**3. Case Study**

This Section is organised as follows. Section 3.1 provides a sensitivity analysis of the impact of different frequency controllers/machine parameters using a complete SUC, a DUC, a simplified SUC and an alternative SUC model, and different scheduling time periods. This analysis allows comparing and drawing conclusions on the effect of these parameters on the dynamic performance of the system. Section 3.2 compares the impact that different models of SUC have on the dynamic performance of power systems.

A modified New England IEEE 39-bus system [29] is used in all simulations, where the data of the SUC are taken from [30]. In particular, the value of load curtailment (present in the reference model of SUC) is taken equal to \$1000/MWh [21]; the cost of wind is assumed zero; the value of the fixed (*CFg* ) and variable (*CVg* ) cost coefficients is taken equal to the fixed and proportional cost coefficients *a* (\$/h) and *b* (\$/MWh), respectively, in [30]. The focus of this work is on the first 4 h of the planning horizon. Furthermore, the wind profile is modeled as in [13].

Wind power uncertainty, volatility and rolling planning horizon within the SUC are modeled as in [13]. In particular, three wind power scenarios are considered, namely, low, medium, and high. The medium scenario considers a 25% wind penetration level, while the low and high scenario are built according to the maximum variation width. Moreover, three wind power plants are considered and connected at bus 20, 21 and 23, respectively, with a nominal capacity of 300 MW each. It should be noted that this number of scenarios is considered sufficient as it was shown in [13] that increasing the number of scenarios leads to similar effect on the dynamic performance of the system.

The results of the case study are based on a Monte Carlo method, where 50 simulations are considered for each scenario. Moreover, the standard deviation of the frequency of the COI, *σ*COI, is computed as the average of the standard deviation obtained for each trajectory. The SUC models (6)–(24), (25)–(29) and (30)–(39) are modeled using the Gurobi Python interface [28], whereas all simulations are obtained using a Python-based software tool for power system analysis, called DOME [27].

#### *3.1. Sensitivity Analysis of the Impact of Different Frequency Controllers/Machine Parameters*

In this Section, we vary (one at a time) three relevant frequency control/machine parameters and observe their impact on the standard deviation of the frequency.

The following base-case scenario is considered: the value of the gain of the AGC is taken equal to *K*0 = 50, the value of the droop of the TGs is taken equal to *R* = 0.05, and the value of inertia of the machines is taken equal to the original values. In the scenarios below, the total inertia of the system and the gain of AGC is decreased up to 45% from the base case. Similarly, the aggregated system droop is increased up to 45% from the base case. Finally, as system operators still rely on deterministic UC and different subhourly scheduling time periods [31], this sensitivity is performed using 15- and 5-min time periods and different SUC formulations.

In the following, the subindexes *S* and *D* indicate control parameters for stochastic and deterministic scenarios, respectively.

#### 3.1.1. SUC with 15-min Time Period

In this scenario, we use a 15-min scheduling time period and set the SUC probabilities for the low, medium and high wind power scenario equal to 20%, 60%, 20%, respectively. Same probabilities are used when solving the MC-TDS. Next, the relevant frequency control/machine parameters are varied accordingly up to 45% of their base case value. Figure 4 shows the effect of the variation of these parameters on *σ*COI.

The gain of AGC, *KS*, has the highest impact on *σ*COI. The relationship between the gain *KS*, the droop *RS*, and *σ*COI is almost linear within the used range. This indicates larger frequency deviations as synchronous generators are replaced with RES (assuming that RES will not provide frequency regulation). On the other hand, the inertia *MS* appears to have a small impact on long-term frequency deviation. This result indicates that while the inertia is the main parameter impacting on the frequency dynamics following a major contingency, this is not the case on its impact on the *σ*COI (see also [4], which draws a similar conclusion).

