The most important aspect of the planning of power generation dispatch is its complementary relationship with the power flows on the interconnectors of the transmission network. A plan could become invalid if the power flow created violates the transfer limit of any interconnectors. The long- term dispatch planning is more affected by this because of the relative difficulty in predicting the state of the transmission network in advance. It is generally planned in a trade-based manner, without a means to explicitly compute the power flows. Deviations in the plans are then corrected near the time of dispatch, in the expense of opportunity costs. In Europe, the flow-based market coupling is proposed in the Central Western Europe, which is an effective means for modeling the inter-zonal power flows and transfer capacity allocations. However, its usefulness for long-term planning is still limited. Especially, the Power Transfer Distribution Factors (PTDFs) of its model (key parameter for integrating power flow in dispatch planning) is stochastic and unpredictable. This paper introduces a data-driven approach to construct a zonal model for closing the gap between the long-term and the actual dispatch plan. It is able to reconstruct all zonal PTDFs existed in the system, by inverse modeling the ex-post power flow data. The paper as well presented the validity of decomposing a large zonal model into substantially smaller sub-problems, and the existence of clusters of ex-post power flow cases which share the same zonal PTDFs. These features have greatly simplified the implementation of the method. Keywords: Power generation dispatch, flow-based market model, zonal modeling, inverse modeling, k-nearest neighborhood classification, genetic algorithm I Introduction The European electricity markets are undergoing a steady transformation towards a single market in electricity [1] through the adoption of Flow-Based Market Coupling (FBMC). The FBMC is practical, as it represents each of the existing market zones akin to a nodal element and performs electric power dispatch calculations in a way that resembles the nodal pricing approach. One of the key activities in FBMC is the creation of a common (zonal) network model which ideally represents the real inter-zonal power flows. A stylized example of the zonal modeling is shown in Figure 1. This enables the forward planning of the allocation of inter-zonal transfer capacity, and an optimal power dispatch plan for serving the regional electricity demand. Yet, the zonal modeling in is still imperfect. It has limited usefulness in determining the volume of the inter-zonal power transfers at time much earlier than the actual hour of dispatch (i.e., weeks or months ahead). The values in its zonal Power Transfer Distribution Factors (PTDFs) matrix, ΓA is dependent on the state of the system’s operation. Specifically, they are determined by 2 (i) the reactance of the transmission lines of the detailed network, and (ii) the share of power generation committed by each of the generators within a zone for the zonal production of the next unit of power. The latter is called the Generation Shift Keys (GSKs), Π֕ [2]. Although the reactance of the transmission lines are constant, the Π֕ is stochastic (as the Π֕ is different from time to time). As a consequence, the ΓA is not firm and several of them could exist in a zonal model. The ΓA too is stochastic and indeterminate in value [3]. (a) (b) Figure 1. A stylized example of zonal modeling in FBMC: (a) a 9-bus transmission network model, (b) a 3-bus zonal model of the 9-bus transmission network The actual inter-zonal power flows would deviate from those presumed by the market players. Often, the committed power generation plans has to be corrected (through re-dispatching) to ensure balancing in the supply and demand while not violating the transmission constraints. The zonal model is more firm at times closer to the