Grid inadequacy assessment for high power injection diversity Part II: Finding grid expansion options

Electrical Power and Energy Systems 118 (2020) 105831

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Grid inadequacy assessment for high power injection diversity Part II: Finding grid expansion options

T

⁎

Adonis E. Tioa, , David J. Hilla,b, Jin Maa a b

School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006, Australia Department of Electrical and Electronics Engineering, The University of Hong Kong, Hong Kong

A R T I C LE I N FO

A B S T R A C T

Keywords: Power system planning Future grids Inadequacy metrics Power injection diversity Transmission network expansion

Power injection scenarios will become more diverse in the future as intermittent renewable generation, dynamic loads, and energy storage devices become more prevalent especially under a market environment. Grid planners need new planning tools to find long-term grid expansion options that promote competition and equitable market access by accommodating diverse scenarios. Part I of the paper presented a framework and a set of metrics for measuring inherent grid inadequacy for high power injection diversity using the power flow infeasible set. This paper uses the ideas in Part I to develop an approach that finds grid expansion options by directly minimizing inherent grid inadequacy as objective. We present one implementation using a robust-like optimization model that minimizes the size of a scenario-based representation of the power flow infeasible set. We show using case studies that the proposed approach using inherent grid inadequacy metrics can identify solutions distinct from other approaches and better in some measure.

1. Introduction Increased uncertainty and diversity in future power injection scenarios poses a challenge in expanding transmission networks. One aspect of the Transmission Network Expansion Planning (TNEP) process under a competitive market environment is to ensure that the grid can accommodate spatiotemporal variabilities in bus power injections in the future within reasonable costs and with minimal preferential treatment towards some market players. This is a difficult task because it involves accounting for a diversity of power injection states that are in turn dependent on long-term uncertainties decades into the future [1]. In the operational timescale, there is uncertainty from (1) the inherent variability of intermittent generation and load usage and (2) the unpredictability in the response of market players to upcoming market mechanisms. The latter becomes more challenging to model when energy storage and dynamic loads gets integrated to the grid as market players gain greater flexibility in injecting and drawing power to and from the grid. To hedge against the increased diversity of future power injection, there is emerging interest in designing less inherently inadequate grid structures that can accommodate more operating conditions [2] and co-planning for grid flexibility to adapt to changing power injection circumstances [3,4]. There is limited research on these fields however and more work is needed to explore the merits of ⁎

integrating such ideas in TNEP. Part I of this paper presented a framework and a set of metrics for measuring inherent grid inadequacy using the power flow infeasible set. This work uses the ideas in Part I to explore the problem of finding grid structures that are less inherently inadequate to accommodate power injection diversity. Ref. [2] explored this problem from a graph-theoretic perspective by proposing a graph-based indicator of inherent grid inadequacy. We revisit the problem from a different perspective using an optimization model that directly minimizes inherent grid inadequacy as the objective. We do this by minimizing the size of a scenario-based representation of the power flow infeasible set that we defined in Part I as basis for defining inadequacy indices. We believe that this approach has not been explored before and may give expansion options that are distinct from that of existing approaches and better in accommodating more diverse scenarios. The proposed method is not intended to replace existing methods however, but rather provide a different way of identifying solutions that planners may further explore in later planning stages. The following are the major contributions of this work: a. We present and explore a new framework for finding grid expansion options using a different objective, that is, directly minimizing inherent grid inadequacy to accommodate power injection diversity. b. We propose one implementation using a robust-like optimization

Corresponding author. E-mail addresses: [email protected] (A.E. Tio), [email protected] (D.J. Hill), [email protected] (J. Ma).

https://doi.org/10.1016/j.ijepes.2020.105831 Received 15 August 2019; Received in revised form 5 December 2019; Accepted 3 January 2020 0142-0615/ © 2020 Elsevier Ltd. All rights reserved.

