Flexibility of interconnected power system operation: Analysis, evaluation and prospection

Cross-region power exchange plays a signiﬁcant role in the efﬁcient utilization of power resources. An increased power demand and renewable integration leads to a decreased security margin of power system operation. The operational ﬂexibility in an interconnected power system can be further extended by expanding the operating region of each power system. In this study, the potential ﬂexibility in an interconnected power system is summarized from cross-region and intra-region perspectives. A uniﬁed approach is presented to evaluate the ﬂexibility by multi-parametric programming. An illustrative example and a 661-bus utility system are utilized to validate the proposed uniﬁed approach. A non-iterative framework that considers the ﬂexibility in the interconnected power system operation is presented. Relevant lines of future research have also been indicated for practical implementations of the ﬂexibility.

emergency transmission rates [8]; (iii) for tie-lines, the tie-line scheduling adjustment; and (iv) for demand consumers, their sensitivity to prices [10]. These flexibilities from various parts in the interconnected power system operation are reviewed in Section 2.
Furthermore, the flexibility of different components is represented in different forms (e.g. the flexibility for the demand by consumer is the sensitivity to prices, while the flexibility for the tie-line is the tie-line scheduling adjustment). Therefore, different flexibilities require different evaluation methods, as reviewed in Section 2. The operation pattern of these components is challenging to reasonably compare and optimize. In this study, a unified approach based on multi-parametric programming is proposed for a straightforward evaluation of the flexibility of different components. In the presented unified approach, the flexibility of one component is defined as a feasible region composed of coupling variables connected to other components. In the feasible region, the constraints of the component can be guaranteed unviolated.
Finally, a non-iterative framework is presented in Section 4 to consider the flexibility in the interconnected power system operation. Moreover, future study directions (including the incentive for the flexibility in operation, and practical restrictions of the presented unified evaluation approach) are indicated to transform research advances into practical implementations.

THE FLEXIBILITY OF THE INTERCONNECTED POWER SYSTEM OPERATION
The feasible operating region of one certain regional system in an interconnected system is presented in Section 2.1. The flexibility that affects the feasible operating region is summarized in Section 2.2, together with evaluation methods.

2.1
Feasible operating region of one regional system In an interconnected system, the feasible operating region of one certain regional system is reflected by its required constraints. Assuming the efficiency and robustness of the DC power flow model [11], the following linear constraints are adopted in power industries to reflect the feasible operating region: (i) Coupling constraints. A particular regional system is connected with other systems via tie-lines for the power balance, that is, where P G , P B , and P D are the generation level, tie-line power and demand, respectively; e G , e B , and e D are "allone" column vectors associated with P G , P B , and P D , respectively. (ii) Flow constraints relevant to tie-line power. The tie-line power should not exceed its limitations, that is, where P B and P B are lower and upper bounds, respectively. (iii) Generator capacity constraints. The generation level P G should not exceed its upper bound P G and lower bound P G , that is, (iv) Flow constraints relevant to transmission lines in the regional system. The flow of transmission lines P L should not exceed its upper bound P L and lower bound P L , that is, where S is the power transfer distribution factor (PTDF) matrix; M G , M D , and M B are incident matrices associated with P G , P B , and P D , respectively.

2.2
The flexibility from different perspectives [1][2][3][4][5], constraints (1) and (2) are relevant to the tie-line power from cross-region power systems, while constraints (1) and (2) are only relevant to components in the focused regional power system. Consequently, the flexibility is revisited in Section 2.2 from the cross-region and intra-region perspectives, as shown in Figure 1.
From the cross-region perspective, the flexibility in relation to the tie-line power is focused on and from the intra-region perspective, the flexibility in relation to the generator side, transmission side and demand side is reviewed. Furthermore, the evaluation of the flexibility of different components is reviewed. Note that the flexibility in an interconnected power system is the union of the flexibility in each regional power system. Thus, the flexibility of a regional power system is discussed in this study.