dispatch, but imprecise for long-term dispatch planning. As pointed out by [4], trading long-term contracts in FBMC is difficult because of the inability to forecast future transmission capacity in the zonal model. Also, the FBMC currently implemented in CWE is only able to publish day-ahead ΓA for the reference for the market players (the traders cannot anticipate the future ΓA) [4]. In the authors’ opinion, the restriction of having a proper zonal ΓA information is caused by the model-based (bottom-up) calculations of zonal modeling. Although the details of the transmission network (the impedance and transfer limit of the transmission lines) are known to the Transmission System Operators (TSOs), they cannot regulate a fixed Π֕ for all of the zone in advance. Also, the model-based approach is unable to consider all of the Π֕ because they could theoretically exist in an infinite number of combinations. This paper describes a zonal modeling alternative [5] that could close the gap between the long-term planning and the real-time market performance. It is a data-driven (top-down) inverse modeling approach which construct an equivalent zonal model based on the ex-post power flow data (the historical inter-zonal power flows data e.g., published by the TSOs). The most important steps in the inverse modeling is the discovery of the virtual reactance of the inter-zonal interconnectors. 3 The method then compute all of the ΓA existed in the ex-post data using the virtual reactance values. Each of the ΓA is implicitly related to a Π֕ at the time of their presence. As the elements in ΓA, 𝛽֕⟶֕ɱӴֈA are sensitivity values associated to the change in an inter-zonal interconnector created by the power generation at a zone, they could be directly used for computing the inter-zonal transfer scenarios associated to a long-term dispatch plan. For instance, one could assess the best and worst case impact of a power generation plan on the inter-zonal bottlenecks using the ΓA found, by computing the lower and upper bound of power flow on them. The paper is organized as follows. Section II introduces the theory of the inverse modeling, and two unique characteristics of the linear power flow formulations that lead to the substantial enhancement of the effectiveness of the method: (i) the valid decomposition of a larger network model into smaller sub-problems; and (ii) the existence of groups of power flow cases that could be discovered by a simple heuristic approach, the cases in each of the groups share an approximately identical ΓA. Section III describes the complete implementation of the inverse modeling. Section IV demonstrates the test of the method on a sub-problem of the Norwegian transmission system comprises of three market zones. Finally, Section V presents the discussions and conclusions. II The inverse modeling Any real-world problems possess (i) a set of observed data, d and; (ii) a set of model parameters, M. A forward operator, G maps the M to d, forming a forward problem [6] [7]: 𝑑=𝐺(𝑀) (1) For an inverse problem, the d is known, while M is unknown [8]. Thus, a solution to the inverse problem is achieved by reconstructing the M based on the known d: 𝑀=𝐺−φ(𝑑) (2) where, 𝐺−φ is the inverse operator of the problem. In the inverse modeling of a transmission network, the d is the ex-post power flow on the transmission line k, 𝐹ֆ; and M is the set of reactance values of the transmission lines, X where 𝑥ֆ∈𝑋 (𝑥ֆ is the reactance of transmission like k). The G is the linear power flow theory [9] that relates the X to the 𝐹ֆ as follows 𝐹ֆ=Bքօ(𝜃ք−𝜃օ) (3) where, the i and j are the indices of the buses connected by the transmission line k; the 𝜃ք and 𝜃օ are the voltage angle at bus i and j; the Bքօ is the admittance of the transmission line k. The 𝜃ք and 𝜃օ are derived by solving the following equation Θ = B−φP (4) where, P = [𝑃φ,…,𝑃λ]⊺ contains the net (active) power generation at each bus, and Θ = [𝜃φ,…,𝜃N]⊺ contains the bus voltage angles. The B is a two-dimension matrix defined based on the susceptance of the transmission lines, where 𝑏քօ= 1𝑥քօ⁄ (the 𝑥քօ is the reactance of transmission line k which connects bus i and j), such that Bքօ= Bօք= −𝑏քօ; ∀𝑖≠𝑗 and Bքք=∑𝑏քօN օ≠ք. The 𝑃ք (in (4)) is also the sum of net export of the transmission lines connected to bus i. Clearly, as d is well-defined (i.e., by the ex-post power flow data) while the M is unknown, the problem is an inverse modeling problem (as in (2)). The problem is solved by adopting a 𝐺−φ for finding the value of X (the M) by performing the following iterative steps: (i) find the approximate 4 X (henceforth denoted as 𝑋∗), which comprises the estimated value of 𝑥ֆ of the transmission lines, and compute the B matrix, (ii) calculate the Θ and the estimated 𝐹ֆ (henceforth denoted as 𝐹ֆ∗) using (3), and (iii) compare the 𝐹ֆ∗ with the known 𝐹ֆ. A solution (the optimum 𝑋∗) is found when either all of the found 𝐹ֆ∗ is equal to the 𝐹ֆ, or deviate from it by a small error margin, 𝜖. The objective function, 𝜙 of the inverse problem is as follows 𝜙=ం|𝐹ֆ∗−𝐹ֆ|≈0έ ֆ=φ (5) The authors adopted the Genetic Algorithm (GA) [10] as the solver of the mentioned iterative steps because the problem is nonlinear [5]. Both 𝑋∗ and Θ are unknown, and has to be solved in an iterative manner. The GA is thus the 𝐺−φ. Once the 𝑋∗ is derived, we can calculate the PTDF matrix of the transmission network, Γ as follows Γ=BցB֍−φ (6) where, B֍−φ is the inverse of the reduced B matrix (as in (4)), within which the Bքօ associated with an arbitrarily selected reference bus (bus 𝑖Ј) is removed. The Bց is a two-dimension matrix where each of its row is associated with transmission line k; in each row, the value is 𝑏քօ at column i, and −𝑏քօ at column j. The column associated with the reference bus is as well removed. The derived Γ is a two-dimension matrix where the row index corresponse to transmission line k and each column corresponds to bus i. As the Γ is computed using the 𝑋∗, it is as well an approximated parameter. It is henceforth denoted as Γ*. The derived Γ* is a two-dimension matrix where the row index corresponds to transmission line k and each column corresponds to bus i. The elements of Γ* are henceforth denoted as 𝛽ք⟶քɱӴֆ. They represent the change on transmission line k, when a unit of power is dispatched from bus i to bus 𝑖Ј. For an inversed zonal modeling, the computation of (6) would yield the ΓA,*, within which are all of the 𝛽֕⟶֕ɱӴֈA values, which are the linear sensitivity factor which reflect the change in the power flow on the interconnector m when a unit of power is dispatched from zone z to a reference zone, 𝑧Ј. Subsequently, the calculation of the inter-zonal power flow in a dispatch plan is simplified as follows 𝐹ֈ=ం𝑃֕A𝛽֕⟶֕ɱӴֈAZ ֕=φ (7) where, the 𝐹ֈ is the power flow on an inter-zonal interconnector m; and 𝑃֕A is the net power generation in zone z. 2.1 Decomposition of the inverse problem The main issue of using the GA as the 𝐺−φ is the stochastic nature of its implementation (and its solution), and the rapid increase in its run-time when solving larger problems. The GA is an approximate method, which is an analytical procedure for developing solutions that are close to the exact but unknown solution of the nonlinear problem [11]. Thus, the solution derived at the end of its implementation may vary in different repetition of its implementation. Nonetheless, the performance of GA is often comparable to other approach when it is solving a small problem. An example is given in Figure 2, which shows (i) the run-time (in seconds) and (ii) the sum of differences 5 in power flow on the transmission lines (i.e., the 𝜙) derived from testing the GA in solving the inverse problems of size 3 to 9. The GA is repeated 30 times when solving each of the problems. The median of both run-time and 𝜙 increases with the size of the problem, N solved. The range