Two-stage Transient-stability-constrained Optimal Power Flow for Preventive Control of Rotor Angle Stability and Voltage Sags

Jorge Uriel Sevilla-Romero; Alejandro Pizano-Martínez; Claudio Rubén Fuerte-Esquivel; Reymundo Ramírez-Betancour

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Two-stage Transient-stability-constrained Optimal Power Flow for Preventive Control of Rotor Angle Stability and Voltage Sags PDF

- ORCID：
Jorge Uriel Sevilla-Romero ¹
✉
- ORCID：
Alejandro Pizano-Martínez ²
✉
- ORCID：
Claudio Rubén Fuerte-Esquivel ¹
✉
- ORCID：
Reymundo Ramírez-Betancour ³
✉

1. Faculty of Electrical Engineering, Universidad Michoacana de San Nicolás de Hidalgo, 58030 Morelia, México； 2. Department of Electrical Engineering, Universidad of Guanajuato, 36787 Salamanca, México； 3. Department of Electrical and Electronic Engineering, Universidad Juárez Autónoma de Tabasco, 86040 Villahermosa, México

Updated：2024-09-24

DOI：10.35833/MPCE.2023.000461

OUTLINE

Abstract

In practice, an equilibrium point of the power system is considered transiently secure if it can withstand a specified contingency by maintaining transient evolution of rotor angles and voltage magnitudes within set bounds. A novel sequential approach is proposed to obtain transiently stable equilibrium points through the preventive control of transient stability and transient voltage sag (TVS) problems caused by a severe disturbance. The proposed approach conducts a sequence of non-heuristic optimal active power re-dispatch of the generators to steer the system toward a transiently secure operating point by sequentially solving the transient-stability-constrained optimal power flow (TSC-OPF) problems. In the proposed approach, there are two sequential projection stages, with the first stage ensuring the rotor angle stability and the second stage removing TVS in voltage magnitudes. In both projection stages, the projection operation corresponds to the TSC-OPF, with its formulation directly derived by adding only two steady-state variable-based transient constraints to the conventional OPF problem. The effectiveness of this approach is numerically demonstrated in terms of its accuracy and computational performance by using the Western System Coordinated Council (WSCC) 3-machine 9-bus system and an equivalent model of the Mexican 46-machine 190-bus system.

Keywords

Dynamic security assessment; transient stability; transient voltage sag (TVS); optimal power flow (OPF).

I. Introduction

THE occurrence of severe disturbances in power systems may lead to large excursions of rotor angles of the generator that cause transient instability and bus transient voltage sag (TVS) problems. The transient instability involves an irrevocable deviation among the transient trajectories of rotor angles of the generator that cause the loss of synchronism of generators. In contrast, TVS could trigger load shedding control actions as an emergency countermeasure against transient instability problems. Hence, the electric power system operating at a given equilibrium point can be declared insecure if the transient trajectories of rotor angles and bus voltage magnitudes are not bounded in response to a specific severe disturbance [

1]. However, the transient evolution of trajectories can be bounded through effective control actions, which are preventively assessed and executed to lead the system to be a secure steady-state operating point. The active power re-dispatch of generators is one of the most effective control actions to ensure the security of power systems [1], [2]. Furthermore, this control action can be achieved in the most economical way by solving the transient-stability-constrained optimal power flow (TSC-OPF) problem [3], [4]. Based on the above consideration, this work focuses on determining an economically optimal equilibrium point that ensures the transient trajectories of rotor angles and voltage magnitudes are maintained within bounds when the electric power system is subjected to a severe disturbance.

Strictly speaking, the TSC-OPF problem is formulated as a semi-infinite optimal problem, where the objective is to minimize the active power re-dispatch cost subjected to equality and inequality constraints [

5], [6]. The equality constraints represent the steady-state and dynamic operating state of the power system. In contrast, the set of inequality constraints keeps the transient trajectories of rotor angles and nodal voltage magnitudes within bounds [7]. The solution to this challenging problem focuses on deterministic- and evolutionary-based TSC-OPF approaches [3], [4], with the former being adopted in this work. The deterministic-based TSC-OPF approaches must transform the TSC-OPF model into a non-linear optimal problem that is solved by using existing non-linear programming methods [5], [6]. Depending on the transformation strategy adopted, different TSC-OPF approaches have been reported, with a comprehensive classification given in [3], [8], and [9]. These research works clearly show that only a few deterministic-based approaches, which are classified as simultaneous discretization (SD) [3], [7], and [10], multiple shooting (MS) [11], and single shooting (SS) [12] approaches, have considered the transient stability and TVS constraints for formulating the TSC-OPF problem.

The SD approaches discretize the dynamic, transient stability, and TVS constraints at each time step associated with the numerical discretization of the stability study period. The entire set of constraints is directly included in the conventional optimal power flow (OPF) formulation, which results in a discrete non-linear TSC-OPF model that is solved as a single problem for control parameters, steady-state variables, and dynamic-state variables. Since the number of discretized constraints is proportional to the number of integration steps, the dimension of the TSC-OPF model is several orders higher than that of the traditional OPF model. Furthermore, since the set constraints must be simultaneously satisfied for every time step of the entire transient stability experimental period, the TSC-OPF problem suffers from enormous complexity and computational burden, so the solution may become intractable even for small-scale electric power systems [

13].

MS [

11] and SS [12] approaches simplify the solution of the TSC-OPF problem by replacing the dynamic constraints with time domain (TD) simulations, which are performed in each iteration of the optimization solution process to obtain the system dynamics. Based on the resulting system dynamics, the transient stability and TVS constraints are evaluated, and a sensitivity analysis is performed to numerically assess their corresponding gradients for solving the optimization problem. The MS approach has a moderate convergence rate and may impose an enormous computational burden because of the massive execution of TD simulations and sensitivity analysis [3]. In addition, the SS approach shows a slow convergence rate, and it fails in cases where the TD simulation and trajectory sensitivity analysis are ill-conditioned because of unbounded state trajectories [11].

An attractive strategy for solving the TSC-OPF problem is proposed by deterministic sequential approaches introduced in [

13] and [14]. These approaches reformulate the TSC part as active power re-dispatch constraints. The TSC-OPF problem is then decomposed into two mutually connected subproblems: one associated with the OPF problem incorporating the active power re-dispatch constraints, and the other one with the transient stability assessment that obtains stability status of the transient trajectories of rotor angles, as well as the information required to assemble those re-dispatch constraints. Consequently, both subproblems are sequentially solved to force active power re-dispatches that gradually cancel out the rotor angle instability without including discretized constraints in the OPF problem. Therefore, the problem dimension, computational burden, and complexity are much lower than those in the SD, MS, and SS approaches. The drawback is that the formulation of the TSC part is based on heuristic criteria for active power re-dispatch, which may lead to suboptimal solutions [4]. Considering the suboptimal solution, the improved deterministic sequential approaches focused on the preventive control of the transient trajectories of rotor angles have been recently proposed, where the most economical active power re-dispatch is performed by considering a non-heuristic stability criterion during the preventive control process, e.g., in [15]-[21]. Even though the idea behind the deterministic sequential approaches is simple and intuitive, the approaches based on this concept have only focused on the preventive control of transient stability, e.g., without taking care of TVS in bus voltage magnitudes. The lack of control, however, for the transient evolution of voltage magnitudes within specified limits might activate load shedding schemes because of TVS. From the mathematical perspective, sequential approaches overlook TVS problems because the optimal problem is formulated solely in terms of steady-state variables. As a result, current sequential approaches cannot control TVS in voltage magnitudes. Thus, the proposed approach in this paper overcomes this problem, making such control possible by taking into account that, for large disturbances, TVS in voltage magnitudes is associated with significant excursions of generator rotor angles, as demonstrated in [1]. Consequently, the dynamics of voltage magnitudes can be controlled through active power re-dispatch.

Based on the preceding discussion, a deterministic non-heuristic sequential TSC-OPF approach is proposed, wherein both transient stability and TVS problems are seamlessly addressed. Therefore, the existence of two feasible operating regions is assumed: the transient stability region and admissible transient voltage sag (ATVS) region, which are composed of equilibrium points where the system subjected to a specified contingency scenario exhibits bounded evolutions of rotor angles and transient voltages. Furthermore, as directly inferred from the research works reported in [

7], these two regions intersect in the parameter space of active power generation. Hence, to move the operating state of the system from a transiently infeasible operating point to an equilibrium point inside the intersection of both feasible operating regions, a sequence of non-heuristic active power re-dispatch is performed to initially steering the operating state of the system into the transient stability region and then into the ATVS region. These active power re-dispatches are performed by using the concept of projection of a point onto a set so that the proposed approach is composed of two projection stages: the rotor angle control (RAC) stage and the transient voltage control (TVC) stage [22], [23].

The approach reported in [

19] has been adopted and reformulated to express the projection operation as an active power re-dispatch problem. The unique features and contributions of the proposed approach are the following.

1) The prevention of RAC stability and the TVS is performed by the proposed approach. The goal is achieved by introducing the concept of projection onto sets. The application of projections steers the operating state of the system towards an operating point where the transient trajectories of rotor angles and voltage magnitudes are bounded within pre-established limits.

