0 Introduction For a large interconnected power system, accurately determining the power transmission capacity between power system regions to maximize the fulfillment of load demands in each region while meeting safety and reliability constraints has become an urgent research topic in the power system field. The transmission capacity between system regions is a crucial measure for assessing the reliability of an interconnected system. System planners can use this measure as an indicator to evaluate the interconnection strength and compare the merits of different transmission system structures; while system operators can use it as a vital basis for real-time assessment of the power exchange capacity between different regions of the interconnected system. Solving the problem of calculating the transmission capacity between power system regions is a highly challenging task. Its difficulty is reflected in two aspects: first, the power system itself is a complex nonlinear dynamic system. As the amount of power exchange between system regions increases, typical phenomena in nonlinear dynamic systems, such as Hopf bifurcation or saddle-point bifurcation, may occur, thereby compromising system safety; second, this problem must consider not only the normal operating mode of the system but also the impact of various fault conditions. Currently, research on the transmission capacity between system regions mainly adopts the Optimal Power Flow (OPF) method, which has been discussed in detail in the literature. These methods have two main drawbacks: first, they cannot consider dynamic constraints such as system stability; second, the obtained result is an ideal target solution, and it is uncertain how to transition from the existing solution to the target solution, or even whether such a transition is possible. To overcome these shortcomings, the literature introduces a method for analyzing inter-regional transmission capacity based on continuous power flow calculations, and briefly illustrates the advantages of this method through numerical examples. This paper, building upon the literature, provides a more comprehensive and systematic analysis of the proposed method, establishes a more specific transmission capacity analysis model, proposes corresponding solution methods, and evaluates the method using actual system examples. The analysis shows that the method based on continuous power flow calculations can consider both static safety constraints such as voltage levels and line and equipment overloads, as well as voltage stability constraints caused by saddle point bifurcation in the power flow equation solution and the effects of other dynamic constraints, thus possessing significant practical value. **[b]1. Concept and Definition of Inter-regional Power Exchange Capacity[/b]** According to the definition given by the North American Power System Reliability Committee (NERC) in 1995, the inter-regional power exchange capacity of a system refers to the maximum power that can be transferred from one region to another through all transmission loops between the two regions, provided that at least the following three constraints are met: a. Under normal, fault-free conditions, the load and voltage levels of all equipment (including lines) in the system are within their rated ranges; b. Under fault conditions where a single component (such as a transmission line, transformer, or generator) is out of service, the system should be able to absorb dynamic power oscillations and maintain system stability; c. When the incident described in constraint b occurs and the system power oscillation subsides, before the dispatcher adjusts the fault-related system operation mode, the power and voltage levels of all equipment (including transmission lines) should be within their rated ranges under the given emergency conditions. The NERC definition clearly demonstrates the complexity of determining the power transmission capacity of a power system. This problem requires consideration not only of static safety constraints such as system voltage levels and line load levels, but also dynamic constraints such as stability; it requires consideration not only of the system's normal operating mode, but also of the impact of various fault conditions. To illustrate the concept of inter-regional power transmission capacity, a simplified interconnected system as shown in Figure 1 is used for further discussion. In Figure 1, regions A, B, and C represent three subsystems containing generation, transmission networks, and loads, respectively. Two subsystems can be connected by one or more transmission lines. TXY represents the active power flow at the interface from region X (representing A, B, and C) directly to region Y (representing A, B, and C). [img=315,148]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6.gif[/img][align=left] Fig.1 Simplified interconnected system One method to define the power exchange between regions of a power system is to use the net output power of a study region as a measure of its power exchange with external systems. For example, in Fig.1, the net output power of study region A, i.e., the power flowing from region A into regions B and C (TAB+TAC), can be used to describe the power exchange between this region and external systems. Another definition method is to use