2.2.1
The flexibility from the cross-region perspective The generation level P G changes when P B is adjusted. The flexibility from the cross-region perspective is reflected in P B .
In the existing methods, the maximum summation of the tieline power is adopted for flexibility measurement. The maximum summation of the tie-line power based on a system flow method is solved in [12][13][14], while the optimization problem is solved in [15],where the objective function is the maximum summation of the tie-line power over system constraints.
However, the above representation of the flexibility relevant to the tie-line power ignores the coupling of tie-line power among border buses. To accurately reflect the flexibility relevant to the tie-line power, the feasible region that comprises all combinations of tie-line power with which constraints of the regional system can hold is developed in [9] and [16][17][18]. Ref. [9] adopts multi-parametric programming to enumerate combinations of active and inactive constraints when tie-line power varies. Each combination reflects one sub-region of the feasible region and thus the union of all sub-regions is precisely the feasible region. By only enumerating the combinations of active and inactive constraints that are relevant to the boundary of the feasible region, Ref. [16] modifies multi-parametric programming to accelerate the identification of the feasible region. Based on [9], the link between the day-ahead market and the long-term market is considered for the feasible region of the tieline power in [17,18] identifies the feasible region by exploring vertices.

The flexibility from the intra-region perspective
The flexibility based on the intra-region is reviewed from the generator side, transmission side and demand side.

Generator side
The typical flexibility in the generator side is the deep peak regulation of generators, and is achieved by adjusting lower bound P G . For the conventional generators, the lower bound P G can be adjusted by technologies (e.g. regular peak regulation and deep peak regulation without oil) [19]. For coal-fired power plants, lower bound P G is determined by heat supply because the steams both produce generation levels and supply heat demand [20].

Transmission side
The typical flexibility in the transmission side lies in upper bound P L and lower bound -P L in (4). When flow constraints are overloaded, the transmission lines can be damaged because of the heat cumulative process [21,22]. When overloaded, the transmission lines are not immediately damaged, implying the overloads can temporally hold. This indicates that bounds P L and -P L can be temporally extended for more flows to pass through the transmission lines. This flexibility has been implemented in some power systems, for example, adaptive emergency transmission rates in ISO-NE [8]. For the measurement of bounds P L and -P L , they are subjected to the quantity of current and temperature passing through the transmission lines for specific periods, as shown in [8].

Distribution side
The typical flexibility in the demand side is the demand response that indicates that demand consumers are sensitive to the local marginal prices (LMPs) in electricity markets. The consumed demand can be formulated as a function of LMPs and can be modelled based on historical data, then be adopted to efficiently reflect the flexibility of one demand consumer by the relationship between the demand and LMPs [23]. However, in distribution systems, several consumers with different functions exist and are aggregated to one equivalent demand that respects the constraints in the distribution systems. The function that reflects the relationship between the equivalent demand and the LMPs should be studied further. The aforementioned issue can be resolved by the proposed unified approach in Section 3.

A UNIFIED APPROACH OF THE FLEXIBILITY EVALUATION
The flexibility of different components requires different evaluation methods and thus the existing evaluation of the flexibility is complicated. Furthermore, the evaluation of the flexibility that reflects the relationship between the LMPs and the equivalent demand in the distribution system remains unresolved.
In Section 3.1, a unified approach is presented to easily evaluate the flexibility of different components. The compatibility of the unified approach is discussed in Section 3.2. Finally, an illustrative example and one 661-bus utility system are shown in Section 3.3.

Framework of the unified approach
As stated in Section 2, the flexibility is the adjustment ability when the focused constraints hold. Thus implies, all combinations of parameters that describe the adjustment ability with which the focused constraints hold can be regarded as the flexibility.
Consequently, the flexibility in the unified approach is redefined as a feasible region that comprises all combinations of those parameters w with which the focused constraints hold. To obtain the flexibility, we consider the following compact model: where e is the "all-one" column vector whose dimension is consistent with that of w; A, B, and C are constant matrices that describe the focused constraints; x represents continuous variables in the focused constraints.
For illustration purposes, we assume the flexibility from the cross-region perspective to explain the compact model (6). The flexibility from the cross-region perspective is described by tieline power; thus, the parameter w is selected as tie-line power P B . Inequality Ax+Bw ≤ C represents constraints in the connected regional power systems where x is the generation level P G .
To obtain the feasible region based on (6), the Karush-Kuhn-Tucker (KKT) conditions are adopted, resulting in: (i) Stationary conditions: where λ is a dual variable. (ii) Complementary conditions: If the optimal solution of (6) is x * and the active and inactive constraints are known, the KKT conditions in (7)- (9) can be reformulated as follows: (i) Linear equations (ii) Linear inequalities Based on (10)- (14), it is found that x * is only determined by (11). When A I is non-singular, x * can be formulated as a func-tion of w based on (11), that is, The solution in (15) should satisfy (2), that is, Consequently, when one combination of active and inactive constraints is known, a polytope described by (16) in space w can be found. The key challenge of the proposed approach lies in the enumeration of the combinations of the active and inactive constraints. The numeration process is a multiparametric programming process as shown in Algorithm 1. When all combinations of active and inactive constraints are obtained, the union of the corresponding polytopes is the feasible region over w. Note that A I can be singular because of the degeneracy. The method in [24] can be applied to solve such situations.