2) The projection operation is formulated as a transiently constrained active power re-dispatch problem, so the projection corresponds to the solution of a slightly extended conventional OPF model, referred to as the TSC-OPF model, which has a dimension, complexity, and computational burden similar to that of a traditional OPF model.

3) The active power re-dispatch is generated non-heuristically by minimizing the transient excursions of rotor angles and voltage magnitudes to the maximum rate of change with respect to the specified reference values. The non-heuristic active power re-dispatch also avoids the system overstabilization because the operating equilibrium point sought is projected in close proximity to the boundary of the ATVS region, as shown by the numerical results.

The rest of this paper is organized as follows. Section II provides the fundamentals of the proposed approach. Section III provides the formulation of the projection operation for performing the active power re-dispatch of generators. Section IV shows the proposed TSC-OPF approach, while case studies are presented in Section V. Lastly, Section VI reports the conclusions of this paper.

II. Fundamentals of Proposed Approach

The proposed approach consists of sequentially solving the transient stability and TSC-OPF problems until obtaining an equilibrium point that supports a specific disturbance, while maintaining the dynamics of rotor angles and transient voltage magnitudes is bounded within acceptable values according to their corresponding transient stability and TVS indices. This results in a dynamic system response, where the generators remain in synchronism without low transient voltage magnitudes, which causes the system to reach a secure steady-state equilibrium point.

In the proposed approach, the results obtained from the transient stability simulation when the system transient response is insecure because of the loss of transient stability or unbounded transient voltage magnitudes, provide the information needed to assemble the TSC-OPF model. This model is then solved to non-heuristically assess the optimal generation re-dispatch that steers the system to a transiently stable equilibrium point. In the sequential solution process, the equilibrium point obtained from the TSC-OPF model is provided as an initial operation condition for performing the transient simulation that allows determining if this equilibrium point is transiently stable.

A. Transient Stability and Trajectory Sensitivity Analysis

From the preventive security perspective, the power system at a given equilibrium point $O P_{β}$ and subjected to a specified contingency scenario is declared transiently secure in terms of dynamics of rotor angles and voltage magnitudes if the following two criteria are simultaneously satisfied in a transient stability study [

1]:

η_{R A C}

and

η_{T V C}

Stability criterion of RAC $η_{R A C}$ states that the system synchronism is maintained when subjected to a severe disturbance if the transient trajectories of rotor angles do not surpass $δ_{m a x}$ with regard to $δ_{C O I} (t)$ during T, as given by (1) [

12]. In this case,

δ_{C O I}

is given by (2) [19].

|δ_{i} (t) - δ_{C O I} (t)| \leq δ_{m a x} \forall t \in T, i = 1,2, \dots, n_{g}

(1)

δ_{C O I} (t) = \frac{\sum_{i = 1}^{n_{g}} H_{i} δ_{i} (t)}{H_{T}} = \frac{\sum_{i = 1}^{n_{g}} H_{i} δ_{i} (t)}{\sum_{i = 1}^{n_{g}} H_{i}}

(2)

TVS in voltage magnitudes is associated with the rotor angle displacements occurring during a large disturbance [

1]. Hence, to ensure a transiently secure response of the power system, the transient evolution of voltage magnitudes in all buses must be bounded within a pre-established limit [1]. In this paper,

η_{T V C}

states that voltage magnitudes

V (t)

are within secure bounds if their minimum values are greater than V_min during T [7], as given by (3).

V_{k} (t) > V_{m i n} \forall t \in T, k = 1,2, \dots, n_{b}

(3)

where $V_{k} (t)$ is the element of $V (t)$ .

To assess the transient response of the electric power system operating at $O P_{β}$ , the transient trajectories and their sensitivities with regard to a control variable are obtained by combining a TD simulation and the staggered direct method (TD-SDM) [

24]. TD-SDM analysis determines if

O P_{β}

satisfies (1) and (3) for a specified disturbance, and it also provides the information of evaluating the dynamic sensitivities of rotor angles and voltage magnitudes with regard to the active power of each generator, i.e.,

\partial δ_{i} (t) / \partial P_{g_{i}}

and

\partial V_{k} (t) / \partial P_{g_{i}}

, respectively. These sensitivities are used to formulate the non-heuristic active power re-dispatch in the TSC-OPF model.

In TD simulation, the power system dynamics are formulated by the set of differential-algebraic equations (DAEs).

\{\begin{array}{l} \frac{d x (t)}{d t} = f (x (t), y (t), u, τ) \\ f (\cdot) : R^{n_{x} + n_{y} + n_{u} + n_{τ}} \to R^{n_{x}} x \in X \subset R^{n_{x}}, y \in Y \subset R^{n_{y}}, u \in U \subset R^{n_{u}} \\ 0 = g (x (t), y (t), u, τ) \\ g (\cdot) : R^{n_{x} + n_{y} + n_{u} + n_{τ}} \to R^{n_{y}} τ \in Γ \subset R^{n_{τ}} \end{array}

(4)

In (4), the set of differential equations associated with the generators and their control units is denoted by the diffenertial functions $f (\cdot)$ , while the stator algebraic equations and power flow mismatch equations are represented by the functions $g (\cdot)$ .

The formulations of how sensitive the trajectories of state variables are with regard to the changes in the active power produced by the i^th generator are given by (5) [

24]. The derivation of this set of equations is detailed in [25].

\{\begin{array}{l} \frac{d x_{P_{g_{i}}}}{d t} = \frac{\partial f (\cdot)}{\partial x} \frac{\partial x}{\partial P_{g_{i}}} + \frac{\partial f (\cdot)}{\partial y} \frac{\partial y}{\partial P_{g_{i}}} + \frac{\partial f (\cdot)}{\partial P_{g_{i}}} \\ x_{P_{g_{i}}} (t_{c l}) = 0 \\ \frac{\partial g (\cdot)}{\partial x} \frac{\partial x}{\partial P_{g_{i}}} + \frac{\partial g (\cdot)}{\partial y} \frac{\partial y}{\partial P_{g_{i}}} + \frac{\partial g (\cdot)}{\partial P_{g_{i}}} = 0 \\ g_{P_{g_{i}}} (t_{c l}) = 0 \end{array} i = 1,2, \dots, n_{g}

(5)

where $x_{P_{g_{i}}} (\cdot) = \partial x (\cdot) / \partial P_{g_{i}}$ and $y_{P_{g_{i}}} (\cdot) = \partial y (\cdot) / \partial P_{g_{i}}$ are the sensitivities of dynamic and algebraic states with regard to the changes of the active power output of the $i^{t h}$ generator, respectively; and $d x_{P_{g_{i}}} / d t$ is a vector representing the dynamic evolution of sensitivities in time.

Under a given contingency scenario and a given equilibrium $O P_{β}$ associated with either the RAC stage or TVC stage, the TD-SDM analysis conducts a step-by-step integration process, which solves (4) to obtain the time evolution of x(t) and y(t). The solution is performed during $T = [t_{0}^{+}, t_{c l}] ⋃ (t_{c l}, t_{e n d}]$ . At each time step $t \in (t_{c l}, t_{e n d}]$ of the integration process, the trajectory sensitivities are also calculated from (5), as detailed in [

25], and

η_{R A C}

η_{T V C}

is checked as appropriate: (1) in the RAC stage or (3) in the TVC stage. If

η_{α}

is satisfied during T, the TD-SDM analysis ends at the time step

t = t_{e n d}

, and

O P_{β}

is declared transiently stable. If not,

O P_{β}

is declared transiently unstable at the first time step

t \in (t_{c l}, t_{e n d}]

in which

η_{α}

is not satisfied. In this case, the time step is considered as

t_{u}

, and the TD-SDM analysis stops to avoid further integration time steps and reduce the computational burden. Under this unstable ending condition, the TD-SDM results below are used to perform the optimal active power re-dispatch through the TSC-OPF model.

1) In the $R A C$ stage: ① $t = t_{u}$ ; ② the values of the rotor angles of each $i^{t h}$ generator at t_u, $δ_{i} (t_{u}) \in x$ ; and ③ the sensitivities $\partial δ_{i} (t) / \partial P_{g_{i}} |_{t_{u}} \in x_{P_{g_{i}}}$ , $i = 1,2, \dots, n_{g}$ .

2) In the TVC stage: ① $t = t_{u}$ ; ② the value of nodal voltage magnitudes at $t_{u}$ , $V (t_{u}) \in y$ ; and ③ the sensitivities $\partial V_{k} (t) / \partial P_{g_{i}} |_{t_{u}} \in y_{P_{g_{i}}}$ , $k = 1,2, \dots, n_{b}$ , $i = 1,2, \dots, n_{g}$ .