the power flow across a specific interface in the system. For example, in Fig.1, the lines between region A and region B, and between region A and region C, together form a power flow interface, and the active power flowing on it (TAB+TAC) can be used as a measure of the power exchange between region A and the other two regions. For the system shown in Figure 1, it can be seen that the two definition methods are completely the same, but for complex systems, they have different emphases and are not entirely equivalent. Given a definition method for inter-regional power exchange, the maximum value of this allowed power exchange under various security constraints reflects the maximum power exchange capacity between the corresponding regions. [b]2 Mathematical Model Description of Maximum Inter-regional Power Exchange[/b] By selecting a definition method for inter-regional power exchange, a mathematical description can be given for obtaining the maximum allowable power exchange between relevant regional subsystems. Taking the system shown in Figure 1 as an example, it is assumed that the power transmission capacity from region A to regions B and C is being studied. For this problem, the two definition methods for power exchange are equivalent, both being TAB+TAC. Given a fault set Rc={c1, c2, ..., cm}, let P_max_A be the maximum power exchange between region A and regions B and C defined relative to the fault set, and let JA, JB, JC represent the bus sets of regions A, B, and C respectively. Then, the following mathematical description can be given: a. Objective function [img=297,38]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6-01.gif[/img] （1） where i∈JA∪JB∪JC. b. Static safety constraints Power flow equation constraints: f(x, pgi, pL, qi) = 0 (2) Under normal operating conditions, voltage, line current and equipment load constraints: [/align][align=left][img=233,75]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6-02.gif[/img] (3) After an accident occurs and the system power oscillation subsides, before the dispatcher adjusts the system operation mode related to the fault, voltage, line current and equipment load constraints: [img=248,83]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6-03.gif[/img] (4) Where ck∈Rc. c. Dynamic safety constraints Small disturbance power angle stability constraints: [img=264,48]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6-05.gif[/img] (5) Voltage stability constraints: [img=259,94]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6-06.gif[/img] (6) Transient stability and transient process voltage constraints: [img=259,94]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/6-06.gif[/img] (7) where ck∈Rc. In the above mathematical description, x represents the system state vector; pgi and qi represent the active power injection of the generator on bus i (zero if it is not a generator bus) and the reactive power injection of the reactive power compensation device (zero if there is no reactive power compensation device); pL represents the given load level (or a set of load levels); the specific expression of the constraint function gi (i=1,…,5) depends on the stability analysis method selected. In the objective function of equation (1), only pgi and qi are selected as the adjustment variables of the optimization function. In fact, such as the tap position of the transformer or phase shifter can also be used as adjustment variables. Equation (7) gives the transient process voltage constraint condition, which refers to the condition that each bus voltage should meet before the fault occurs and the system power oscillation subsides. [t0,te] represents the transient time period under consideration. The mathematical model description of the maximum exchange power between regions given here can only be regarded as a conceptual illustration of the complex problem of maximum exchange power calculation. The factors to be considered in the actual system may be much more complex than those involved in the model. If the analysis is not about the power transmission capacity from region A to region B and region C, but only about the power exchange capacity between region A and region B, then region C is a transit system where power flows from region A to region B. In addition to the above constraints, special constraints related to region C may also need to be added, making the problem more complex. [b]3 Analysis of inter-regional power exchange capacity based on continuous power flow calculation method[/b] The power exchange capacity between system regions is directly related to the bus load and generator power change patterns of related subsystems. In the actual system, since the bus load and generator power changes vary greatly, it is unrealistic and unnecessary to consider all change patterns. Doing so may also give overly conservative results. A more practical approach would be to select a subset from the set of all possible variation patterns of system bus load and generator power for analysis. In the actual operation of a power system, the system bus load and generator power at future times can be obtained through load forecasting and economic dispatching programs. After obtaining the system bus load and generator power at a future time, whether the system at that time meets the safety requirements, and