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allow renewable curtailment in some scenarios. As with [14] and [15], results in [19] and [20] can be sensitive to the chosen scenario pool. Ref. [21] used uncertainty intervals to constrain renewable and load variability instead of identifying discrete scenarios. The proposed model finds a solution that is robust against the worst-case curtailment scenario, which is also claimed to be robust against all other scenarios within the uncertainty interval. All three approaches in [19,20], and [21] however, assume that other generators are freely dispatchable and hence, do not attempt to find solutions that hedge against the diversity in bid-based market dispatch. With the increased penetration of intermittent generation, energy storage, and dynamic load technologies in the future, grid planners need to adapt planning tools to account for even greater power injection diversity. Emerging ideas in expansion planning research include the design of grid structures that are less inherently inadequate against a wide array of scenarios [2] and the co-planning of built-in flexibility to allow the grid to adapt to changing power injection circumstances [3,4]. In this work, we look at the former and explore the latter in a future work. Recognizing that expansion planning based on a limited number of scenarios may be inadequate, Ref. [2] proposed a different way that instead improves a grid’s intrinsic ability to absorb and deliver power. The work argued that maximizing the determinant of a matrix derived from the DC power flow susceptance matrix also maximizes inherent grid adequacy under special circumstances. This graph-theoretic framework that relates network inadequacy solely on a property of the susceptance matrix, however, ignores many practical constraints that limits its practical use. For example, the framework ignores the spatiotemporal distribution of the power injections and the flow capacity of lines – two important parameters that determine the network power flows and dispatch feasibility that the susceptance matrix alone cannot capture. Moreover, the work assumes that the susceptance and capacity of new lines can take on continuous numerical values. In practice, however, standard line configurations limit parameter values to discrete options. This work revisits the idea of inherent grid inadequacy and proposes an alternative implementation of finding less inherently inadequate grid structures that capture practical constraints ignored in [2]. We use the definition of inherent grid inadequacy that we presented in Part I based on the power flow infeasible set to develop an alternative objective in finding grid expansion options. Since the model focuses on minimizing inherent grid inadequacy, we cannot ensure that the solutions found are the best fit to other circumstantial technical and nontechnical requirements specific to each grid. However, we believe that there is inherent value in finding these solutions that planners or other researchers may adjust to their specific needs or find useful on their own.

model that minimizes the size of a scenario-based representation of the power flow infeasible set which can be solved through mixedinteger linear programming (MILP). c. We add to the limited literature of finding less inherently inadequate grid structures that we hope will motivate future work in this field. The rest of the paper is organized as follows. Section 2 reviews existing methods for finding grid expansion options that specifically deal with power injection diversity. Section 3 reviews the idea of inherent grid inadequacy and its relation to the power flow infeasible set, explores how grid expansion affects the size of the infeasible set, and presents an MILP model that minimize the size of the infeasible set to find grid expansion options. Sections 4 and 5 provide illustrative examples that showcase the merits of the proposed approach. Section 6 concludes and identifies areas for future work. 2. Related work Finding good grid expansion options is a difficult combinatoric problem with a large scenario space [5], a large solution space, and several conflicting objectives [6]. The problem is generally known as the TNEP problem in the literature even though in practice, TNEP is a much larger exercise wherein shortlisting of expansion options is only one step in a multi-stage process of increasing rigor and is often periodically repeated [7]. Interested readers may refer to [8–12] for extensive literature reviews of TNEP approaches with varying model complexities. We limit our scope to those that focus on operational diversity in power injection scenarios. Centrally-operated grids have been traditionally expanded using the worst case power injection scenario, e.g. the peak load dispatch, such as in [13]. Cost minimization subject to reliability constraints was a common optimization objective [8,9]. Back then, planning against the peak load was enough because the centralized dispatch gave operational control flexibility in avoiding grid congestion in other conditions. With market liberalization, generator dispatch was decentralized using bids from self-interested power producers. Moreover, public policy and technological advancements pushed for the integration of more intermittent renewable resources to the grid. Grid planners needed to update planning practices to include the provision of nondiscriminatory open-access. Different approaches were proposed. In [14], relevant scenarios are first identified using expert judgment or forecasting techniques, the least-cost expansion solution for each scenario is found, and a minimax regret approach is used to choose the best solution. Ref. [15] extended this approach to make a cost-risk curve that allows the planner to choose acceptable investment and risk levels. Since the scenarios are determined a priori, the approaches in [14] and [15] can be extended to a larger scenario pool by updating the scenario selection methodology. Other approaches integrate scenario identification inside a multilevel optimization model by simulating generation dispatch or market dispatch. Identification of solutions follow different criteria that often involve minimization of expected costs, minimization of maximum cost, or minimization of costs subject to a robustness criterion. Alternatively, social welfare is considered in some works instead of cost. Refs. [16,17] integrated a bid-clearing model inside a stochastic optimization model that maximizes expected social welfare. Ref. [18] used the worst-case market Nash equilibria to inform planning. These approaches allow for market modeling but the results are sensitive to market model and data assumptions and the models require large computational resources. Other approaches use simpler dispatch models to identify power injection scenarios. Ref. [19] used combinatoric sampling to identify renewable and load coincidence scenarios. Then, a cost minimization model was used to find expansion options that are robust against all identified scenarios. Ref. [20] used a clustering approach on historical data to identify renewable and load scenarios. The model relaxes the strict robustness requirement in [19] by adding chance constraints that

3. Proposed approach 3.1. Definition and scope In Part I, we noted that there are different levels of rigor to grid inadequacy assessment and planning depending on the intended purpose of the study, power system model assumptions, and system states selected. In this work, we assume that long-term uncertainties in expansion planning are implicitly embedded in the following given parameters obtained from a separate study: a reference grid topology, location of buses with positive, negative, or mixed power injections, minimum and maximum power injections at each bus, and a list of candidate line additions. The scope of grid inadequacy assessment and planning then hinges on power system modeling assumptions and state selection preferences. These include the following considerations: the degree of diversity of power injection scenarios assumed to describe system states, whether operational interventions such as load shedding, generation redispatch, 2