B. Projection-based Optimal Power Re-dispatch

When the power system is operating at $O P_{U}$ in a given contingency scenario, the preventive control performed by the proposed approach strives to assess a steady-state $O P_{T V C}$ , where $η_{R A C}$ and $η_{T V C}$ are simultaneously satisfied. Therefore, this approach considers the existence of two feasible subsets $S_{R A C}$ and $S_{T V C}$ composed of operating points in the parameter set $S_{F}$ , where criteria (1) and (3) are satisfied, respectively, when performing the TD-SDM analysis. Reference [

7] shows that an operating point satisfying

η_{T V C}

also satisfies

η_{R A C}

, which means that

S_{T V C}

associated with admissible TVS can be considered as a subset of

S_{R A C}

S_{T V C} \subseteq S_{R A C}

. Hence, the operating point OP_TVC sought must lie in the intersection of

S_{T V C}

and

S_{R A C}

O P_{T V C} \in

{S_{T V C} ⋂ S_{R A C}}

. The general approach of projecting

O P_{U}

onto S_TVC to obtain

O P_{T V C}

is explained below.

In a general context, the exact projection of $O P_{α}$ onto a subset $S_{α}$ assesses $O P_{h}$ on the hull of $S_{F}$ closest to $O P_{α}$ [

22]. This projection is performed at the maximum rate of change through a projection operation

P_{C} (\cdot)

, i.e.,

P_{C} (\cdot) : O P_{α} \to O P_{h}

, by using the concepts of directional derivatives and gradients [22]. In the case of multiple sets, an alternating projection method that repeatedly executes exact projection operations is applied to obtain an operating point in the hull of the region defined by the intersection of multiple sets [22], [23]. Since S_TVC is a subset of S_RAC [7], the general alternating projection method is directly simplified as a two-stage projection method. A general description of the projection method is shown in Fig. 1. Without loss of generality, Fig. 1 shows this projection method based on the parameter space S_F of the active power output of two generators. The operating points

O P_{U}

and

O P_{R A C}

are projected onto

O P_{R A C}

and

O P_{T V C}

, respectively, by performing an optimal re-dispatch of the active power of generators, which corresponds to

P_{C} (\cdot)

Fig. 1 General description of projection method.

As the stated above, the proposed approach is formulated in two general sequential projection stages: the RAC stage and TVC stage. The RAC stage first projects OP_U onto the hull of the subset $S_{R A C}$ to obtain the OP_RAC, where $η_{R A C}$ is satisfied. OP_RAC is then projected onto the hull of the subset S_TVC in the TVC stage, which obtains the transiently stable OP_TVC sought. These two general stages are expressed in compact form by (6).

O P_{α} = P_{C α} (O P_{j})

(6)

where $O P_{j} = O P_{U}$ when $α = R A C$ , and $O P_{j} = O P_{R A C}$ when $α = T V C$ .

The projection operation $P_{C α} (\cdot)$ is formulated as a TSC-OPF problem based on a non-heuristic generation dispatch, as detailed in Section III.

III. Formulation of Projection Operators for Non-heuristic Generation Re-dispatch

In the $α$ stage, the exact projection $P_{C α} (\cdot)$ given by (6) cannot be directly performed since the subsets $S_{R A C}$ and $S_{T V C}$ are not known in advance. Hence, the general projection $P_{C α} (\cdot)$ is achieved by executing the two correlated projection sequences, which are referred to as the over-relaxed sequence of $α$ ( $O$ - $S E Q_{α}$ ) and the under-relaxed sequence of $α$ ( $U$ - $S E Q_{α}$ ), respectively. The $O$ - $S E Q_{α}$ and $U$ - $S E Q_{α}$ sequences are shown in Fig. 2.

Fig. 2 O-SEQ_α and U-SEQ_α.

The $O$ - $S E Q_{α}$ sequence recursively executes $P_{C α}^{r O} (\cdot)$ to gradually displace the starting point OP_j until obtaining a first point $O P_{j + (n + 1)}$ inside $S_{α}$ , as denoted by (7), where $η_{α}$ is satisfied.

O P_{β + 1} = P_{C α}^{r O} (O P_{β}) β = j, j + 1, \dots, j + n

(7)

Since the operating point obtained in the $O$ - $S E Q_{α}$ sequence is not generally on the hall of $S_{α}$ , which means that the system is over-stabilized, one must project this point onto that hall of $S_{α}$ through the $U$ - $S E Q_{α}$ sequence. This goal is achieved by considering the last two operating points of the $O$ - $S E Q_{α}$ sequence that define an interval with the last unstable operating point and the first stable operating point, i.e., $O P_{s} = O P_{j + n}$ and $O P_{i n, α} = O P_{j + (n + 1)}$ , respectively, which bracket a critically stable $O P_{α}$ on the hull of $S_{α}$ . This interval is recurrently bisected by using projection operation $P_{C α}^{r U} (\cdot)$ in the $U$ - $S E Q_{α}$ sequence, as indicated by (8), until obtaining a point $O P_{(s + m) + 1}$ on the hull of $S_{α}$ that corresponds to the sought $O P_{α}$ , where $η_{α}$ is satisfied.

O P_{γ + 1} = P_{C α}^{r U} (O P_{γ}) γ = s, s + 1, \dots, s + m

(8)

The projection operations $P_{C α}^{r O} (\cdot)$ and $P_{C α}^{r U} (\cdot)$ involved in (7) and (8), respectively, are formulated in the following subsections as an active power re-dispatch problem.

A. Projection Operation $P_{C α}^{r O} (\cdot)$ in O-SEQ_α Sequence

When the $O$ - $S E Q_{α}$ sequence performed in the $α$ stage, a new point $O P_{β + 1}$ is obtained from a current point $O P_{β}$ through $Δ P_{g, β}^{α} \in R^{n_{g}}$ , where $Δ P_{g, β}^{α}$ represents the difference between the active power output of generators at $O P_{β}$ and $O P_{β + 1}$ : $Δ P_{g, β}^{α} = P_{g, β + 1}^{α} - P_{g, β}^{α}$ , $P_{g, β}^{α}, P_{g, β + 1}^{α} \in R^{n_{g}}$ . In the re-dispatch, some generators will decrease their active power output, and others will increase their generation level to satisfy the nodal balance of active power at the new point $O P_{β + 1}$ [

19].

The active power re-dispatch $Δ P_{g, β}^{α}$ , which corresponds to the projection operation $P_{C α}^{r O} (\cdot)$ , is represented by its magnitude $Δ P_{g, β}^{α} = | | Δ P_{g, β}^{α} | |$ and a unit vector $Δ {\hat{P}}_{g, β}^{α}$ in the direction of $Δ P_{g, β}^{α}$ , where $Δ {\hat{P}}_{g, β}^{α} = Δ P_{g, β}^{α} / | | Δ P_{g, β}^{α} | |$ [

19]. In this case, the magnitude

Δ P_{g, β}^{α}

corresponds to the Euclidian norm or distance between

O P_{β}

and

O P_{β + 1}

in the parametric space of active power generation. Furthermore, the element values in the unit vector

Δ {\hat{P}}_{g, β}^{α}

indicate how that magnitude is distributed among the re-dispatchable generators to satisfy

Δ P_{g, β}^{α} = | | Δ P_{g, β}^{α} | | (Δ P_{g, β}^{α} / | | Δ P_{g, β}^{α} | |)

[19]. The best active power re-dispatch corresponds to the one performed in the scheduled direction

Δ {\hat{P}}_{g, β}^{α}

and the scheduled magnitude

Δ P_{g, β}^{S c h_{α}}

, in which

η_{R A C}

and

η_{T V C}

are maximally improved. This theoretical condition is mathematically formulated by making the direction and magnitude of

Δ P_{g, β}^{α}

Δ {\hat{P}}_{g, β}^{S c h_{α}}

and

Δ P_{g, β}^{S c h_{α}}

, respectively. These two references are derived, evaluated, and included in the TSC-OPF model as described below, which allows performing a projection operation

P_{C α}^{r O} (\cdot)

. The projection operation in

O - S E Q_{α}

is shown in Fig. 3.

Fig. 3 Projection operation for O-SEQ_α.

1) Formulation of Scheduled Direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$

The criterion $η_{α}$ is best improved through the active power re-dispatch when the formulation of the scheduled direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ is based on the gradient of the performance index $φ_{β t_{u}}^{α}$ at $t_{u}$ .

In the RAC stage, $φ_{β t_{u}}^{α}$ is referred to as the transient stability index $φ_{β t_{u}}^{R A C}$ .

Note that $φ_{β t_{u}}^{R A C}$ quantifies the level of coherence of the transient trajectories of rotor angles $δ (t) \in R^{n_{g}}$ at t_u, where $η_{R A C}$ given by (1) is not satisfied.

φ_{β t_{u}}^{R A C} = \sum_{i = 1}^{n_{g}} (δ_{i} (t_{u}) - δ_{C O I} (t_{u} {))}^{2}

(9)

The performance index $φ_{β t_{u}}^{α}$ given by (10) corresponds to the TVC stage. In this case, $φ_{β t_{u}}^{T V C}$ is referred to as the ATVS index and quantifies the deviation level for the trajectories of nodal transient voltages $V (t) \in R^{n_{b}}$ at $t_{u}$ with regard to a value of 1 p.u., where $η_{T V C}$ given by (3) is not satisfied .