how much margin there is for the power exchange between system regions according to the variation pattern of load and generator power from the present to the given time, that is, how much maximum power exchange capacity between system regions will be if the current load and generator power variation pattern continues to develop, will be a very important issue for system operators. The selected system bus load and generator power variation pattern will have very important practical significance. In reference [3], we discussed the method of describing the variation of system bus load and generator power using the equation shown in equation (8). This method can meet the above requirements. f(x, λ) = g(x) - λb = 0 (8) In equation (8), b is called the direction vector, which determines the corresponding variation pattern of system bus load and generator power when the bus injection condition number λ changes. As the parameter λ changes, equation (8) represents a set of system power flow equations under specific bus load and generator power change modes. It can be well used to solve the problem of maximum power exchange capacity between regions. As can be seen from the mathematical description of solving the problem of maximum power exchange capacity between regions given in the previous section, it is impossible to solve this problem using conventional optimization methods due to many complex constraints such as voltage stability. Using the power flow equation description method given in equation (8), and appropriately defining the direction vector b, regardless of the definition of the power exchange quantity, the power exchange quantity between system regions can be equivalently described by the node injection change condition number λ. Solving the problem of maximum power exchange capacity under safety constraints can be transformed into the following problem to characterize it: the maximum value of the node injection change condition number λ; the constraints are equations (2) to (7). The inter-regional power exchange capacity analysis method based on continuous power flow calculation is not an optimization method in principle. Instead, it determines a trajectory describing the change of the steady-state operating point of the system by using Equation (8) based on the change pattern of the system bus load and generator power. That is, the solution curve of Equation (8). By testing whether the constraint conditions Equations (2) to (7) are satisfied at the points on this solution curve, the maximum value of parameter λ that satisfies the system safety constraint conditions along the direction vector b is obtained. Based on this, the maximum allowable power exchange between regions is obtained according to the corresponding definition. This is an operating point detection method, so it is easy to consider the influence of various constraint conditions. When using the inter-regional power exchange capacity analysis method based on continuous power flow calculation to analyze the power transmission capacity from region X to region Y, a fault set is first selected in the system, and the power output (and/or load demand) of region X and region Y are adjusted according to the pattern determined by the direction vector b under each fault state, so that there is a power surplus in region X and a power shortage in region Y. In this way, a power exchange is naturally formed between region X and region Y. Continuously increasing such adjustments between the two regions will continuously increase the power exchange between region X and region Y until the system reaches the limit values ​​defined by equations (2) to (7). A set of limit values ​​for power exchange can be obtained for the fault set, where the minimum limit value is the maximum power transfer allowed from region X to region Y, and the corresponding fault is called the most severe fault. [b]4 Actual System Example[/b] The example given here is for an actual large-scale power system, where the constraints only consider the voltage stability constraints induced by the saddle-shaped bifurcation point of the power flow equation. A brief description of the system equipment is shown in Table 1. Table 1. Components of an example system [/align][table][tr][td]Equipment Name[/td][td]Quantity[/td][td]Equipment Name[/td][td]Quantity[/td][/tr][tr][td]Bus[/td][td]15005 [/td][td]Generator[/td][td]2412 [/td][/tr][tr][td]Load Bus[/td][td]7570 [/td][td]Non-adjustable Compensator[/td][td]2390 [/td][/tr][tr][td]Line[/td][td]16081 [/td][td]Adjustable Compensator[/td][td]762 [/td][/tr][tr][td]Non-adjustable Phase Shifter[/td][td]50 [td][td]ULTC Phase Shifter[/td][td]50[/td][/tr][tr][td]Non-Adjustable Transformer[/td][td]4287[/td][td]ULTC Transformer[/td][td]3081[/td][/tr][/table][align=center] The generator, acting as an active and reactive power source with terminal voltage control, controls its voltage by adjusting the reactive power output. In calculations, it is generally considered as a PV bus; when the reactive power reaches its limit, it is treated as a PQ bus, where the fixed Q is the corresponding reactive power limit. A constant PQ load model is used. The considered control equipment includes an online adjustable parallel compensator, a ULTC transformer, and a ULTC phase shifter. Interface power flow is used as a measure