4.1.1. Objective Function

The primary objective is minimizing the total control costs of multiple resources, and the secondary objective is minimizing the total weighted deviations of the frequency and voltage. The two objectives are normalized and combined through a weighted coefficient to formulate the objective function *f*, as shown in the following equations:

$$\min f = f\_1 + w\_0 f\_{2'} \tag{1}$$

$$\min f\_1 = (\sum\_{i=1}^{N\_{\rm D}} \Delta p\_i^{\rm DC} + \sum\_{j=1}^{N\_{\rm S}} \Delta p\_j^{\rm pump} \cdot x\_j + \sum\_{k=1}^{N\_{\rm L}} \Delta p\_k^{\rm load}) / S^{\rm base} \tag{2}$$

$$\min f\_2 = \varepsilon\_f \sum\_{\mathcal{S}} \frac{\Delta f\_{\mathcal{S}}(p)}{\sigma\_{\mathcal{S}} f^{\text{base}}} + \varepsilon\_v \sum\_{\text{m}} \frac{\Delta V\_m(p)}{V^{\text{base}}} K\_{V\prime} \tag{3}$$

$$
\Delta f\_{\mathcal{S}}(p) = \Delta f^{\sf s}(p) + \Delta f\_{\mathcal{S}}^{\sf cl}(p), \tag{4}
$$

$$
\Delta V\_m(p) = \Delta V\_m^\circ(p) + \Delta V\_m^d(p), \tag{5}
$$

where *f*1 is the primary objective; *f*2 is the secondary objective; ω0 is the weighted coefficient; *N*D, *N*S, and *N*L are the numbers of HVDCs, pumped storages, and interruptible loads in emergency resources; <sup>Δ</sup>*p*DC*i* is the power adjustment of HVDC *i*; <sup>Δ</sup>*p*pump *j* is the consumed power of the tripped pumped storage *j*; *xj* is a 0–1 variable, 1 represents tripping the pumped storage *j* while 0 represents keeping the original state; <sup>Δ</sup>*p*load *k* is the power adjustment of load *k*; *S*base is the base value of the power system capacity; ε*f* and ε*v* are the weighted coefficients of the frequency and voltage; Δ*fg* and <sup>σ</sup>*g* are the frequency deviation and coefficient of the primary frequency adjustment at the generator *g*, respectively; *f* base is the reference frequency; Δ*Vm* is the voltage deviation of bus *m*; *KV* is the voltage regulation factor; *V*base is the reference voltage; Δ*f* s is the steady-state frequency deviation of the system; Δ*f* d*g* is the transient-state frequency deviation of bus *g*; and Δ*V*<sup>s</sup>*m* and <sup>Δ</sup>*V*d*m* are the steady-state and transient-state voltage deviations of bus *m*.

Equation (1) is the objective function, in which the weighted coefficient ω0 is defined by users. Equation (2) is the primary objective, with <sup>Δ</sup>*p*DC*i* , *xj*, and <sup>Δ</sup>*p*load *k* as decision variables. Equation (3) describes the secondary objective. Considering that the power imbalance of the receiving-end system due to the HVDC blocking event will seriously affect the system frequency, assume ε*f* > ε*v*. Equations (4) and (5) represents the frequency deviation and voltage deviation, respectively.

It should be noted that the priority of three kinds of control resources is different, which is reflected by the control action time in the control strategy. Taking the control speed and control cost into account, the action sequence adopted here is HVDCs, pumped storages, and interruptible loads. Considering the communication delay and control device response time, the control action time of HVDCs is 100 ms after the security or stability issue occurs, and the pumped storages and interruptible loads are followed, which are 300 and 500 ms, respectively [53]. Therefore, the control action time for control resources is fixed and not taken as the decision variable in the decision-making.

## 4.1.2. Adjustment Amount Constraints

The adjustment amount of each equipment should not exceed its maximum power capacity, such as the maximum active power of HVDC can be increased up to being 1.1 times the rated capacity [54]. Therefore, the emergency control strategies should meet the following constraints:

$$p\_i^{\rm DC,min} - p\_i^{\rm DC} \le \Delta p\_i^{\rm DC} \le p\_i^{\rm DC,max} - p\_i^{\rm DC} \ (i = 1, \cdots, N\_{\rm D}), \tag{6}$$

$$p\_k^{\text{load,min}} - p\_k^{\text{load}} \le \Delta p\_k^{\text{load}} \le p\_k^{\text{load,max}} - p\_k^{\text{load}} \ (k = 1, \dots, N\_{\text{L}}), \tag{7}$$

where *<sup>p</sup>*DC*i* is the transmission power of HVDC *i*; *p*DC,max *i* and *p*DC,min *i* are the transmission power limits of HVDC *i*; *p*load *k* is the power of load *k*; and *p*load,max *k* and *p*load,min *k* are the LS amount limits of load *k*.

Equation (6) is the power adjustment amount limits of HVDC and Equation (7) is the LS limits.
