional and integral gains, respec- (33) tively. If and are selected as In (32) and (33), and are the state variables and the outputs, (39) and are the control inputs, and and are the distur- bance inputs. Due to the factor , the dynamics of and are coupled and nonlinear. To decouple and linearize the dynamics, (40) Authorized licensed use limited to: POLITECNICO DI BARI. Downloaded on November 16, 2009 at 03:23 from IEEE Xplore. Restrictions apply.