of the data also increased with N. A significant deviation in 𝜙 is seen in the 8-bus and 9 bus problems. A greater spread in run-time too is seen in the 9-bus problem. Ideally, the 𝜙 of the solution should be kept as small as possible (or 𝜙<𝜖) (as in (5)). The figure has also showed that the quality of the solution is poor when the problem size is larger than six. The best performance is achieved for the 3-bus problem, as its run-time is minimum and its 𝜙≈ 0. In other words, problems of smaller size are more tractable. (a) (b) Figure 2. The performance (run-time and quality of solutions found) of GA for solving the inverse problems of different size: (a) the run-time required by the solver for finding an optimal solution; (b) the sum of differences in power flow at the interconnectors for the solutions found (a) (b) (c) Figure 3. A simple example of decomposition performed on a 4-bus network model. The model is separated as two sub-problems (SP1 and SP2) of three buses: (a) the original 4-bus transmission network model, (b) one of the decomposed sub-problems, created by virtually removing bus 4 from the original network, and (c) the net power generation at bus 1 is readjusted after the decomposition by subtracting 𝐹φΚ from its original value 6 Fortunately, the inverse problem can always be decomposed as several small sub-problems. As shown in Figure 3, a 4-bus network model can be separated as two sub-problems (SP1 and SP2). The ex-post power flow on all of the transmission lines is same after the separation. The net power generation on each of the common buses is adjusted by subtracting the power flow of the “disconnected” transmission lines from the original value. For example, the net power generation at bus 1 (as in Figure 3 (a)), 𝐹φϵ+ 𝐹φϯ+ 𝐹φΚ becomes 𝐹φϵ+ 𝐹φϯ in SP1 (Figure 3(b)) as the transmission line connecting bus 1 and bus 4 is virtually removed from it after the decomposition (Figure 3 (c)). Its validity could be explained by observing the 𝑃ք at a bus that co-exists in a sub-problem before and after the decomposition based on (3): 𝑃ք= ంBֆΔ𝜃ֆK ֆ=φ (8) where, k is the index of transmission line connecting bus i and j, and Δ𝜃ֆ is the difference in voltage angle of transmission line k, or Δ𝜃ֆ= 𝜃ք−𝜃օ. Before a decomposition is conducted, 𝑃ք at the bus is as follows 𝑃ք = BφΔ𝜃ք … + BK−φΔ𝜃K−φ+BKΔ𝜃K (9) The net power generation at the bus is reformulated as below after the decomposition as transmission line K is disconnected from it. The 𝑃ք is therefore, reduced by BKΔ𝜃K: 𝑃ք−BKΔ𝜃K = BφΔ𝜃ք … + BK−φΔ𝜃K−φ+BKΔ𝜃K−BέΔ𝜃έ (10) = BφΔ𝜃ք … + BK−φΔ𝜃K−φ As the Bֆ value is constant, the Δ𝜃ֆ must be equal in both (9) and (10). All of the remaining transmission line that are connected with bus i would retain the same value of 𝐹ֆ after the decomposition. Thus, each of the sub-problems can be solved independently. The fact that the power flow remain the same on the transmission line that co-exist in the sub-problems too allows the recombination of the solutions of the sub-problems to reconstruct the complete zonal model. This further justified the validity of the decomposition strategy. 2.2 The clustering of ex-post power flow data Another issue of the inverse modeling is the existence of multiple sub-optimal 𝑋∗ which fulfills the requirement of 𝜙≈ 0 in (5). Only one of them is the real solution of the problem. This is because of the existence of multiple 𝑋∗ that could yield the same B matrix used in (3). The existence of multiple solutions for a power flow problem is also a well-known issue [12]. This can be solved by including more than one ex-post power flow cases in (5) as follows [5] [13] 𝜙=ంం|𝐹ֆ∗Ӵ֎−𝐹ֆ֎|≈0έ ֆ=φϢ ֎ (11) where, s is the index of a power flow case. The cases has to be selected carefully. This is because, several ΓA actually exist in the zonal model. Each of them is implicitly related to a unique Π֕ [13]. In other words, several clusters of cases could be found in the ex-post data. The