φ_{β t_{u}}^{T V C} = \sum_{i = 1}^{n_{b}} (V_{i} (t_{u} {) - 1)}^{2}

(10)

Based on the above, the most significant improvement in the transient evolution of the system is achieved when the active power re-dispatch at the current point $O P_{β}$ is performed in the direction that reduces the value of performance index $φ_{β t_{u}}^{α}$ at the maximum rate of change. Hence, the scheduled direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ is mathematically defined by the unitary vector given in [

19].

Δ {\hat{P}}_{g, β}^{S c h_{α}} = - \frac{\nabla_{P_{g, β}} φ_{β t_{u}}^{α}}{| | \nabla_{P_{g, β}} φ_{β t_{u}}^{α} | |}

(11)

Since $φ_{β t_{u}}^{α}$ is not explicitly expressed in terms of the active power output of generators, as clearly shown in (9) and (10), $\nabla_{P_{g, β}} φ_{β t_{u}}^{α}$ is attained by using the chain rule given by (12), where the settings of elements $ε_{k} (t)$ , $φ_{β t_{u}}^{α}$ , and u_b depend on the control stage that are being performed.

\nabla_{P_{g, β}} φ_{β t_{u}}^{α} = {(\frac{\partial φ_{β t_{u}}^{α}}{\partial ε_{k} (t)} \frac{\partial ε_{k} (t)}{\partial P_{g_{i}}})|}_{t_{u}} i = 1,2, \dots, n_{g}

(12)

In the $R A C$ stage, the settings are given by $ε_{k} (t) = δ_{k} (t)$ , $φ_{β t_{u}}^{α} = φ_{β t_{u}}^{R A C}$ , and $u_{b} = n_{g}$ , which results in:

\nabla_{P_{g, β}} φ_{β t_{u}}^{R A C} = \sum_{k = 1}^{n_{g}} {(\frac{\partial φ_{β t_{u}}^{R A C}}{\partial δ_{k} (t)} \frac{\partial δ_{k} (t)}{\partial P_{g_{i}}})|}_{t_{u}} i = 1,2, \dots, n_{g}

(13)

In this case, the first partial derivative corresponds to (14), where $B = 1$ for $i = k$ and $B = 0$ for $\forall i \neq k$ , and it is analytically obtained from (9).

{\frac{\partial φ_{β t_{u}}^{R A C}}{\partial δ_{i} (t)}|}_{t_{u}} = \sum_{i = 1}^{n_{g}} 2 (δ_{i} (t_{u}) - δ_{C O I} (t_{u})) (B - \frac{H_{i}}{H_{T}}) i = 1,2, \dots, n_{g}

(14)

The time evolution of rotor angles $δ (t)$ and their partial derivatives $\partial δ_{i} (t) / \partial P_{g_{i}}$ involved in (14) and (13), respectively, are numerically obtained from the TD simulation and dynamic sensitivity analysis, as explained in Section II-A. In this case, the TD-SDM results used to evaluate (13) and (14) are as follows: ① $t_{u}$ at which the criterion (1) is not satisfied; ② the values of rotor angles of the i^th generator at t_u $δ_{i} (t_{u}) \in x (t)$ ; and ③ the sensitivities $\partial δ_{i} (t) / \partial P_{g_{i}} |_{t_{u}} \in x_{P_{g_{i}}}$ , $i = 1,2, \dots, n_{g}$ .

Similarly, the gradient $\nabla_{P_{g, β}} φ_{β t_{u}}^{α}$ in the $T V C$ stage is directly formulated by setting $ε_{k} (t) = V_{k} (t)$ , $φ_{β t_{u}}^{α} = φ_{β t_{u}}^{T V C}$ , and $u_{b} = n_{b}$ , which results in (15). The partial derivative $\partial φ_{β t_{u}}^{T V C} / \partial V_{k} (t)$ is given by (16), while the dynamics of nodal voltage magnitudes $V (t) \in R^{n_{b}}$ and their sensitivities with regard to the active power generation are obtained from the TD-SDM analysis. Hence, the evaluation of (15) and (16) is based on the following TD-SDM results: ① $t_{u}$ at which the criterion (3) is not satisfied; ② the value of nodal voltage magnitudes at t_u, $V (t_{u}) \in y$ ; and ③ the sensitivities $\partial V_{k} (t) / \partial P_{g_{i}} |_{t_{u}} \in y_{P_{g_{i}}}$ , $i = 1,2, \dots, n_{g}$ , $k = 1,2, . . ., n_{b}$ .

\nabla_{P_{g, β}} φ_{β t_{u}}^{T V C} = {(\frac{\partial φ_{β t_{u}}^{T V C}}{\partial V_{k} (t)} \frac{\partial V_{k} (t)}{\partial P_{g_{i}}})|}_{t_{u}} i = 1,2, \dots, n_{g}

(15)

{\frac{\partial φ_{β t_{u}}^{T V C}}{\partial V_{k} (t)}|}_{t_{u}} = 2 (V_{k} (t_{u}) - 1) k = 1,2, \dots, n_{b}

(16)

2) Formulation of Scheduled Magnitude $Δ P_{g, β}^{S c h_{α}}$

The value of $Δ P_{g, β}^{S c h_{α}}$ can be obtained from (17).

Δ P_{g, β}^{S c h_{α}} = λ Δ P_{g, β, m a x}^{α} = λ | | Δ P_{g, β, m a x}^{α} | |

(17)

From a mathematical viewpoint, the active power re-dispatch $Δ P_{g, β, m a x}^{α}$ is obtained from the solution to the constrained optimial problem (18). In this model, the objective function maximizes the dot product representing the scalar projection of $Δ P_{g, β, m a x}^{α}$ onto the scheduled direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ , subject to satisfying the lossless active power balance and the limits of active power generation. In this proposed formulation, $Δ P_{g, β, m a x}^{α} = P_{g, β, m a x}^{α} - P_{g, β}^{α}$ such that the vector element $P_{g_{i}, β, m a x}^{α} \in P_{g, β, m a x}^{α}$ is the active power output of the $i^{t h}$ generator with lower and upper active power limits $P_{g_{i}}^{L}$ and $P_{g_{i}}^{U}$ , respectively.

\{\begin{array}{l} \underset{P_{g}}{m i n} f (\cdot) = - Δ P_{g, β, m a x}^{α} \cdot Δ {\hat{P}}_{g, β}^{S c h_{α}} \\ s . t . \sum_{i = 1}^{n_{g}} P_{g_{i}, β, m a x}^{α} - \sum_{i = 1}^{n_{l}} P_{l_{i}} = 0 \\ P_{g_{i}}^{L} \leq P_{g_{i}, β, m a x}^{α} \leq P_{g_{i}}^{U} i = 1,2, \dots, n_{g} \end{array}

(18)

Lastly, to avoid generators operating close to one of their limits and the computation of a transiently stable operating point far from the region boundary, which both can result from the projection operation $P_{C α}^{r O} (\cdot)$ , $λ$ is set at a small value as 5%.

3) Formulation of $P_{C α}^{r O} (\cdot)$ as TSC-OPF Problem

To project $O P_{β}$ onto the feasible subset $S_{α}$ , the conventional OPF model is slightly extended to force the re-dispatch $Δ P_{g, β}^{α}$ to be performed with a magnitude $Δ P_{g, β}^{S c h_{α}}$ and direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ in the parametric space of generation. Hence, the resulting TSC-OPF model, which corresponds to the projection operation $P_{C α}^{r O} (\cdot)$ described in (7), is given as follows.

\{\begin{array}{l} \underset{P_{g, β + 1}, V_{β + 1}, θ_{β + 1}}{m i n} f (\cdot) = f_{E} (P_{g, β + 1}) - \frac{Δ P_{g, β}^{α}}{| | Δ P_{g, β}^{α} | |} Δ {\hat{P}}_{g, β}^{S c h_{α}} \\ s . t . G (V_{β + 1}, θ_{β + 1}, P_{g, β + 1}) = 0 \\ H (V_{β + 1}, θ_{β + 1}, P_{g, β + 1}) \leq 0 \\ | | Δ P_{g, β}^{α} | |^{2} - (Δ P_{g, β}^{S c h_{α}})^{2} = 0 \end{array}

(19)

where $Δ P_{g, β}^{α} = P_{g, β + 1}^{α} - P_{g, β}^{α}$ .

Furthermore, the second term in the objective function forces the active power re-dispatch to be as close as possible to the scheduled direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ , in which the system transient response is improved at the maximum rate of change. The last equality constraint assures that the Euclidian norm of the total amount of active power re-dispatched equals $Δ P_{g, β}^{S c h_{α}}$ .