#### 3.1.2. DUC with 15-min Time Period

As discussed above, system operators still rely on a DUC formulation when scheduling the system. Therefore it is important to compare the impact of the variation of the relevant control/machine parameters on *σ*COI, using a SUC and a DUC. With this aim, we set the DUC probabilities for the low, medium and high wind power scenarios equal to 0%, 100%, 0%, respectively. Thus, perfect forecast, which corresponds to the medium scenario, is assumed. However, when solving the MC-TDS, 20%, 60%, 20% probabilities are used to generate the three wind power scenarios. This creates a mismatch between forecast and actual wind variations and allows evaluating the robustness of the SUC and DUC formulations.

Figure 4 shows the impact on *σ*COI of varying control/machine parameters. A relevant difference with respect to the scenario above is that the droop *RD* has the highest impact on *σ*COI when its value is ≥ 30% with respect to the base case. On the other hand, the gain of the AGC, *KD* and the inertia *MD* have similar impact on *σ*COI as in the previous scenario. It appears that, the impact of different control parameters on *σ*COI depends on the UC formulation (deterministic or stochastic). It is thus not obvious which one (i.e., *R* or *K*) has the highest impact on *σ*COI. Figure 4 compares the results of scenario 1 and 2 and shows that using a SUC leads to lower variations of the frequency. Furthermore, it should be noted that the differences between scenarios in Figure 4, for example, *RS* and *RD* after 28%, are due to the fact that both models, namely, SUC and DUC, produce different schedules for generators.

**Figure 4.** 15-min time period—*<sup>σ</sup>*COI as a function of different frequency controllers/machine parameters using the complete SUC and DUC models.

Finally, although the focus of this paper is on the impact of different SUC models on the dynamic behaviour of the system, it is relevant to show the total operating cost for each model. With this aim, Table 1 shows the total operating cost for each subhourly UC model. For the considered case, the complete SUC and DUC produce very similar operating costs, namely, 412,000\$ and 411,580\$, respectively. While the simplified SUC model provides a lower estimate of the operating costs, namely, 398,000\$, as it is less constrained. The alternative SUC model results in lower operating costs compared to others models (complete SUC, simplified SUC and DUC), namely, 339,470\$, mainly due to the fact that it has a slightly different objective function, e.g., it does not include the fixed cost.

**Table 1.** 15-min time period—total operating costs for different UC models.


3.1.3. Sensitivity Analysis Using the Simplified and Alternative SUC, and 15-min Time Period

Here, we perform the same sensitivity analysis as above, but this time using the simplified and alternative SUC models, respectively. Such an analysis allows comparing and drawing conclusions on the impact of different frequency control/machine parameters on *σ*COI, using different subhourly SUC models. With this aim, Figure 5 shows the relevant results of the sensitivity analysis. Note that the subindexes *Sim* and *Alt* indicate control parameters for simplified and alternative SUC scenarios, respectively. As expected, results are, in general, very similar to the ones discussed in previous sections. The gain of the AGC, in this case, *KSim* and *KAlt*, has most of the time the highest impact on *σ*COI, as well as the inertia of the machines appears to have a small impact. Moreover, using an alternative SUC model and varying its relevant parameters leads to lower variations of the frequency. This is due to the fact that the alternative SUC model schedules more generators to be online compare to the other three UC models. Thus, depending on the SUC model, different control parameters (e.g., *K* or *R*) can have different impact on *σ*COI.

**Figure 5.** 15-min time period—*<sup>σ</sup>*COI as a function of different frequency controllers/machine parameters using the simplified and alternative SUC models.

#### 3.1.4. SUC with 5-min Time Period

This scenario investigates whether a shorter scheduling time period of SUC, namely, 5 min, changes the results and conclusions drawn for the 15-min time period. Due to the shorter time period, the wind power uncertainty level within SUC is lower compared to the previous section. Both SUC and MC-TDS probabilities for the low, medium and high scenarios are set equal to 20%, 60% and 20%, respectively.

