Improving security and economy of interconnected power network through explicit feasible region of tie-line power transfer
Introduction
Interconnections of regional networks promote the optimal utilization of power resources. The increased social welfares have been observed in many interconnected power systems [1], [2]. For example, HVDC tie lines across China deliver the rich renewable power in western China to satisfy the power demand centralized in eastern China [3]. European countries (e.g. Demark, Norway) accommodate high-penetration renewables via tie-line power exchanges in the European interconnection [4]. Some American independent system operators (e. g. PJM, MISO) coordinates power resources via tie lines under a joint operation framework [5].
For an interconnected power network, the regional networks are connected through tie-lines. For each regional power network, the tie-line power has a clear impact on the operation pattern of the inner power network. If the tie-line power is inappropriately determined, the operational constraints of the inner power network, including the steady-state constraints and transient constraints, will be violated. All combinations of PB and QB within which the operation of each regional power network is feasible are described as a set of operational constraints in coordinates of tie-line power (i.e., PB and QB). In this paper, such operational constraints are also referred as the feasible region of tie-line power transfer. For the interconnected power network, an intersection of all feasible regions of tie-line power transfer defines the transfer capability of tie-lines.
Currently, system operators firstly consider steady-state requirements associated with active power to calculate a feasible region of tie-line power transfer and then check its feasibility of transient requirements. For violated transient requirements, system operators will adjust the previous region in ad-hoc manners to eliminate violations. However, difficulties in finding a feasible solution rise because the security region of tie-line power transfer becomes smaller with increasing power demands and renewables. Consequently, a tie-line scheduling in existing practical industries is determined in ad-hoc manners, which could sacrifice the economy and security in the interconnected power networks.
In the existing studies and practical industries, a feasible region of tie-line power transfer is generally described by the available power transfer capability (ATC), which gives the maximum tie-line power at border nodes and interfaces. For example, considering state-steady requirements, Refs. [6], [7], [8] utilize a network flow method to capture the maximum tie-line power at border nodes and interfaces. Ref. [9] calculates the maximum tie-line power at border nodes and interfaces by solving several optimization problems. However, only certain assumed facets are used to describe the feasible region of tie-line power transfer, which leads to inaccuracy in the characterization. Also, these references describe the feasible region of tie-line power transfer by the line constraints in the equivalent model of a regional network. Line constraints are calculated by keeping the consistency of the power transfer capability among borer nodes before and after equivalence. However, the generator capacity constraints are neglected and the identified region is relevant to the current operating points, which can affect the accuracy in the identification.
To overcome these defects, some methods have been proposed to improve the accuracy in the feasible region of tie-line power transfer. Refs. [10], [11] investigate the feasible region of tie-line power transfer via multi-parametric programming. When the tie-line power varies, the combination of active and inactive constraints of the optimization problem in one certain regional network will change. If combinations of active and inactive constraints can be enumerated, the feasible region of tie-line power transfer is obtained. To accelerate the calculation in Refs. [10], [11], Ref. [12] proposes fast multi-parametric programming by only exploring the combinations of active and inactive constraints related to boundaries of the feasible region of tie-line power transfer. Ref. [13] determines the feasible region of tie-line power transfer by enumerating the vertices of the feasible region of tie-line power transfer. Vertices are calculated based on repeatedly solving optimization problems. However, the above references only focus on the feasible region of tie-line power transfer considering constraints associated with active power. Once the voltage magnitude and reactive power are incorporated, a large number of introduced variables and constraints could affect the computational efficiency of the identification.
Few studies are reported for the feasible region of tie-line power transfer considering transient constraints, although some studies have focused on the feasible region in the pre-fault generator-demand space that satisfies transient constraints [14], [15]. Refs. [14], [15] point out that the feasible region in the pre-fault generator-demand space can be approximated by a convex hull. Refs. [14], [15] adopt the energy function as the transient requirement. Given the operating point with the critical energy function, Ref. [14] performs a linearization on the critical energy function to approximate a boundary of the feasible region. In Ref. [15], the boundary is approximated based on solving nonlinear equations associated with the critical energy function. However, it is tricky to find the operation point corresponding to the critical energy function because the operating point can be only explored by repeated trials, particularly in multimachine systems. Also, in the existing practical industries, one common transient requirement is measured by the rotor angles relevant to the center of inertia, which is adopted in this paper. However, the tie-line power has not been incorporated in transient requirements and thus few studies have been reported for the tie-line schedule considering transient constraints.