B. Projection Operation $P_{C α}^{r U} (\cdot)$ in $U$ - $S E Q_{α}$

Once the point $O P_{i n, α} = O P_{j + (n + 1)}$ inside $S_{α}$ has been obtained, it is projected onto the hull of this feasible region through the projection operation $P_{C α}^{r U} (\cdot)$ . This projection is performed in $U$ - $S E Q_{α}$ of the $α$ based on the point $O P_{γ}$ and $O P_{i n, α}$ that bracket point $O P_{α}$ on the hull of $S_{α}$ : $T_{γ} = [O P_{γ}, O P_{i n, α}]$ such that $O P_{α} \in T_{γ}$ . Note that these operating points are known from the $O$ - $S E Q_{α}$ sequence. Moreover, the direction and magnitude of the active power re-dispatch that take the power system from the point $O P_{γ}$ to $O P_{i n, α}$ are also known from (7) of the $O$ - $S E Q_{α}$ sequence, and they are denoted by $Δ {\hat{P}}_{g, γ}^{S c h_{α}} = Δ {\hat{P}}_{g, j + n}^{S c h_{α}}$ and $Δ P_{g, γ}^{S c h_{α}} = Δ P_{g, j + n}^{S c h_{α}}$ , respectively.

Based on the information mentioned above, $P_{C α}^{r U} (\cdot)$ determines the magnitude and the direction in which the active power re-dispatch must be performed from the current operating state $O P_{γ}$ to obtain the new $O P_{γ + 1}$ . The flow chart of projection operation for $U$ - $S E Q_{α}$ is shown in Fig. 4.

Fig. 4 Flow chart of projection operation in U-SEQ_α.

The point $O P_{γ + 1}$ is located in the middle of the interval $T_{γ}$ , reducing the search interval for the subsequent execution of $P_{C α}^{r U} (\cdot)$ . Thus, similar to the projection operation $P_{C α}^{r O} (\cdot)$ explained in Section III-A, the scheduled magnitude $Δ P_{g, γ + 1}^{S c h_{α}}$ , the scheduled direction $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}}$ , and the formulation of the projection $P_{C α}^{r U} (\cdot)$ in $U$ - $S E Q_{α}$ are described as below.

1) Formulation of Scheduled Magnitude $Δ P_{g, γ + 1}^{S c h_{α}}$

The formulation and evaluation of the scheduled magnitude $Δ P_{g, γ + 1}^{S c h_{α}}$ are achieved by halving the known magnitude $Δ P_{g, γ}^{S c h_{α}}$ .

Δ P_{g, γ + 1}^{S c h_{α}} = \frac{Δ P_{g, γ}^{S c h_{α}}}{2}

(20)

2) Formulation of Scheduled Direction $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}}$

For $P_{C α}^{r U} (\cdot)$ , $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}}$ is fixed to the one that performs the last projection operation $P_{C α}^{r O} (\cdot)$ in $U$ - $S E Q_{α}$ : $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}} = Δ {\hat{P}}_{g, γ}^{S c h_{α}}$ .

3) Formulation of $P_{C α}^{r U} (\cdot)$ as TSC-OPF Problem

Based on $Δ P_{g, γ + 1}^{S c h_{α}}$ and $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}}$ , the TSC-OPF model (19) is assembled and solved to obtain $O P_{γ + 1}$ . Thus, $β$ in (19) must be replaced by $γ$ , which results in (21), where $Δ P_{g, γ}^{α} = P_{g, γ + 1}^{α} - P_{g, γ}^{α}$ .

\{\begin{array}{l} \underset{P_{g, γ + 1}, V_{γ + 1}, θ_{γ + 1}}{m i n} f (\cdot) = f_{E} (P_{g, γ + 1}) - \frac{Δ P_{g, γ}^{α}}{| | Δ P_{g, γ}^{α} | |} Δ {\hat{P}}_{g, γ}^{S c h_{α}} \\ s . t . G (V_{γ + 1}, θ_{γ + 1}, P_{g, γ + 1}) = 0 \\ H (V_{γ + 1}, θ_{γ + 1}, P_{g, γ + 1}) \leq 0 \\ | | Δ P_{g, γ}^{α} | |^{2} - (Δ P_{g, γ}^{S c h_{α}})^{2} = 0 \end{array}

(21)

4) Adjustment of Interval $T_{γ + 1}$

$η_{α}$ is tested through the TD-SDM analysis applied to $O P_{γ + 1}$ . If $η_{α}$ is not satisfied, $O P_{α}$ is inside the interval $T_{γ + 1}$ defined by $O P_{γ + 1}$ and $O P_{i n, α}$ such that $T_{γ + 1} = [O P_{γ + 1}, O P_{i n, α}]$ . If $η_{α}$ is satisfied, $O P_{α}$ is inside the interval defined by the previous point $O P_{γ}$ and the new point $O P_{γ + 1}$ . Hence, $T_{γ + 1} = [O P_{γ}, O P_{γ + 1}]$ .

Lastly, the TD-SDM analysis is used to assess the transient evolution of the system at $O P_{γ + 1}$ , which does not require the sensitivity assessment for performing $P_{C α}^{r U} (\cdot)$ . This is because the scheduled direction remains fixed in $U$ - $S E Q_{α}$ . Hence, the TD-SDM must only integrate the set of (4) to verify $η_{α}$ and determine if $O P_{γ + 1}$ is transiently stable.

C. Comparison with Other Models

The size and complexity comparison of TSC-OPF model is shown in Table I, which reveals the theoretical merits of the proposal and advances in reducing the size and complexity of the optimal problem regarding other models that perform system stability based on transient stability and TVS criteria. In this case, the conventional OPF problem is extended with only two constraints in the TSC-OPF problem, so both formulations have similar dimensions. Furthermore, the dimension of the TSC-OPF problem remains similar to the conventional OPF problem regardless of the size of the power system to be studied. In this paper, unlike the models reported in [

7], [11], and [12], those two constraints do not depend on n_b, n_g, and N_s. Hence, the proposed approach avoids including the discrete-time equations of the multi-machine system in the OPF problem. The TD simulations required to compute the transient stability and TVS constraints are performed for a short study time period given by

T = [t_{0}^{+}, t_{u}] :

t_{e n d} = t_{u}

. Note that these simulations are executed outside the TSC-OPF problem and only once at each iteration of the stabilization process. This is not the case for the proposals reported in [11] and [12], where the TD simulations must be executed at each iteration of the optimization process to obtain their corresponding transient stability and TVS constraints.

TABLE II Results of Stabilization Process for WSCC System

α	Sequence	Point	$P_{g_{1}}$ (MW)	$P_{g_{2}}$ (MW)	$P_{g_{3}}$ (MW)	Cost ($/hour)
α	Base point	OP_U	105.94	113.04	99.24	1132.2
RAC	O-SEQ_RAC	OP_in_,RAC	121.17	101.91	94.83	1134.8
RAC	U-SEQ_RAC	OP_RAC	118.24	104.36	95.35	1133.9
TVC	O-SEQ_TVC	OP_in_,TVC	160.33	82.90	74.36	1163.4
TVC	U-SEQ_TVC	OP_TVC	160.12	83.05	74.41	1163.1

Model	t_end	N_s	Number of transient stability constraints^*	Number of dynamic constraints	Number of TVS constraints	Heuristic stability criterion
SD [7]	Arbitrary	$N_{s} = (t_{e n d} - t_{c l}^{+}) / Δ t$ arbitrary selected	N_s	(2n_b+2n_g)N_s	n_bN_s	No
SS [12]	Not required	0	1	0	n_b	No
MS [11]	Arbitrary	N_s arbitrary selected	n_gN_s	(2n_b+2n_g)N_s	n_bN_s	No
Proposed	Not required	0	2	0	2	No

Note： ^* means considering classical generator model.

IV. TSC-OPF Approach for Preventive Control

The proposed approach is formulated by expressing the RAC and TVC stages in terms of $O$ - $S E Q_{α}$ and $U$ - $S E Q_{α}$ given by (7) and (8), respectively.

Considering $O P_{j} = O P_{U}$ as the starting point, (7) and (8) are performed once to achieve the RAC stage, which obtains the point $O P_{α} = O P_{R A C}$ . OP_RAC is then considered as the starting point $O P_{j}$ in the TVC stage to obtain the point $O P_{α} = O P_{T V C}$ through a new application of $O$ - $S E Q_{α}$ and $U$ - $S E Q_{α}$ .

The step-by-step procedure of the proposed approach for solving the TSC-OPF problem is given as follows.

Step 1: for $α = R A C$ or $α = T V C$ , do the following.

Step 2: set the starting point OP_j in $O$ - $S E Q_{α}$ as $O P_{j} = O P_{U}$ when $α = R A C$ . Conversely, set $O P_{j} = O P_{R A C}$ when $α = T V C$ .

Step 3: perform $O$ - $S E Q_{α}$ . In this case, the projection operation $O P_{β + 1} = P_{C α}^{r O} (O P_{β})$ must be conducted for $β = j$ , $j + 1$ , $\dots$ , $j + n$ , describing as follows.

Step 3-1: execute the TD-SDM analysis for the specified contingency scenario to verify compliance with $η_{α}$ for point $O P_{β}$ . If $η_{α}$ is satisfied, go to Step 3-6. Otherwise, the TD-SDM analysis provides the following results: ① the values of t_u; ② $δ_{k} (t_{u})$ and $\partial δ_{k} (t) / \partial P_{g_{i}} |_{t_{u}}$ , $k = 1,2, . . ., n_{b}$ , $i = 1,2, \dots, n_{g}$ when $α = R A C$ or $t = t_{u}$ ; and ③ $V_{k} (t_{u})$ and $\partial V_{k} (t) / \partial P_{g_{i}} |_{t_{u}}$ , k=1,2, $\dots$ , $n_{b}$ , $i = 1,2, \dots, n_{g}$ when $α = T V C$ .