Compatibility of the unified approach
All flexibility in Section 2.2 can be incorporated into the presented unified approach: (i) The flexibility from the cross-region perspective can be represented by the tie-line power region that provides feasible combinations of tie-line power with which the constraints of the connected regional power systems hold. Tie-line coupling variables are selected as parameter w in (6). Decision variables of the connected regional power systems are selected as x (i.e. generation levels) in (6), and the constraints of the connected regional power systems are selected as the focused constraints in (6). (ii) The flexibility from the intra-region perspective on the generator side can be represented by the adjustment amount of P G . For the conventional generators, the lower bound P G can be adjusted by the deep peak regulation. Consequently, the adjustment amount is regarded as parameters w in (6). The decision variables in (6) are selected as technologies in the deep peak regulation (e.g. the amount of oil added). The constraints in (6) are selected as the linearized relationship between the adjustment amount and the adopted technologies.
For the coal-fired power plants, the flexibility is still represented by the adjustment amount of P G . Consequently, the adjustment amount is also regarded as parameter w in (6). The decision variables in (6) are selected as the heat supply that affects generation levels. The constraints in (6) are selected as the linearized relationship between the heat supply and the adjustment amount of P G .
(i) The flexibility from the intra-region perspective on transmission side lies in its emergency transmission rates. This flexibility is only relevant to environmental elements (e.g. the amount of current and temperature passing through the transmission lines for particular periods) [8]. Consequently, its flexibility cannot be adjusted by manual decisions and thus the x in (6) ceases to exist. This indicates the feasible region has been described as one point that is calculated by the relationship between the emergency transmission rate and environmental elements. Accordingly, this is a specific case of the presented unified approach. (ii) The flexibility from the intra-region perspective on the demand side lies in the relationship between the LMPs and equivalent demand. Therefore, in the presented unified approach, parameter w in (6) should be selected as both LMPs and the equivalent demand. Decision variable x in (6) is the consumed load when facing the prices, together with distributed generation levels. The constraints in (6) should reflect the relationship between LMPs and the consumed load. Moreover, the distribution system constraints should be incorporated into (6).
Because the transmission network is affected by demand from distribution networks, the flexibility in the demand side should initially be obtained as a boundary, and then other types of flexibility (e.g. the flexibility from the cross-region perspective) can be calculated.

Case studies
In Section 3.3.1, the proposed unified approach is initially verified by calculating the flexibility in the demand side in one threebus distribution network. In Section 3.3.2, the proposed unified approach is further verified by calculating the flexibility from the cross-region perspective in a 661-bus utility system.

Verification of the flexibility in the demand side
In this section, a simple three-bus distribution network, shown in Figure 2, is used to illustrate the presented unified approach in the demand side. This example has been uploaded to [25]. Bus 1 belongs to the transmission system. For bus 1, there is an equivalent demand d e and LMP λ from regional transmission operators (RTOs). Buses 2 and 3 belong to the distribution system. Loads d 2 and d 3 are at buses 2 and 3, respectively. Distributed generators p 2 and p 3 are at buses 2 and bus 3, respectively. The relationships between loads (d 2 and d 3 ) and LMP (λ) are shown in Figure 3.
The model in (6) can be formulated superficially for this illustrative example as follows: where the linear model of the distribution system is adopted to describe the constraints (18)-(25): constraints (18) and (19) describe the power balance and flow constraints in the distribution system, respectively; constraints (20) and (21) describe the relationships between the LMPs and demand; constraints (22) and (23) describe the bounds of demand; constraints (24) and (25) are limitations of distributed generators. By adopting Algorithm 1, the flexibility described in the space (λ, d e ) can be identified as the feasible region shown in Figure 4.