Consequently, three main problems in existing methods will be addressed in this paper: Problem 1 ): How to consider steady-state constraints with voltage magnitude and reactive power in a feasible region of tie-line power transfer? Problem 2 ): How to incorporate transient constraints in a feasible region of tie-line power transfer? Problem 3 ): How to explicitly perform a tie-line scheduling when both steady-state and transient constraints are considered?
Regarding the importance of the feasible region of tie-line power transfer and defects of the existing studies, this paper presents a method to capture an explicit feasible region of tie-line power transfer with steady-state and transient constraints, and then focuses on its application in the tie-line scheduling, which improves the security and economy of the interconnected power networks. The contributions of this paper are summarized as follows:
This paper proposes a determination method of the feasible region of tie-line power transfer considering steady-state and transient constraints. The feasible region of tie-line power transfer is explicitly identified based on a vertex search method. Steady-state and transient constraints that are appropriate to the vertex search method are derived for the first time: 1) a power transfer distribution factor (PTDF) matrix of a linear power flow model with reactive power and voltage magnitude is derived for the first time to equivalently reflects the used linear model with fewer variables and constraints; 2) the transient constraints of the tie-line power are derived based on the swing and current injection equations, and then converted from the continuous-time domain to a vector space using a constraint transcription technology. Also, the Jacobian matrices of converted constraints are provided to guarantee their compatibility in the existing determination method. Furthermore, a linear programming (LP) model is proposed on the basis of the feasible region of tie-line power transfer to efficiently and explicitly perform the tie-line scheduling that satisfies both state-steady constraints and transient constraints in interconnected networks. The proposed tie-line scheduling model can provide an accurate dispatch boundary for each regional network and then promotes the optimal utilization of generation dispatches in interconnected power networks.
Section snippets
Overview of the proposed determination method
A feasible region of tie-line power transfer considering steady-state and transient constraints can be regarded as an intersection of two regions: one with steady-state constraints and one with transient constraints. For the identification of each region with different constraints, one promising idea can be to explore the vertices of each region. However, it requires solving optimization problems with steady-state or transient constraints to find different vertices. Consequently, the main tasks
Problem formulation
Taking the advantages of efficiency and computational convergence brought by the linear power flow model in existing practical industries, this paper adopts the linear power flow model in [16], which considers the reactive power and voltage magnitude, to construct steady-state constraints. Considering the tie-line power PB and QB, the inner optimization problem of one certain regional network can be given as follows:
subject to
Feasible region of Tie-Line power transfer with transient constraints
When a contingency occurs, a power system will come into a transient state. The transient constraints should be satisfied to guarantee the security operation of power systems when a contingency occurs [17]. Consequently, it is a vital issue to consider transient constraints in the feasible region of tie-line power transfer.
However, the traditional transient constraints are described as a series of differential–algebraic equations (DAEs) relevant to generators only, as given in subsection 4.1.1.
Tie-Line scheduling based on the feasible region of Tie-Line power transfer
Interconnected power networks are generally operated by different system operators. One typical management style is that the different operators are coordinated by a center coordinator through tie-line scheduling. However, the existing tie-line scheduling is generally made in an ad-hoc manner: the tie-line scheduling is artificially set and then is iteratively checked and adjusted for the feasibility of steady-state and transient constraints among regional networks. Consequently, the tie-line
Case studies
The proposed methods are verified in the IEEE 9-bus and IEEE 118-bus test systems.
Conclusions
The feasible region of tie-line power transfer can provide a feasible region of the tie-line power. Both steady-state constraints and transient constraints can significantly affect the feasible region of tie-line power transfer. This paper proposes a determination method of the feasible region of tie-line power transfer considering steady-state constraints and transient constraints. To ensure the feasible region of tie-line power transfer can be explicitly identified by the existing vertex
CRediT authorship contribution statement
Wei Lin: Methodology, Software, Investigation, Formal analysis, Writing - original draft. Zhifang Yang: Methodology, Conceptualization, Writing - original draft. Juan Yu: Methodology, Investigation, Formal analysis, Conceptualization. Wenyuan Li: Methodology, Writing - original draft. Yu Lei: Writing - review & editing, Funding acquisition.
Acknowledgements
This work was supported by National Natural Science Foundation of China (51807014), State Grid Corporation of China (“The key technology and application of active and reactive power scheduling based on nonlinear projection”), Fundamental Research Funds for the Central Universities (2019CDXYDQ0010) and China Scholarship Council.
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