Step 3-2: evaluate the gradient $\nabla_{P_{g, β}} φ_{β t_{u}}^{α}$ from (13) when $α = R A C$ or from (15) when $α = T V C$ .

Step 3-3: use $\nabla_{P_{g, β}} φ_{β t_{u}}^{α}$ to assess the scheduled direction $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ from (11), which in turn is used to formulate (18). The solution of (18) obtains $Δ P_{g, β, m a x}^{α}$ . Lastly, evaluate (17) to obtain the scheduled magnitude $Δ P_{g, β}^{S c h_{α}}$ .

Step 3-4: based on $Δ {\hat{P}}_{g, β}^{S c h_{α}}$ , $Δ P_{g, β}^{S c h_{α}}$ , and $O P_{β}$ , the TSC-OPF model (19) is formulated and solved to obtain the new point $O P_{β + 1}$ .

Step 3-5: let $β = β + 1$ , and go to Step 3-1.

Step 3-6: the $O$ $- S E Q_{α}$ sequence ends, and point $O P_{j + (n + 1)}$ corresponds to the first point $O P_{i n, α}$ , which is inside the subset $S_{α}$ . Additional results are $O P_{γ} = O P_{j + n}$ , the interval $T_{γ} = [O P_{γ}, O P_{i n, α}]$ , the scheduled direction $Δ {\hat{P}}_{g, γ}^{S c h_{α}} = Δ {\hat{P}}_{g, j + n}^{S c h_{α}}$ , and the scheduled magnitude $Δ P_{g, γ}^{S c h_{α}} = Δ P_{g, j + n}^{S c h_{α}}$ .

Step 4: perform the $U$ - $S E Q_{α}$ sequence in the $α$ stage. In this case, the projection operation $O P_{γ + 1} = P_{C α}^{r U} (O P_{γ})$ must be performed as $γ = s, s + 1, \dots, s + m$ , as described as follows.

Step 4-1: evaluate $Δ P_{g, γ + 1}^{S c h_{α}}$ according to (20), and set the direction $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}}$ as $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}} = Δ {\hat{P}}_{g, γ}^{S c h_{α}}$ .

Step 4-2: use $Δ {\hat{P}}_{g, γ + 1}^{S c h_{α}}$ , $Δ P_{g, γ + 1}^{S c h_{α}}$ , and $O P_{γ}$ to formulate and solve the TSC-OPF problem (21). The solution corresponds to the $O P_{γ + 1}$ .

Step 4-3: execute the TD-SDM analysis to test $η_{α}$ at $O P_{γ + 1}$ and to obtain the new interval $T_{γ + 1}$ , as reported in Section III-B. In addition, if $η_{α}$ is not satisfied, set point $O P_{γ}$ as $O P_{γ} = O P_{γ + 1}$ and increase $γ$ as $γ = γ + 1$ ; otherwise, set point $O P_{i n, α}$ as $O P_{i n, α} = O P_{γ + 1}$ , whereas the point $O P_{γ}$ and the index $γ$ maintain their previous values.

Step 4-4: assess the length of the interval $T_{γ + 1}$ as $ε = T_{γ + 1}$ . If $ε$ is greater than a specified tolerance Tol, go back to Step 4-1. Otherwise, the $U$ - $S E Q_{α}$ sequence ends, and the current point $O P_{i n, α}$ inside the subset $S_{α}$ corresponds to the point $O P_{α}$ on the hull of the subset $S_{α}$ .

Step 4-5: if the $U$ - $S E Q_{α}$ sequence ends for $α = R A C$ , the point $O P_{α}$ is considered as the point OP_RAC. In this case, $α$ must be updated as $α = T V C$ , and the stabilization process goes back to Step 2. If the $U$ - $S E Q_{α}$ sequence ends for $α = T V C$ , the point $O P_{α}$ is set as point OP_TVC. This point is the solution to the TSC-OPF problem, and thus the proposed approach ends.

The procedure first performs the RAC stage and then executes the TVC stage.

In the stage $α$ , the $O$ - $S E Q_{α}$ and $U$ - $S E Q_{α}$ sequences are performed to obtain the point $O P_{α}$ on the hull of the subset $S_{α}$ . Therefore, the stage $α$ starts at point OP_j, as given in Step 2 and conducts the $O$ - $S E Q_{α}$ sequence to gradually move the point towards the transient stablity region by recursively executing projection operation $P_{C α}^{r O} (\cdot)$ , as indicated from Step 3-1 to Step 3-5. Hence, each projection operation $P_{C α}^{r O} (\cdot)$ obtains a new point closer to the hull of the $S_{α}$ . When the system satisfies $η_{α}$ at a given new point, as tested in Step 3-1, the $O$ - $S E Q_{α}$ sequence ends, and the new point is set as the first point $O P_{i n, α}$ inside the set $S_{α}$ . In addition, the point $O P_{i n, α}$ and the last point $O P_{γ}$ outside the subset $S_{α}$ define the upper and lower end points of the interval $T_{γ}$ that bracket point $O P_{α}$ , respectively. The further results needed to perform the $U$ - $S E Q_{α}$ sequence are given in Step 3-6.

The $U$ - $S E Q_{α}$ sequence performs a bisection having process where the projection operation $P_{C α}^{r U} (\cdot)$ is recurrently executed to assess new points inside the interval $T_{γ}$ , as performed from Step 4-1 to Step 4-3. This interval is reduced throughout the process by adjusting its endpoints with those assessed points. In addition, the system stability at those points must be tested, as indicated in Step 4-3. The $U$ - $S E Q_{α}$ sequence ends when the length of the interval is lower than a specified tolerance Tol, as verified in Step 4-4. With the success of this verification, the current point inside the stability region is set as the point $O P_{α}$ since it satisfies $η_{α}$ and is very close to the hull of the subset $S_{α}$ .

When the procedure described above is satisfied in the $R A C$ stage, the result is the point OP_RAC on the hull of the subset $S_{R A C}$ . Thus, the $T V C$ stage must be started at Step 2. When the $T V C$ stage is satisfied, the transiently stable point OP_TVC is known.

V. Case Studies

To numerically illustrate the effectiveness of the proposed approach in solving the TSC-OPF problem, the Western System Coordinated Council (WSCC) 3-machine 9-bus system [

24] and the Mexican 46-machine 190-bus equivalent system [16] are considered in this paper. The classical generator model and constant impedance loads are considered in the TD-SDM analysis, while loads are modeled as constant power for optimization studies. However, the proposed approach is entirely general, and the model used for representing a power system component is not a constraint imposed by the proposed formulation. For the TD-SDM analysis, the integration time step is 0.01 s.

η_{R A C}

and

η_{T V C}

are set as

δ_{m a x} = 120 °

and

V_{m i n} = 0.85

p.u. for the transient limits of rotor angles and TVS, respectively. Lastly, the percentage

λ

is fixed at a value of 5% for the

O

S E Q_{α}

sequence, whereas the convergence tolerance for the

U

S E Q_{α}

sequence is set as

T o l = 0.01

A. WSCC 3-machine 9-bus System

For WSCC 3-machine 9-bus system, the likely contingency scenario is given by a permanent three-phase-to-ground fault to ground incepted at $t = 0$ s at bus 7 and cleared at $t = 0.35$ s by tripping the line connecting bus 7 and bus 5. The study time period is $T = [0,1] s$ . The results of the stabilization process for WSCC system are reported in Table II. A conventional OPF analysis provides the point OP_U with active power re-dispatch given in row 2, columns 4 to 6, and the total generation cost reported in row 2, column 7, of Table II. The procedure is applied as follows.

Step 1 sets $α = R A C$ , which starts at the point OP_U, as indicated in Step 2. For the first projection operation $P_{C R A C}^{r O} (\cdot)$ of the O-SEQ_RAC sequence performed in Step 3-1, the TD-SDM analysis detects that the system operating at OP_U does not satisfy $η_{R A C}$ given by (1) at $t_{u} = 0.48$ s. In this case, OP_U is outside the $S_{R A C}$ . Projections onto hulls of the subsets $S_{R A C}$ and S_TVC are shown in Fig. 5, such that $δ_{2}$ surpasses the limit $δ_{m a x} = 120 °$ during the transient simulation. Rotor angles at the point OP_U in RAC stage are shown in Fig. 6. Accordingly, the TSC-OPF model (19) is assembled and solved in Step 3-4, with a solution given by OP_U₊₁ that remains outside $S_{R A C}$ . Based on this $O P_{U + 1}$ , and according to Step 3-5, a new projection operation $P_{C R A C}^{r O} (\cdot)$ is performed. In the third projection operation of the O-SEQ_RAC sequence, the TD-SDM analysis performed in Step 3-1 detects that the first point OP_U₊₃ inside the subset $S_{R A C}$ is obtained. According to Step 3-6, the point is set as OP_in_,RAC=OP_U+3. The active power re-dispatch of each generator and the total generation cost at opint $O P_{i n, R A C}$ are given in row 3 of Table II. The O-SEQ_RAC sequence ends according to Step 3-6.

