*3.1. Macro Parameters Histories*

The runaway dynamic characteristics of the four pump-turbines are shown in *n*11-*Q*<sup>11</sup> plane in Figure 2, in which the unit parameters are defined as *n*<sup>11</sup> = *nD*1/ √ *H* and *Q*<sup>11</sup> = *Q*/ *D*2 1 √ *H* , where *H* = *E*1–*E*2, with *E*1 and *E*2 the total energy values at the spiral-casing inlet and runner outlet, respectively. Comparing the computed results (red lines) of the four pump-turbines, we know that the dynamic trajectories of PT-1 and PT-2 have very high amplitudes in high frequency pulsation signals in the *n*11-*Q*<sup>11</sup> plane, while those of PT-3 and PT-4 are relatively smaller and become obvious only near the runaway points. In addition, the low-pass filtered curves (green lines) of the original data do not go along the static characteristic curves (black lines) obtained from the model tests, however, they have good agreements before entering the S-shaped region. Once entering the S-shaped region, the dynamic curves deviate from the measured static ones. These deviations have been analyzed in [27], in which the influences of the sections for head definition, the water inertia in pipes and the rotational inertia of unit on the dynamic trajectory were discussed. In this paper, due to neglecting water inertia in pipes and choosing the same rotational inertias, the deviations are different. In fact, the simulating rotational inertia is based on the actual value of PT-1, therefore, the actual rotational inertia of PT-2 is much larger, and those of PT-3 and PT-4 are much smaller. For PT-2, small simulating rotational inertia will lead to large speed increasing rate, then the dynamic trajectory is on the right side of the static curve obviously, which is opposite to the phenomenon in PT-3 and PT-4. To verify the rationality of the above settings and results, we take reference [28] as an example, in which the influence of the inertia of rotating part has been well explained, and it shows that the dynamic trajectories affected by different rotating part inertia in *n*11-*Q*<sup>11</sup> plane are very similar with those in this paper. In addition, there is no very large deviation in the dynamic trajectories, though the pulsations in the *n*11-*Q*<sup>11</sup> plane and variation period of rotational speed are different. From the above analysis, we know that the results of transient process are quite different from the static ones and it is necessary to consider the dynamic effect in transient simulations.

The time histories of the main macro parameters during the runaway processes are also shown in Figure 2. Generally speaking, the dynamic histories of PT-1 and PT-2 show damped oscillations, while those of PT-3 and PT-4 demonstrate undamped oscillations. The working points of PT-1 and PT-2 go through the turbine (T) and turbine braking (TB) modes, but do not enter the reverse pump (RP) mode, and the macro parameters fluctuate in the T and TB regions with gradually decreasing amplitudes. On the other hand, the working points of PT-3 and PT-4 not only go across the T and TB modes, but also go down to the RP mode, and fluctuate periodically in these three modes. Overall, the fluctuation periods of the macro variables of the four pump-turbines are about 11.5, 10, 14.4, and 9.6 s, respectively, though the inertia values of rotating parts are the same (Table 1) in the simulations. The periods are also influenced by the rated rotating speed, discharge, and output. In addition, the maximum rotational speeds are heavily affected by the above factors [27,28], and can reach more than 1.4 times that of the initial value in PT-1 but less than 1.2 times in PT-4.

**Figure 2.** Working point trajectories and parameter histories of the four pump-turbines: (**a**) PT-1, (**b**) PT-2, (**c**) PT-3, (**d**) PT-4.