selected cases must be originated from the same cluster. Otherwise, the GA would converge to a solution with poor 𝜙 value in (11). Theoretically, the 𝛽֕⟶֕ɱӴֈA is calculated as follows 7 𝛽֕⟶֕ɱӴֈA=ం ঁం𝜋ւӴ֕𝛽ք∗⟶քɱӴֆֆং,∀𝑘∈𝐾ֈ,𝑧∈{𝑧Ј,𝑧} ւ,𝑧≠𝑧Ј (12) where, 𝜋ւӴ֕∈ Π֕ is the share of power generation committed by generator g in zone z to the generation of the next unit of power in the zone; 𝑖∗ is the index of the bus which is connected with generator g, 𝑧Ј is the index of the reference zone, and 𝐾ֈ is the set of indices of all of the constituting transmission lines of interconnector m. Because of the existence of varying Π֕ ,various 𝛽֕⟶֕ɱӴֈA (and thus the ΓA) could exist. Nevertheless, only a finite number of Π֕ exists in the actual operation of the transmission network and they tend to repeat themselves over the time. Consequently, the domain of the ΓA is as well finite. A simple heuristics approach has been developed in [13] for selecting cases that belong to a same cluster. It is formulated based on the characteristics of the cases of the cluster as follows a) Each of the zones in the cases would have an approximately identical value in power generation ratio, 𝑤֕P compare to their counterpart in the other cases of the same cluster. For each case, the 𝑤֕P is calculated by dividing the net power generation at zone z, 𝑃֕A by a weighting factor, W which is the largest absolute value of all of the 𝑃֕A in the case. For a case denoted with s, the 𝑊֎ is calculated as follows 𝑊֎=max(|𝑃֕AӴ֎|); ∀𝑧∈{1,…,Z} (13) Hence, the 𝑤֕PӴ֎ is defined as: 𝑤֕PӴ֎=𝑃֕AӴ֎ 𝑊֎ (14) The cases could potentially be part of the same cluster if they meet the following criteria: 𝑤֕PӴ֎≈𝑤֕PӴ֎ref; ∀𝑠∈{1,⋯,S}; ∀𝑧∈{1,⋯,Z} (15) where, 𝑠ref is an arbitrarily selected case within the same cluster. b) The cases that met the criteria of (15) could be part of the same cluster if they have an approximately identical value in their power flow ratio, 𝑤ֈΑӴ֎ compared with their counterpart in the other cases of the same cluster. The ratio is calculated by dividing the 𝐹ֈ֎ by the 𝑊֎ in (13): 𝑤ֈFӴ֎=𝐹ֈ֎ 𝑊֎ (16) The cases from the same cluster (possessing the same ΓբӴ∗) would meet the following criteria: 𝑤ֈFӴ֎≈𝑤ֈFӴ֎ref; ∀𝑠∈{1,…, S}; ∀𝑧∈{1,…, Z} (17) For instance, the 𝑤ֈF of I1, 𝑤φF of Figure 1 (b) of two arbitrary cases are formulated as follows. For the first case (case 1): 𝑤φFӴφ =𝐹φφ 𝑊φ (18) =𝑃ϵAӴφ𝛽ϵ⟶φӴφAӴφ+𝑃ϯAӴφ𝛽ϯ⟶φӴφAӴφ 𝑊φ =𝑤ϵPӴφ𝛽ϵ⟶φӴφAӴφ+ 𝑤ϯPӴφ𝛽ϯ⟶φӴφAӴφ and, the second case (case 2) 𝑤φFӴϵ =𝐹φϵ 𝑊ϵ (19) 8 =𝑃ϵAӴϵ𝛽ϵ⟶φӴφAӴϵ+𝑃ϯAӴϵ𝛽ϯ⟶φӴφAӴϵ 𝑊φ =𝑤ϵPӴϵ𝛽ϵ⟶φӴφAӴϵ+ 𝑤ϯPӴϵ𝛽ϯ⟶φӴφAӴϵ In (18) and (19), if 𝑤φFӴφ≈𝑤φFӴϵ and [𝑤ϵPӴφ,𝑤ϯPӴφ ]⊺≈ [𝑤ϵPӴϵ,𝑤ϯPӴϵ]⊺, then [𝛽ϵ⟶φӴφAӴφ,𝛽ϯ⟶φӴφAӴφ]⊺= [𝛽ϵ⟶φӴφAӴϵ,𝛽ϯ⟶φӴφAӴϵ]⊺. Both case 1 and 2 would have the same ΓAӴ∗. This is a strict clustering approach because the cases in each cluster must fulfil (14) and (16). In reality, the ΓAӴ∗ of several of such clusters are identical. This issue can be handled by testing a new case with ΓAӴ∗ of identified clusters. The cases belong to the same (identified) cluster if its ΓAӴ∗ could reproduce the 𝐹ֈ of all of the interconnectors in the new case. III The implementation The inverse modeling is implemented in five steps: (i) decomposition is performed by splitting the complete zonal network model into small sub-problems; (ii) the ex-post power flow data is collected for each of the sub-problems, and the 𝑤֕PӴ֎ and 𝑤ֈFӴ֎ is calculated for each of them; (iii) groups of (ex-post) power flow cases possessing closest possible 𝑤֕PӴ֎ and 𝑤ֈFӴ֎ values are selected from the ex-post power flow data; (iv) inverse modeling is performed on each of the groups to derive their respective 𝑋∗ and ΓAӴ∗ and (v) the ΓAӴ∗ derived is tested on all of the other (untreated) cases; and the cases which also achieved 𝜙≈ 0 in (5) when computed using the ΓAӴ∗ is automatically included as part of the group, thus expanding the size of the cluster. Figure 2. The workflow of the inverse modeling for computing ΓAӴ∗ The authors applied the k-Nearest Neighborhood (KNN) classification algorithm [14] in step (iii) for selecting the groups of power flow cases. The KNN is an efficient supervised machine learning algorithm for identifying cases of closest proximity in their 𝑤֕PӴ֎ and 𝑤ֈFӴ֎. The steps are repeated until all of the ΓAӴ∗ existed in the zonal model are fully identified. The complete work-flow for the inverse modeling is shown in Figure 2. 