Fig. 5 Projections onto hulls of subsets of S_RAC and S_TVC.

Fig. 6 Rotor angle at OP_U.

The $U$ - $S E Q_{R A C}$ sequence is now executed by using the projection operation $P_{C R A C}^{r U} (\cdot)$ to formulate and solve the TSC-OPF model (21) in Step 4-2. The operating point obtained is provided to Step 4-3, where the TD-SDM analysis declares that the system satisfies $η_{R A C}$ . In addition, the criterion stated in Step 4-4 is satisfied for the specified tolerance Tol. Based on these results, the operating point is declared as point $O P_{R A C}$ , thus ending both $U$ - $S E Q_{R A C}$ and RAC stage. The active power re-dispatch and total generation cost for $O P_{R A C}$ are given in row 4 of Table II. Rotor angles at OP_RAC are shown in Fig. 7. Figure 7 clearly shows that the limit $δ_{m a x} = 120 °$ is satisfied for point OP_RAC, which is located on the hull of the subset S_RAC, as shown in Fig. 5.

Fig. 7 Rotor angle at OP_RAC.

According to Step 4-5, point OP_RAC is used to start the TVC stage in Step 2. In Step 3-1 of the first projection operation $P_{C T V C}^{r O} (\cdot)$ of the O-SEQ_TVC sequence, the TD-SDM analysis indicates that the system operating at OP_RAC and subjected to the contingency scenario does not satisfy $η_{T V C}$ given by (3) at $t_{u} = 0.36$ s. In this case, there are transient trajectories of voltage magnitudes surpassing $V_{m i n} = 0.85$ p.u.. Transient voltages at OP_RAC are shown in Fig. 8. Hence, point OP_RAC is outside S_TVC, as clearly shown in Fig. 5. To achieve a transiently stable operating point, the $O$ - $S E Q_{T V C}$ sequence performs a total of 10 single projection operations $P_{C T V C}^{r O} (\cdot)$ to assess the first point inside the subset $S$ _TVC, which is set as $O P_{i n, T V C} = O P_{R A C + 10}$ in Step 3-6. The active power re-dispatch and the total generation cost for $O P_{R A C + 10}$ are given in row 5 of Table II.

Fig. 8 Transient voltages at OP_RAC.

The $O$ - $S E Q_{T V C}$ sequence results reported above are transferred to the $U$ - $S E Q_{T V C}$ sequence, which starts in Step 4. In this case, three single projection operations $P_{C T V C}^{r U} (\cdot)$ are performed to assess point OP_TVC on the hull of the subset $S_{T V C}$ . Hence, the sequence and the proposed approach end in Step 4-5 of the third projection operation. Rotor angle at $O P_{T V C}$ and transient voltages at $O P_{T V C}$ , as shown in Fig. 9 and Fig. 10, reveal that both criteria $η_{R A C}$ and $η_{R A C}$ are satisfied, respectively. Furthermore, the active power re-dispatch required to achieve this transiently stable operating point and its associated total generation cost are given in row 6 of Table II.

Fig. 9 Rotor angles at OP_TVC.

Fig. 10 Transient voltages at OP_TVC.

Lastly, the comparison of results with the proposed approach and the global approach [

7] is shown in Table III, which compares the active power re-dispatch, total generation cost, and CPU time related to the solution to the TSC-OPF problem. Note that the active power re-dispatch and total generation cost compare well, whereas the total CPU time required by the proposal in this paper is 95.4% lower than that required by the global approach, which clearly shows the prowess of the proposed approach.

TABLE III Comparison of Results with Proposed Approach and Global Approach

Approach	$P_{g_{1}}$ (MW)	$P_{g_{2}}$ (MW)	$P_{g_{3}}$ (MW)	Cost ($/h)	Total CPU time (s)
Global	157.94	82.57	77.27	1161.40	201.04
Proposed	160.33	82.90	74.36	1163.40	9.18

B. Mexican 46-machine 190-bus Equivalent System

The proposed approach is applied to a reduced model of the Mexican interconnected power system (MIPS) composed of 46 generators, 91 loads, and 265 transmission components, with its topological structure and nomenclature reported in [

16]. The representative diagram of MIPS is summarized in Fig. 11.

Fig. 11 Representative diagram of MIPS.

For the MIPS, the contingency scenario is given by a permanent three-phase-to-ground fault incepted at bus 182 at $t = 0$ s and cleared at $t = 0.15$ s by tripping the line connecting buses 182 and 86. The study period is $T = [0,5] s$ . A conventional OPF analysis is executed to obtain the base point OP_U, which results in the total generation cost given in row 2, column 4 of Table IV.

TABLE IV Costs of OPS for MIPS

α	Sequence	Point	Cost ($/hour)
α	Base point	OP_U	21093.4
RAC	O-SEQ_RAC	OP_in_,RAC	21408.8
RAC	U-SEQ_RAC	OP_RAC	21106.4
TVC	O-SEQ_TVC	OP_in_,TVC	21448.5
TVC	U-SEQ_TVC	OP_TVC	21111.0

Step 1 sets $α = R A C$ and point OP_U is used to start the $O$ - $S E Q_{R A C}$ sequence from Step 2. The TD-SDM analysis related to the first projection operation $P_{C R A C}^{r O} (\cdot)$ is executed in Step 3-1 and detects that some transient trajectories of rotor angles surpass the limit $δ_{m a x} = 120 °$ . Rotor angles at point OP_U are shown in Fig. 12. The system is declared unstable at $t_{u} = 0.59$ s. The TSC-OPF problem is then formulated and solved in Step 3-4 to obtain the point OP_U₊₁, where the total generation cost is given in row 3 of Table IV. The $O$ - $S E Q_{R A C}$ sequence returns to Step 3-1, where the TD-SDM determines that $η_{R A C}$ is satisfied. The $O$ $- S E Q_{R A C}$ sequence ends in Step 3-6 with the system operating point given by $O P_{i n, R A C} = O P_{U + 1}$ .

Fig. 12 Rotor angles at OP_U.

The projection operation $P_{C R A C}^{r U} (\cdot)$ is now applied to perform the $U$ $- S E Q_{R A C}$ sequence in Step 4. After seven executions of this operation, OP_RAC is obtained on the hull of the subset $S$ _RAC with the total operating cost given in row 4 of Table IV. The corresponding rotor angles at OP_RAC and transient voltages at OP_RAC are shown in Figs. 13 and 14, respectively. It is noted that the rotor angle limit $δ_{m a x} = 120 °$ is satisfied. The transient voltage limit $V_{m i n} = 0.85$ p.u., however, is not satisfied at $t_{u} = 0.96$ s.

Fig. 13 Rotor angles at OP_RAC.

Fig. 14 Transient voltages at OP_RAC.

The stabilization process for voltage magnitudes is now performed through the TVC stage by considering OP_RAC as the starting point of this process. When $α = T V C$ , the $O$ - $S E Q_{T V C}$ sequence performs one projection operation $P_{C T V C}^{r O} (\cdot)$ to obtain the point OP_RAC₊₁, where $η_{T V C}$ is satisfied. This point is set as $O P_{i n, T V C} = O P_{R A C + 1}$ in Step 3-6 with a generation cost given in row 5 of Table IV. Lastly, the $U$ - $S E Q_{T V C}$ sequence is executed and seven projection operations $P_{C T V C}^{r U} (\cdot)$ are conducted to obtain point OP_TVC. The system operating at OP_TVC and subjected to the specified contingency scenario satisfies both criteria $η_{R A C}$ and $η_{T V C}$ , which is corroborated by the rotor angle at OP_TVC and transient voltages at OP_TVC, as shown in Figs. 15 and 16, respectively.

Fig. 15 Rotor angles at OP_TVC.

Fig. 16 Transient voltages at OP_TVC.

Furthermore, the corresponding generation cost of point OP_TVC is given in row 6 of Table IV.

Finally, the total generation costs shown in Table IV clearly reveal that the most economic generation cost corresponds to the transiently unstable point OP_U.

Point OP_RAC only ensures the rotor angle stability with a cost increase of 0.06% with regard to the cost associated with point OP_U. Lastly, the criteria of rotor angles and voltage magnitudes are simultaneously satisfied for the specified contingency scenario when the system is operating at point OP_TVC, which increases the generation cost of point OP_U by only 0.08%. Results show that the generation cost slightly increases with the improvement of the power system security through the proposed approach, although the total power re-dispatch for assessing point OP_TVC is 56.6 MW. The CPU time required to solve the TSC-OPF problem is 218.89 s.