of inter-regional power exchange. Based on the actual geographical distribution of the system, the system is divided into a western system, a central system, and an eastern system. Two power transmission interfaces (i.e., the western interface and the eastern interface) are defined as shown in Figure 2. [/align][img=315,157]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/8-1.gif[/img] Fig.2 The interface definition of the example system for transfer capability analysis [b]The calculation process consists of the following steps:[/b] a. Determine the power exchange scheme. The load center of the system under study is mainly concentrated in the east. Under normal circumstances, power is supplied from the west to the east through the central system. The eastern system consists of 10 subsystems. It is now assumed that the generator output power in region 4 of the eastern system gradually decreases, while the generator output power in region 25 of the western system gradually increases. The excess power generated by the generators in the western system will be transmitted to the east through the central system to compensate for the insufficient generator output power of the system caused by the decrease in the generator power in the east. In the analysis of the actual system power exchange capacity, the specific power exchange scheme should be determined by the future system load forecast results and the generator economic dispatch results. The change direction of generators and loads will be directly reflected in the direction vector b of equation (8). b. Selection of severe faults. There are a large number of different fault modes in the actual power system. It is unrealistic and unnecessary to analyze all faults. It is necessary to first identify severe faults and find the fault set that has a greater impact on the power exchange between regions. Here, the method introduced in reference [5] is used to identify 18 severe faults for key analysis. c. Solution curve tracing under ground state and fault conditions. For the determined power exchange scheme and fault set, given the voltage level and equipment load level constraints, the continuous method is used to trace the system solution curve along the direction of increasing node power change condition number until a certain static safety constraint condition in the system is violated or the solution curve reaches the saddle bifurcation point of the power flow equation, and the corresponding interface power flow is determined. The calculation results are shown in Table 2. d. Analysis of maximum power exchange capacity between regions. As can be seen from Table 2, relative to the given fault set and power exchange scheme, the maximum active power allowed to be exchanged at the eastern interface should be 5398.20 MW, and the maximum active power allowed to be exchanged at the western interface should be 5133.23 MW. Table 2. Maximum permissible interface real power flow corresponding to different contingencies under ground state and fault conditions. [table][tr][td]Faulted Line Start-End[/td][td]Eastern Interface Power Flow/MW [/td][td]Western Interface Power Flow/MW [/td][/tr][tr][td]Ground State[/td][td]6301.33 [/td][td]6025.70 [/td][/tr][tr][td]900-906 [/td][td]6003.20 [/td][td]5133.23 [/td][/tr][tr][td]904-906 [/td][td]6010.45 [/td][td]5282.30 [/td][/tr][tr][td]900-904 [/td][td]6107.62 [/td][td]5604.58 [/td][/tr][tr][td]904-911 [/td][td]6202.00 [/td][td]5888.55 [/td][/tr][tr][td]4548-911 [/td][td]6008.54 [/td][td]6008.54 [/td][/tr][tr][td]896-87904 [/td][td]5872.22 [/td][td]5812.77 [/td][/tr][tr][td]4548-919 [/td][td]5711.72 [/td][td]5773.99 [/td][/tr][tr][td]898-899 [/td][td]6180.41 [/td][td]924.43 [/td][/tr][tr][td]899-4548 [/td][td]5897.53 [/td][td]5894.98 [/td][/tr][tr][td]905-4548 [/td][td]5398.20 [/td][td]5558.13 [/td][/tr][tr][td]905-924 [/td][td]6133.16 [/td][td]5936.83 [/td][/tr][tr][td]896-897 [/td][td]5968.18 [/td][td]5787.92 [/td][/tr][tr][td]907-924 [/td][td]6209.82 [/td][td]5954.25 [/td][/tr][tr][td]896-903 [/td][td]6251.06 [/td][td]6020.77 [/td][/tr][tr][td]902-903 [/td][td]6100.62 [/td][td]5882.15 [/td][/tr][tr][td]909-924 [/td][td]6239.37 [/td][td]5998.71 [/td][/tr][tr][td]916-917 [td]6237.50 [td]5954.02 [td]898-920 [td]6289.08 [td]6007.44 [td]6007.44 [td]6237.50 [td]5954.02 [td]898-920 [td]6289.08 [td]6007.44 [td] [img=315,259]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/dlxtzdh/dlxtzdh99/dlxtzdh9904/image/8-2.gif[/img] Fig.3 Voltage magnitude at some buses versus node condition number 5 Conclusion This paper establishes a mathematical model describing the power exchange capacity between regions of an interconnected power system based on a continuous power flow calculation method, and introduces a method for calculating the maximum power exchange between regions. It can consider both static safety constraints such as voltage levels, line and equipment overloads, and voltage stability constraints caused by saddle point bifurcation of the power flow equation solution, as well as other dynamic constraints, thus possessing significant practical value. Although the calculation speed of this method cannot yet meet the requirements of online power system analysis, it can serve as a very effective offline analysis tool. The analysis of actual system examples fully demonstrates this point.