9 IV Test case The authors applied the method to a test case of three market zones in the south of Norway. They are market zone NO1, NO3, and NO5. The three zone network is a sub-problem decomposed from the Nordic zonal model published by Statnett [15]. The hourly data of the first week of 2020 is used as the input. In the week, the areas have a total of 37.3% of all of the hydro-reservoir of the Norwegian system [16], as shown in Table I. Table I. Hydro-reservoir in the Norwegian markets within the first week of 2020 Market zone Hydro-reservoir (in GWh) NO1 4007 NO3 5594 NO5 11182 All market zones 55699 Share (in %) 37.3% An example of a group of three power flow cases identified by the KNN is given in Table II and III. The tables respectively shows the net power generation and the inter-zonal power flow of the cases. They are the ex-post power flow data on January 2, in hour 11, 13 and 14) as well as their respective 𝑤֕PӴ֎ and 𝑤ֈFӴ֎. Evidently, the KNN algorithm could find cases with very close 𝑤֕PӴ֎ and 𝑤ֈFӴ֎ values. Table II. Net zonal power generation of the KNN identified cases Zone 𝑃֕A,1 𝑃֕A,2 𝑃֕A,3 𝑤֕PӴφ 𝑤֕PӴϵ 𝑤֕PӴϯ NO1 -1304.51 -1316.38 -1374.84 -1.00 -1.00 -1.00 NO3 601.54 609.30 648.04 0.46 0.46 0.47 NO5 702.97 707.08 726.80 0.54 0.54 0.53 Table III. Inter-zonal power flow of the cases in Table II Interconnector Start End 𝐹ֈφ 𝐹ֈϵ 𝐹ֈϯ 𝑤ֈFӴφ 𝑤ֈFӴϵ 𝑤ֈFӴϯ I1 NO1 NO3 -312.36 -314.24 -328.71 -0.24 -0.24 -0.24 I2 NO1 NO5 -992.15 -1002.14 -1046.13 -0.76 -0.76 -0.76 I3 NO3 NO5 289.18 295.06 319.33 0.22 0.22 0.23 The 𝑋∗ found by GA is shown in Table IV. The ΓAӴ∗ calculated with 𝑋∗ is shown in Table V. The 𝐹ֈ, 𝐹ֈ∗, and 𝜙 of three of the cases calculated using the 𝑋∗ is given in Table VI. The solution is tested on the other cases and found 14 additional cases that belong to the same cluster. All cases in the same cluster and their respective maximum error in approximation, 𝛾= max (|էՒ∗−էՒ| էՒ×100)) are listed in Table VII. The entire domain of ΓAӴ∗ could be fully identified by repeating the steps. As shown, the inverse modeling had identified a cluster in the ex-post power flow data of the first week of 2020. The cluster comprises 17 cases with highest 𝛾 of 8.03%. All of them share the 10 same ΓAӴ∗ that is given in Table V. The inverse modeling is repeated until all of the clusters are fully identified. It is evident that the ΓAӴ∗ has repeated itself within the zonal model as it appears in several consecutive hours of two occasions (from hour 9 to 21 in January 2, and from hour 9 to 11 in January 3). Because of the validity of the decomposition, the 𝑋∗ found in this sub-problem remain valid as they could be recombined with the 𝑋∗ of the other sub-problem to form the complete zonal model of the Nordic transmission system. Table IV. The 𝑋∗ found by GA for the KNN identified cases Interconnector 𝑋∗ I1 1.0000 I2 0.3099 I3 0.0139 Table V. The ΓAӴ∗ for the three bus model computed using 𝑋∗ (in Table IV) Zone Interconnector 𝛽֕⟶֕ɱӴֈA NO1 I1 - NO1 I2 - NO1 I3 - NO3 I1 -0.2446 NO3 I2 -0.7553 NO3 I3 0.7553 NO5 I1 -0.2341 NO5 I2 -0.7658 NO5 I3 -0.2341 Table VI. The 𝐹ֈ, 𝐹ֈ∗, and 𝜙 of the cases identified by KNN Case Interconnector 𝐹ֈ∗ (MW) 𝐹ֈ (MW) |𝐹ֈ∗−𝐹ֈ| (MW) 1 I1 -311.76 -312.36 0.60 1 I2 -992.76 -992.15 0.61 1 I3 289.79 289.18 0.61 2 I1 -314.24 -314.62 0.38 2 I2 -1002.14 -1001.77 0.37 2 I3 295.06 294.68 0.38 3 I1 -328.71 -328.71 0 3 I2 -1046.13 -1046.13 0 3 I3 319.33 319.33 0 𝜙 2.95 11 Table VII. All other cases in the same cluster of the ones shown in Table II and III Case