VI. Conclusion

A sequential TSC-OPF approach is proposed to accurately assess the most economical operating point that simultaneously satisfies $η_{R A C}$ and $η_{T V C}$ when a power system is subjected to a specified contingency scenario. In this preventive control, the evolution of the transient trajectories corresponding to rotor angles and voltage magnitudes is bounded through an economic active power re-dispatch, which is performed to guide the system to a normal operating state. For this purpose, the TSC-OPF problem is formulated as two mutually connected subproblems. The first is associated with a conventional OPF problem extended with only two additional active power re-dispatch constraints based on projections onto sets. The second relates to the transient stability assessment that determines the evolutions of rotor angles and voltage magnitudes and provides the information required to assemble the active power re-dispatch stability constraints. In this paper, the TSC-OPF model has a dimension, complexity, and computational burden similar to that of a conventional OPF model because including discretized constraints in the formulation is unnecessary.

Numerical results clearly demonstrate the effectiveness of the proposed approach in solving the TSC-OPF problem and avoiding system unstable operation. Concerning the 3-machine 9-bus system, the cost of performing the preventive control through the proposed approach and the global SD approach is 2.73% and 2.58% higher than the base cost associated with the transiently insecure base operating point, respectively. In case studies, the computational time required to achieve the preventive control is 9.18 s and 201.04 s for the proposed approach and the global SD approach, respectively. Hence, the proposed approach is only 0.1464% more expensive than that obtained by the SD approach, saving 95.4% of CPU time. The solution obtained by the proposed approach compares well with that provided by the global SD approach but with the advantage of avoiding the resolution of an optimal problem of enormous dimension, complexity and computational burden. The case study associated with the MIPS shows that the cost of generation re-dispatch for maintaining the system secure operation is only 0.08% more expensive than the corresponding transiently insecure base operating point. The proposed approach achieves the preventive control of this 46-machine 190-bus system in 218.89 s, which is very similar to the CPU time required by the global SD approach to stabilize the 3-machine 9-bus system, which is 201.04 s. This comparison of computational performances verifies the computational savings achieved by the proposed approach.

Since the proposed approach is based on the well-known projections onto set concept, two different feasible regions have been straightforwardly addressed to solve the TSC-OPF problem. Hence, a basis for handling the multi-contingency case is provided because the stability region for each contingency could be defined as a set of points for the non-heuristic solution to the multi-contingency TSC-OPF problem. This is an important topic that the authors will address in a forthcoming publication.

Nomenclature

Symbol	——	Definition
A.	——	Constants
δ_max	——	Limit of rotor angle
α	——	Rotor angle control (RAC) stage or transient voltage control (TVC) stage
λ	——	Percentage of magnitude of the maximum permissible active power re-dispatch in over-relaxed sequence of α (O-SEQ_α)
τ	——	System parameters
Δt	——	Time step
η_α	——	Security criterion of α
η_RAC	——	Stability criterion of RAC
η_TVC	——	Admissible criterion of TVC
H_i	——	Inertia constant of the i^th generator
n_b	——	Number of buses
n_g	——	Number of generators
n_l	——	Number of loads
N_s	——	Number of integration steps Δt
$P_{l_{i}}$	——	Active power consumed by the i^th load $l_{i}$
t_cl	——	Fault clearing time
$t_{0}^{+}$	——	Disturbance inception time
t_end	——	End of experimental period
T	——	Study time period
V_min	——	Limit of deep voltage
B.	——	Functions
$f (\cdot)$	——	Vector of differential functions
$f_{E} (\cdot)$	——	Cost function of generators
$g (\cdot)$	——	Vector of algebraic functions
$G (\cdot)$	——	Vector of power mismatch equations
$H (\cdot)$	——	Vector of physical and operative limits of components
$P_{C α} (\cdot)$	——	Projection of infeasible operating point in α
$P_{C} (\cdot)$	——	Projection operation
C.	——	Sets and Operating Points
$R^{(\cdot)}$	——	Set of real numbers
$Γ$	——	Set of time-invariant system parameters
$θ_{β}$	——	Set of voltage phase angles at OP_β
OP_α	——	Operating point where η_α is satisfied
OP_β	——	Operating point in O-SEQ_α
OP_h	——	Operating point in hull of S_α
OP_j	——	Infeasible operating point of P_Cα $(\cdot)$
OP_in,α	——	First stable operating point in O-SEQ_α
OP_s	——	Last unstable operating point in O-SEQ_α
OP_U	——	Transiently unstable operating point
OP_γ	——	Operating point in under-relaxed sequence of α (U-SEQ_α)
$P_{g, β}$	——	Vector of active power of generators at OP_β
S_F	——	Set of active power output of generators
S_α	——	Subset of S_F where η_α is satisfied
S_TVC	——	Subset of transient stability region in TVC stage
S_RAC	——	Subset of transient stability region in RAC stage
U	——	Vector of control variables
$V_{β}$	——	Vector of voltage magnitudes at OP_β
X	——	Vector of dynamic state variables
Y	——	Vector of algebraic state variables
D.	——	Variables
δ(t)	——	Dynamics vector of rotor angles at time t
δ_i(t)	——	Rotor angle of the i^th generator at time t
δ_COI(t)	——	Rotor angle of center of inertia δ_COI at time t
$φ_{β t_{u}}^{α}$	——	Index in O-SEQ_α
$Δ P_{g, γ}^{α}$	——	Vector of active power re-dispatch in U-SEQ_α
$Δ P_{g, β}^{α}$	——	Vector of active power re-dispatch in O-SEQ_α
$Δ P_{g, β, m a x}^{α}$	——	Vector of the maximum permissible active power re-dispatch in O-SEQ_α
$Δ {\hat{P}}_{g, β}^{α}$	——	Unit vector in direction of $Δ P_{g, β}^{α}$
$Δ P_{g, β}^{S c h_{α}}$	——	Vector of scheduled magnitude in O-SEQ_α
$Δ {\hat{P}}_{g, β}^{S c h_{α}}$	——	Vector of scheduled direction in O-SEQ_α
$Δ P_{g, γ}^{S c h_{α}}$	——	Vector of scheduled magnitude in U-SEQ_α
$Δ {\hat{P}}_{g, γ}^{S c h_{α}}$	——	Vector of scheduled direction in U-SEQ_α
$\nabla_{P_{g, β}} φ_{β t_{u}}^{α}$	——	Gradient of $φ_{β t_{u}}^{α}$ with regard to active power output of generators at current OP_β
j	——	Number of current operating point for O-SEQ_α
m	——	Number of projections in U-SEQ_α
n_x	——	Number of dynamic state variables
n_y	——	Number of algebraic state variables
n_u	——	Number of control variables
n_τ	——	Number of time-invariant system parameters
$P_{g_{i}}$	——	Active power production of the i^th generator
$P_{C α}^{r O} (\cdot)$	——	Projection operation in O-SEQ_α
$P_{C α}^{r U} (\cdot)$	——	Projection operation in U-SEQ_α
$P_{C α}^{r O} (O P_{β})$	——	Projection operation of $O P_{β}$ in O-SEQ_α
$P_{C α}^{r U} (O P_{γ})$	——	Projection operation of $O P_{γ}$ in U-SEQ_α
t_u	——	Time to instability
$T_{γ}$	——	Interval brackets $O P_{γ}$ and $O P_{i n, α}$
Tol	——	Convergence tolerance for U-SEQ_α
u	——	Vector of control variables
V_k(t)	——	Dynamics of voltage magnitudes $V (t)$ at bus k
V(t)	——	Nodal transient voltage magnitudes at time t
x(t)	——	Dynamic state variables x at time t
y(t)	——	Algebraic state variables y at time t

References

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Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Two-stage Transient-stability-constrained Optimal Power Flow for Preventive Control of Rotor Angle Stability and Voltage Sags PDF

Abstract

Keywords

I. Introduction

II. Fundamentals of Proposed Approach

A. Transient Stability and Trajectory Sensitivity Analysis

B. Projection-based Optimal Power Re-dispatch

III. Formulation of Projection Operators for Non-heuristic Generation Re-dispatch

A. Projection Operation $P_{C α}^{r O} (\cdot)$ in O-SEQ_α Sequence

B. Projection Operation $P_{C α}^{r U} (\cdot)$ in $U$ - $S E Q_{α}$

C. Comparison with Other Models

IV. TSC-OPF Approach for Preventive Control

V. Case Studies

A. WSCC 3-machine 9-bus System

B. Mexican 46-machine 190-bus Equivalent System

VI. Conclusion

Nomenclature

References

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Two-stage Transient-stability-constrained Optimal Power Flow for Preventive Control of Rotor Angle Stability and Voltage Sags PDF

Abstract

Keywords

I. Introduction

II. Fundamentals of Proposed Approach

A. Transient Stability and Trajectory Sensitivity Analysis

B. Projection-based Optimal Power Re-dispatch

III. Formulation of Projection Operators for Non-heuristic Generation Re-dispatch

A. Projection Operation PCαrO(⋅) in O-SEQα Sequence

B. Projection Operation PCαrU(⋅) in U-SEQα

C. Comparison with Other Models

IV. TSC-OPF Approach for Preventive Control

V. Case Studies

A. WSCC 3-machine 9-bus System

B. Mexican 46-machine 190-bus Equivalent System

VI. Conclusion

Nomenclature

References

A. Projection Operation $P_{C α}^{r O} (\cdot)$ in O-SEQ_α Sequence

B. Projection Operation $P_{C α}^{r U} (\cdot)$ in $U$ - $S E Q_{α}$