**2. WHIM Modeling**

## *2.1. The HTG Model*

A hydraulic turbine (HT) converts energy from falling water into rotating shaft power [10]. The HT model used in this article is nonlinear, since the simulations consider large variations for the HT operating *Energies* **2020**, *13*, 5937

point. The equations below, in per unit (p.u) values, describe the HT nonlinear model. Their derivation can be obtained from the references [12,20]:

$$q = y\sqrt{h} \tag{1}$$

$$P\_{h-\text{mac}} = A\_l \eta q h - K\_D y(\omega - 1) \tag{2}$$

Equation (1) describes the valve which regulates the flow rate *q* to the HT. In it, *y* is the turbine gate opening position that can vary from 0 (fully closed) to 1 (fully open); *q* = *Q*/*Qbase*, is the flow rate p.u., *Q* being the flow rate passing through the turbine and *Qbase* the turbine flow rate with the gate fully open, *h* = *H*/*Hbase* is the pressure head of water p.u., with *H* as the pressure head in the turbine admission and *Hbase* the total available static head above the turbine [20]. Equation (2) gives the mechanical power p.u. produced by the HT, *Ph*−*mec*, which has two terms. The first term is the power produced by the flow rate p.u *q*, with an effective pressure head of water in p.u. *h* and *η* is the turbine hydraulic efficiency. *At* is a proportionality factor which depends on the HT active rated power and the SM apparent rated power. The second term accounts for a speed deviation effect (*ω* − 1) where *ω* is the HT shaft speed p.u; 1 p.u. is the rated turbine speed and *KD*, the damping torque coefficient [20].

The penstock is the pressure pipe that delivers water from the dam to the turbine admission. The following equation, describing the penstock model, was previously derived in [12,20]:

$$\frac{dq}{dt} = \frac{1}{T\_{\text{IV}}} \left(1 - h - h\_f\right) \tag{3}$$

In it, *hf* is the head loss due to friction in the penstock and *Tw* is the water time constant or water starting time in seconds, defined as:

$$T\_w = \frac{L}{A\mathcal{g}} \frac{Q\_{\text{base}}}{H\_{\text{base}}} \tag{4}$$

where *L*, in m and A, in m2, are the penstock length and area, respectively, and *g* is the gravitational acceleration. Equation (3) is valid for penstocks of short length, in which the pressure wave effects can be neglected.

In WH mode, the HTG must supply the grid with an amount of power equal to the difference between the consumer load and the WTG supplied power. The HT working power range can therefore be very broad and the low load regime can be very common. The efficiency of a HT depends mainly on the flow rate [10]. The HT efficiency is null for flow rates ranging from zero to the no-load flow rate *qnl*. Different types of hydro-turbines, such as Pelton, Crossflow, Francis, Kaplan, etc. have different *qnl* and efficiency curves. Due to the highly variable working regime of the HT in WH mode, a good efficiency for *q* > *qnl* is the main requirement for the HT in the study considered here. Additionally, in WO mode, as commented before, the HT runs with zero flow rate. When the WHIM is in WO mode and the WTG active power falls below the consumer load, the system frequency will fall and the WHIM must transition from WO to WH mode, ordering the HTG to supply power. In order to supply power, the HT speed governor must increase the flow rate from zero. The time spent to take the flow rate from zero to *qnl* is a dead time in which the HT is not producing any power and the system frequency will continue falling. The bigger *qnl*, the longer the dead time, and the worse will be the system transient. Therefore, an HT with a small *qnl* is needed to improve the WO to WH modes transition. Among the different HT types, the ones that most comply with the previous requirements are Crossflow and Pelton impulse type turbines, Pelton type HT having better efficiency figures than Crossflow type HT. A Pelton type HT has, therefore, been considered in this article. Figure 2 depicts the Pelton turbine efficiency curve used in this article, as a function of *q*, in abscissa, where *qnl* is 0.1 p.u.

**Figure 2.** Pelton turbine efficiency vs. water flow rate p.u.

The HT mechanical output power is converted into electrical power by the SM. The SM must be permanently connected to the isolated grid as it generates the isolated grid voltage waveform. To allow the SM automatic voltage regulator control the system voltage within allowable limits, the SM speed has to always be close to its rated speed. The SM nominal power *PSM*−*NOM* is 300 kVA, which is enough to provide the microgrid with all the active and reactive power demanded. Since the system frequency is *f* = 50 Hz and the SM has *p* = 16 pole pairs, calculating the synchronous speed, *n*, as *n* = 60 *f* /*p*, the HTG rated speed, *nref* , results in 187.5 rpm. Inertia constant *H* for low speed HTGs (<200 rpm), ranges between 2–3 s [21], so 2 s is assigned to *HHTG*, as this article's HTG is a low power one.

## *2.2. The WTG Model*

The WTG in Figure 1 is a fixed-pitch WT driving an Induction Generator (IG) directly connected to the autonomous grid. Both elements, WT and IG, form a constant-speed stall-controlled WTG. The mechanical power extracted from the wind by the wind turbine, as stated by [22], is

$$P\_{t-m\text{cc}} = \mathbb{C}\_p \, P\_{wind} = \mathbb{C}\_p \, \frac{1}{2} \rho A v^3 \tag{5}$$

where the power coefficient, *Cp*, represents the ratio of the power extracted from the wind to the power available in the wind. The power available in the wind, *Pwind*, can be calculated as half of the air density *ρ*, times the blade swept area *A*, times the cube of the wind speed *v*. In the absence of pitch regulation and due to the very limited IG speed range variation, *Cp* can be considered, as a first approximation, a function of the wind speed only. As the wind speed is quasi-random, the active power generated by the WTG will be uncontrolled. The stall-controlled WT model simulated here follows the description presented in [23]. The IG has a rated power of 275 kW (WTG rated power *PT*−*NOM* = 275 kW), which is enough to supply all the active power demanded in WO mode, and since it consumes reactive power, a capacitor bank is included in Figure 1 to improve the power factor. Inertia constant *Hw* values for WTGs range from 2 to 6s[24], so 2 s is assigned to *Hw*, as this article's WTG is a low power one.

## *2.3. The DL Model*

The Dump Load (DL) is used in the WHIM to artificially load the isolated grid when the WTG power exceeds the load consumed power in WO mode. The DL shown in Figure 1 comprises a bank of eight three-phase resistors connected in series with Gate Turn-Off (GTO) thyristor-based power switches. The eight resistor values are *R*, *R*/2, *R*/22, ..., *R*/27. With *Vn* the isolated grid rated voltage, *P*0 = *V*2*n* /*R*

*Energies* **2020**, *13*, 5937

is the rated power of the resistor of value *R* and the eight resistors rated powers are *P*0, 2*P*0, <sup>2</sup>2*P*0, ..., <sup>2</sup>7*P*0. If *I*0, *I*1, ..., *I*7 are the opened(0)/closed(1) states of the GTO power switches in series with the resistors, the active power absorbed by the DL can be expressed as

$$(I\_0 + I\_1 \mathbf{2}^1 + \dots + I\_7 \mathbf{2}^7) P\_0 \tag{6}$$

Equation (6) states a DL power variation from 0 to 255*P*0. In this article *P*0 = 1.5 kW and the DL rated power *PD*−*NOM* is 382.5 kW, which is 39% greater than *PT*−*NOM*. This *PD*−*NOM* value guarantees the possibility of WO mode operation, even in the absence of any consumer load and with a WTG power rated above one.