Verification of other types of the flexibility
Various types of flexibility should be tested for the proposed unified approach. For simplicity but without loss of key features of the flexibility, the unified approach is used to describe the flexibility from the cross-region perspective in a 661-bus utility system in China. Three tie-lines are connected at buses 5, 9 and 60, and their tie-line powers are denoted as P 5 , P 9 and P 60 , respectively. The positive P 5 , P 9 , and P 60 indicate the tie-line power injected to this utility system from other connected regional systems, while negative ones indicate the tie-line power delivered to other connected regional systems.
The flexibility from the cross-region perspective is described by the feasible region in the three-dimension space (P 5 , P 9 , and P 60 ). The description of the feasible region that satisfies constraints in (1)-(5) also considers the flexibility in the intra-region perspective: (i) in the generator side, the generation level P G can reach 0.9 P G ; (ii) in the transmission side, the flow limit P L is increased by 10%; (iii) in the demand side, the demand is not fixed at P D while it can be dispatched in an interval [0.9P D , 1.1P D ].
By adopting Algorithm 1, the feasible regions with and without the flexibility in the intra-region perspective are shown in Figures 5 and 6.
The physical limits of tie lines are 8 p.u. From Figure 5, with the flexibility in the intra-region perspective, the total capacities of the tie lines are adequately utilized. However, in Figure 6, it can be observed that the capacities of the tie lines are not Feasible region with more flexibility in the intra-region perspective entirely utilized, that is, when the flexibility in the intra-region perspective is not considered. Consequently, the feasible region without the flexibility in the intra-region perspective is an exact subset of that with the flexibility in the intra-region perspective. Simply, the flexibility in the intra-region perspective can extend the feasible region of tie-line power. It is feasible that more flexible constraints make contributions to more combinations of tie-line power. A large feasible region of tie-line power indicates increased power exchange among regional systems. For example, the maximum summation of tie-line power in Figure 5 is 22.9235 p.u., while in Figure 6 is 20.1627 p.u. Therefore, the unified approach can evaluate the flexibility from the cross-region perspective. Furthermore, the flexibility in the intra-region perspective can definitely expand the feasible region of tie-line power; thus, increased power exchange among regional systems can be implemented to facilitate the utilization of power resources in an interconnected system.
Additionally, the feasible region with increased flexibility in the intra-region perspective is evaluated in Figure 7. To provide more flexibility, flow limit P L is increased by 30%.
The feasible region in Figure 7 is larger than that in Figure 5. This conclusion is intuitive because higher flexibility indicates more flexible constraints that can hold more combinations of the tie-line power. Accordingly, higher flexibility leads to a large feasible region of the tie-line power, hence more power exchange among connected regional systems.

PROSPECTION
Based on the analysis of the flexibility of the interconnected power system operation in Section 2 and the unified evaluation method in Section 3, a non-iterative application framework is proposed in Section 4.1 to consider the flexibility in the interconnected power system operation. Furthermore, relevant lines of future research (including the incentive for the flexibility in operation, and practical restrictions of the proposed unified evaluation approach) are presented in Section 4.2.