Figure 6 shows the effect of the variation of the relevant control parameters on *σ*COI. The most visible effect of reducing the time period to 5 min is that the value of *σ*COI decreases in all scenarios due to a lower wind power uncertainty. In this case, the gain of the AGC, *KS*, has the highest impact on *σ*COI. The relationship is linear in the considered range, which indicates the need to increase the frequency regulation to keep long-term frequency deviations within certain limits. Finally, the inertia has little effect on *σ*COI, supporting the conclusions above.

**Figure 6.** 5-min time period—*<sup>σ</sup>*COI as a function of different frequency controllers/machine parameters.

#### 3.1.5. DUC with 5-min Time Period

Figure 6 shows the effect of the variation of control/machine parameters using a 5-min scheduling and DUC. Results are very similar to the 5-min scheduling SUC. To better show the differences, Figure 6 compares the results of both scenarios, where it can be observed that the SUC leads, in general, to lower frequency variations. In both scenarios, the gain of the AGC has the highest impact on *σ*COI, whereas the inertia has a small impact.

#### *3.2. Comparison of Different SUC Models*

Even though system operators are skeptical regarding the use of SUC approaches due to their complexity and transparency, they acknowledge the need to better represent uncertainty when scheduling the system [24].

For this reason, the objective of this section is to compare the impact that these models have on long-term power system dynamics. Such a comparison is made using a 15-min time period.

We first compare the impact on the dynamic response of the New England 39-bus system of the SUC formulations (6)–(24) (complete SUC) and (25)–(29) (simplified SUC).

**Figure 8.** Trajectories of *<sup>ω</sup>*COI for 15-min time period and simplified SUC.

Figures 7 and 8 show the *ω*COI, of the complete and simplified SUC models, respectively. The two formulations returns an almost identical *ω*COI, namely *σ*COI = 0.000800 pu(Hz) for the complete SUC and, while that of the simplified SUC is *σ*COI = 0.000794. Specifically, the difference is about 1%. Figures 9 and 10 show the mechanical power of the conventional synchronous generators 1, 2 and 4, of the complete and simplified SUC model, respectively. The two models produce similar schedules. In fact, at the beginning of the planning horizon, the simplified SUC model alternates the schedules for generators 2 and 4, and after some time (i.e., after 6000 s) it produces the same schedules as the complete SUC model. It appears, thus, that for this particular system and for normal operation conditions, the differences between using a complete SUC model and a simplified one are negligible. These results sugges<sup>t</sup> thus that an involved UC formulation is not necessarily the best in normal operating conditions of the system.

**Figure 9.** Mechanical power of synchronous generators 1, 2 and 4, for 15-min time period and complete SUC model.

**Figure 10.** Mechanical power of synchronous generators 1, 2 and 4, for 15-min time period and simplified SUC model.

Next, we compare the impact on system dynamics of the problem (6)–(24) (complete SUC) and the problem (30)–(39) (alternative SUC) described in Section 2.6. With this aim, a MC-TDS per each SUC formulation is carried out.

Figure 11 shows *ω*COI for the alternative SUC model. Compared to the complete SUC model (see Figure 7), the alternative formulation leads to lower frequency variations, i.e., *σ*COI = 0.000390. In other words, the differences between the models is about 48%. This is due to fact that the two SUC formulations produce slightly different schedule for generators. Specifically, the alternative SUC model schedules a few more generators (Figure 12) compared to the complete SUC model (Figure 9). This, in turn, implies more regulation, which helps better manage wind uncertainty. This can be observed in Figure 11 where, in the time between two scheduling events—i.e., 15 min—frequency variations are lower due to the increased frequency regulation available in the system.

**Figure 12.** Mechanical power synchronous generators 1, 2 and 4, for 15-min time period and alternative SUC model.

This result had to be expected. From the dynamic point of view, in fact, it is better to schedule more conventional synchronous generators, which provide both primary and secondary frequency regulations, rather than wind generation.