Date Hour 𝛾 (%) 1 2020-01-02 9 5.16 2 2020-01-02 10 2.15 3 2020-01-02 11 0 4 2020-01-02 12 2.12 5 2020-01-02 13 0.21 6 2020-01-02 14 0.13 7 2020-01-02 15 6.82 8 2020-01-02 16 2.64 9 2020-01-02 17 8.03 10 2020-01-02 18 0.04 11 2020-01-02 19 0.06 12 2020-01-02 20 0.57 13 2020-01-02 21 4.16 14 2020-01-03 9 1.88 15 2020-01-03 10 2.59 16 2020-01-03 11 4.57 17 2020-01-04 20 6.03 V Discussions and Conclusions The use of the zonal model with correct representation of the inter-zonal power flow is beneficial to the long-term generation dispatch planning. A correct zonal model would reflect how the local planning of bulk power dispatch would affect the inter-zonal bottleneck in any part of the transmission network. This is very important for minimizing the corrections (and the associated opportunity costs) needed at the time of dispatch. The zonal model is presently the most important component in the FBMC design in Europe. Additionally, the policy makers in Europe prefer to adopt the zonal model for institutional barriers and socio-political reasons [3] [17]. The authors have pointed out that there are two approaches to conduct zonal modeling: (i) the model-based (i.e., bottom-up), and (ii) the data-driven (i.e., top-down) approach. The current state-of-the-arts are mostly model-based approach. Meanwhile, the data-driven approach is relatively unexplored. The usability of the model-based approach is limited for those who do not have the knowledge on the transmission network (especially the impedance of the transmission lines). Also, the direct dependency of the ΓA with the Π֕ could render it impractical. Then, the authors propose a data-driven approach as a better alternative for zonal modeling. The method takes the ex-post power flow data as input, and reconstruct all ΓA (the most important parameter of the zonal model) by means of inverse modeling. It is a highly tractable and feasible method as it inherits two significant characteristics of the linear power flow theory: (i) the ex-post power flow data can be decomposed into small sub-problems; where each of them can be solved independently. The sub-problems could be recombined with the others to reconstruct the complete 12 zonal model; and, (ii) the ex-post power flow data can be grouped into several clusters, where each of the clusters comprises a number of cases that have an approximately identical ΓAӴ∗. The clustering can be done because the ΓAӴ∗ tend to repeat themselves. Also, the ΓAӴ∗ is not sensitive to minor changes in Π֕( see [13] for more information). The use of the linear power flow theory for representing the inter-zonal power flow is reasonable as it deviates from a complete AC power flow analysis by about 5% on average for high-voltage transmissions [3]. The authors have contributed to the field of study by introducing an approach to effectively integrate the power flow characteristics within the zonal model for use in the long-term power dispatch planning. Many long-term simulation tools (e.g., the EMPS model) computes the inter- zonal power exchanges based on the network flow programming (i.e., the available transfer capacity) approach [18]. Although their solutions could be insightful, they are imprecise in reality because the modeling neglected many electrical characteristics [4]. Presently, there is also a need for a formal methodology to determine the future ΓAӴ∗ prediction of zonal prices beyond the day-ahead time- frame [4]. The inverse modeling can effectively identify all ΓAӴ∗ which had physically existed in the past. The trend and probability distribution of the identified ΓAӴ∗ at a certain time in the past (e.g., the seasonal factor) could serve as a valid reference for long-term future price determination. VI Acknowledgements This work would not be possible without the Nordic inter-zonal power flow data kindly published by the system operator of the Norwegian power system, Statnett (www.statnett.no) on the Internet