**4. Results**

As was discussed in Section 3, and with the aim of evaluating frequency oscillations and power system performances under imbalance conditions with different number of areas, two different multi-area power systems were simulated: (i) a two-area power system and (ii) a three-area power system. Both power systems were implemented in Matlab/Simulink c (2016, MathWorks, Natick, MA, USA). Source codes are available under request.

#### *4.1. Two-Area Interconnected Power System*

Firstly, and in order to evaluate the sensitivity of frequency excursions in a multi-area power system, two different imbalance conditions were simulated. In both cases, one area is submitted to imbalances while the other area maintains a balanced condition. A 5% increase in demand of the base power is assumed in all simulations as imbalance power ( Δ *PL*,<sup>1</sup> = Δ *PL*,<sup>2</sup> = 100 MW). Under these scenarios, with an active-power deficit, different frequency control strategies are addressed by the simulations depending on the generation units involved in the frequency response: *case* (1) whole conventional generation units of the multi-area power system; *case* (2) whole conventional generation units and only wind power plants within the area submitted to imbalances; and *case* (3) whole conventional generation units and wind power plants.

Figure 8a,b shows the frequency oscillations in both areas when a power imbalance is applied to Area 1 ( Δ *PL*,1). As can be seen, the maximum nadir is achieved in both areas when *case* (1) is conducted. The nadir values are improved when wind power plants are considered for frequency control: *cases* (2) and (3). Indeed, *case* (2) offers a smoother and less oscillatory response than *case* (3), yielding a stabilization time interval very similar to *case* (1). Moreover, *case* (3) causes three different well-identified frequency shifts: the first one is due to the power imbalance; the second one occurs due to the lack of coordination between power plants as well as the different time response of the supply-side operation units (see Figure 9); and the last one depends on the transition from overproduction mode to recovery mode of the wind power plant located in Area 2 (see Figure 2b and the active power decrease in WPP2 Figure 9).

**Figure 8.** Area 1 under power imbalance (Δ*PL*,1); (**a**) frequency oscillations in Area 1; (**b**) frequency oscillations in Area 2.

**Figure 9.** Area 1 under power imbalance (Δ*PL*,1): generation deviations in Area 2.

A similar study can be carried out by considering power imbalance conditions in Area 2 (Δ*PL*,2). Figure 10 compares the results in terms of nadir for both scenarios (Δ*PL*,<sup>1</sup> and <sup>Δ</sup>*PL*,2) and considering the different frequency control strategies. As can be seen, minor differences are found in both analyses. In addition, Figure 11 compares the tie-line power evolution under both imbalance conditions, <sup>Δ</sup>*PL*,<sup>1</sup> and Δ*PL*,<sup>2</sup> accordingly, and peak-to-peak tie-line power exchange. Subsequently, and according to the generation mix considered in each area, frequency oscillations and active tie-line power results present similar values regardless of the area submitted to imbalances. Based on these results, and taking into account the different frequency control strategies implemented and simulated, lower frequency oscillations are obtained when only wind power plants within the area submitted to imbalance conditions are considered. Therefore, the contribution of wind power plants from other areas under frequency excursions would provide additional oscillation responses.

**Figure 10.** Nadir: Comparison of <sup>Δ</sup>*PL*,<sup>1</sup> and <sup>Δ</sup>*PL*,<sup>2</sup> scenarios. (**a**) Area 1 submitted to imbalance (Δ*PL*,1); (**b**) Area 2 submitted to imbalance (Δ*PL*,2).

**Figure 11.** Active tie-line power evolution: comparison of <sup>Δ</sup>*PL*,<sup>1</sup> and <sup>Δ</sup>*PL*,<sup>2</sup> scenarios. (**a**) Area 1 submitted to power imbalance (Δ*PL*,1); (**b**) Area 2 submitted to power imbalance (Δ*PL*,2); (**c**) Area 1 submitted to imbalance (Δ*PL*,1); (**d**) Area 2 submitted to imbalance (Δ*PL*,2).

#### *4.2. Three-Area Interconnected Power System*

Considering the preliminary conclusion given in Section 4.1, where similar results are obtained independently of the area submitted to imbalances, the authors reduce the number of simulations in this three-area interconnected power system, assuming only that one area is submitted to imbalance conditions. Different frequency control strategies are then simulated by including an active-power deficit applied to Area 1, <sup>Δ</sup>*PL*,1. It is also defined as a step of 5% with respect to the base power (Δ*PL*,<sup>1</sup> = 100 MW).

Figure 12 depicts the frequency deviation of each area and the nadir comparison according to the different frequency control strategies. As can be seen, the results are in line with those obtained previously, when a two-area power system was considered. Therefore, the maximum nadir values are obtained in all areas when wind power plants are not included for frequency control. When wind power plants provide frequency response, the nadir values of all areas are considerably improved. Regarding *case* (2) and *case* (3), nadir values give similar results. However, larger frequency oscillations are identified when *case* (3) is conducted, especially in Area 2 and area 3. This behavior is a consequence of the wind power variations due to the different operation modes of each frequency controller, increasing the tie-line power exchanged between these two areas. Stabilization time presents similar values (*tstab* 100 s) in all cases and areas.

**Figure 12.** Frequency oscillations: Area 1 under power imbalance (Δ*PL*,1); (**a**) frequency oscillations in Area 1 (Δ*f*1); (**b**) frequency oscillations in Area 2 (Δ*f*2); (**c**) frequency oscillations in Area 3 (Δ*f*3); (**d**) nadir values: case comparison.

Figure 13 shows and compares the tie-line power variation and its peak-to-peak value. As can be seen, tie-line power exchange does not overcome the maximum restriction of 10% under any circumstances. Power exchanged between areas 2–3 is practically negligible regardless of the frequency control strategy, as the frequency deviations in these areas are a consequence of imbalances subsequently induced by Area 1 (see Section 3.1). As was previously mentioned, with the use of the wind power plants in all the areas <sup>Δ</sup>*Ptie*2,3 increases due to the wind power plants variations. Actually, <sup>Δ</sup>*Ptie*2,3 *case* (3) doubles the value of *case* (2), subsequently producing more oscillations in frequency deviations in those areas, as depicted in Figure 12. Therefore, *case* (2) is suggested by the authors under imbalance conditions to reduce frequency oscillations and power flow between areas.

**Figure 13.** Area 1 under power imbalance (Δ*PL*,1): tie-line power comparison; (**a**) tie-line power variation between Areas 1 and 2; (**b**) tie-line power variation between Areas 2 and 3; (**c**) tie-line power variation between Areas 3 and 1; (**d**) comparison among peak-to-peak tie-line power variation exchange.