Framework of utilizing the flexibility in the operation
The system operation in the interconnected power system requires a decentralized framework to avoid the intrusion of privacy data of each regional system. A common management style in the interconnected power system is coordinating regional systems by a central coordinator. Additionally, as analyzed in Section 2, the flexibility lies in almost every part of each regional power system. Thus, the application of the flexibility in the operation should be under a decentralized framework. However, the existing decentralized algorithms require the iterative information exchange [26,27]. This may result in communication problems, signal delays, or even convergence issues. To achieve the application of the flexibility under a non-iterative distributed system, the following application framework is presented in Figure 8.
The proposed framework includes the following five stages: Stage 1: Distribution system operators (DSOs) submit the flexibility in the demand side to regional transmission operators (RTOs). Stage 2: Each RTO evaluates the flexibility in its intraregion perspective. Its flexibility on the tie-line power is further evaluated and is then submitted to the central coordinator. From the above five stages, the following three advantages exist in the presented framework: (i) The presented application framework protects the private data of each system. Each system only submits the flexibility used in the operation of other systems and does not submit its detailed parameters to other systems. (ii) The presented application framework is non-iterative.
Stages 1-5 are implemented once. From Stage 1 to Stage 3, the tie-line scheduling among regional systems is determined. The system decisions are obtained in Stages 4 and 5.
Compared with the existing decentralized algorithms, the proposed framework achieves the decentralized solutions when avoiding iterative information exchange. (iii) The flexibility of the interconnected power system operation is fully considered. As shown in Stages 1-5, each system thoroughly considers its flexibility: (a) In Stage 1, the flexibility in the demand side in each DSO is used; (b) In Stage 2, the flexibility in the intra-region

Future directions
In Section 4.2, the future study directions on both, incentivizing the flexibility in the presented non-iterative application framework and the presented unified evaluation approach, are discussed.

Incentive of the flexibility in the presented application framework
The application framework resolves the issue of utilizing the flexibility in the operation of the interconnected power system. Therefore, the precondition of its success is based on the willingness of each participant to be scheduled with their flexibility. This requires a good pricing and settlement mechanism to incentive each participant. If the scarcity rent of the flexibility of each participant is implemented, then each participant will be incentivized for their flexibility, thus promoting more efficient utilization of power resources.
Moreover, the flexibility of more devices should be further mined. The flexibility introduced in this study is common in the interconnected power system. With the fast development of technologies in power systems, many devices will be located in power systems. The flexibility of these new devices (e.g. grid-scale energy storage and electric vehicles) should be considered to further expand the feasible operating region of the interconnected power system.

Future study directions of the unified evaluation approach
Although the presented unified approach can be used to evaluate the flexibility of different devices, there are still some restrictions: (i) The requirement of KKT conditions. The KKT conditions only exist in problems with continuous variables and algebraic constraints; thus, the proposed approach is limited by the application scenarios of the KKT conditions. For example, when stability constraints are considered in the flexibility from the cross-region perspective, the unified evaluation approach is not applicable. Although the feasible region with stability constraints can be approximated as a convex polytope as an alternative by exploring vertices of stability constraints [28], the extension of the proposed unified evaluation approach to the problems without KKT conditions should be further studied. (ii) The particular requirement of linear algebraic constraints.
When active and inactive constraint sets are known, the liner algebraic constraints can be used to generate a feasible region by matrix manipulation. However, some constraints can be non-linear. For example, the power flow constraints in the regional systems and the distribution systems are non-linear. The linearization technologies are adopted to approximate the non-linearization. Therefore, to improve the evaluation accuracy of the presented unified approach, further studies should be conducted to improve linearization technologies.

CONCLUSION
In recent years, the flexibility has been focused to expand the operating region of the power system. This study initially summarizes the flexibility in the cross-region and the intra-region perspectives, together with evaluation methods. It is observed that the flexibility of different components requires different evaluation methods. To simplify the complication caused by various evaluation methods, a unified approach is proposed to easily evaluate the different flexibility. In the proposed unified approach, the flexibility is redefined as a feasible region that can efficiently be solved by multi-parametric programming. A threebus illustrative example and a 661-bus utility system demonstrate the validation the proposed unified approach. Furthermore, numerical results show that the flexibility can expand the feasible region of the tie-line power and thus more power can be exchanged in the interconnected power system operation. Furthermore, a non-iterative application framework is presented to incorporate the flexibility into the operation of an interconnected power system. In addition to utilizing the flexibility, the presented framework allows for coordinating the utilization of power resources among regional power systems and distributed systems in a decentralized way. Moreover, three future directions are pointed out: (i) the pricing and settlement mechanism should be designed to incentivize each participant to be scheduled with their flexibility; (ii) the flexibility of more devices should be further mined; (iii) improvements on the unified evaluation approach: one is to extend the proposed approach to the application scenario where there are no KKT conditions, and the other one is to focus on the linearization technologies on the constraints that affect the flexibility.