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space, via the following relationship that we derived in Part I from the DC power flow equations:

or congestion management interventions are modeled, whether contingency conditions are simulated, and how uncertainties in market bidding and dispatch are considered. We limit our definition of grid inadequacy to that of what we defined in Part I as inherent grid inadequacy. We used this term to refer to a grid’s inherent insufficiency to allow diverse power exchanges without deploying power flow control interventions including, but not limited to, generation or load redispatch, line switching, or Flexible AC Transmission System (FACTS) based control. To assess inherent grid inadequacy, we consider power exchanges in the power injection space bounded by bus power injection limits during normal conditions without line outages. By extension, information about inherent grid inadequacy can be used to qualify inherent grid adequacy. We can use the term global inherent adequacy to refer to the ability of grids to accommodate all possible power injection scenarios in the entire power injection space without intervention. On the other hand, we can use the term relative inherent adequacy to refer to inherent adequacy relative only to a subspace of the entire power injection space.

f

|f

Pis = 0

(1)

P min ≤ P s ≤ P max

(2)

(3)

s, g

| ≤ Cg

(4)

where |f| is the vector of absolute values of f and C is the vector of line capacities. A grid that is globally inherently adequate has no infeasible set. That is, the entire f-space hyperplane lies within the f-space bounding box. On the other hand, a grid that is relatively inherently adequate has a non-empty infeasible set. That is, the grid is only inherently adequate relative to the portions of the f-space hyperplane inside the f-space bounding box.

In this section, we summarize the key ideas relating inherent grid inadequacy and the power flow infeasible set that we discussed in more detail in Part I. Inherent grid inadequacy is closely related to the size of the DC power flow infeasible set. That is, the more inadequate a grid inherently is, the larger the size of the infeasible set. In Part I, we provided a visualization of this set for a three-bus three-line grid that we replicate in Fig. 1 for ease of reference. Similar to the example in the figure, power exchange scenarios s are defined by a vector P s = [P1s , P2s, ⋯, Pns ]T of bus power injections Pi for all i ∈ V g where V g is the set of buses in grid g . Valid power exchanges lie within a hyperplane in the power injection space, P-space, bounded by the following relations: i∈Vg

= X g K g (B g )+P s

where f is the vector of line power flows, X is the diagonal matrix of line susceptances, K is the line-to-bus incidence matrix, B is the DC power flow susceptance matrix, B+ is the pseudo-inverse of B, and the superscripts indicate dependence on s and g . Portions of the f-space hyperplane that fall outside a bounding hyperbox defined by the line capacity constraints comprise the power flow infeasible set. All points in this set violate one or more line flow constraints given by:

3.2. Inherent grid inadequacy and the power flow infeasible set

∑

s, g

3.3. Inherent grid inadequacy and grid expansion We can expand the grid to reduce inherent grid inadequacy by minimizing the size of the power flow infeasible set. By strategically adding new lines to the grid, changes to both the mapping function (3) as well as the f-space bounding boxes defined by C can facilitate the feasibility of previously infeasible power exchanges. It is also possible, however, that some previously feasible power exchanges become infeasible. Fig. 2 illustrates these possibilities by adding new lines to the three-bus three-line grid in Part I. The line susceptances are 1.0 p.u. for Line 1–2 and 0.5 p.u. for Lines 1–3 and 2–3. The power injection limits are 0.0–2.0 p.u. for Bus 1, −1.0–1.0 p.u. for Bus 2, and −1.0–0.0 p.u. for Bus 3. The line capacities are all 1.0 p.u. for existing and new lines alike. We use line loading range diagrams presented in Part I to visualize the feasible and infeasible sets in f-space in lower dimensions and to compare the inherent inadequacy of the different expansion options. Note that in the base case with no new line added, insufficient capacity in Line 1–2 prevents the dispatch of some power exchanges, e.g. total generation in Bus 1 cannot be at 2 p.u. to supply 2 p.u. of load. As such, Line 1–2 can be considered as a grid bottleneck. Adding a duplicate parallel line to either Line 1–2 or Line 1–3 can eliminate the infeasible set but for different reasons: (a) in Fig. 2b, adding Line 1–2 increases the corridor’s unconstrained line loading but the added capacity more than compensates for this increase while (b) in Fig. 2c, adding Line 1–3 reduces the unconstrained line loading of Line 1–2 in direction 1–2 to its capacity limit and eliminates the possibility of infeasible power exchanges. This means that adding another Line 1–2 or another Line 1–3 makes the resulting grid globally inherently adequate, i.e. any power exchange within the power injection limits is feasible even without generation rescheduling, load shedding, or other power flow control interventions. On the other hand, in Fig. 2d, adding Line 2–3 increases the unconstrained line loading in Line 1–2 in direction 1–2 without any additional capacity. This worsens inherent grid inadequacy and results in more power exchange combinations becoming infeasible. For instance, the number of infeasible scenarios increased from 26 in the base case to 31 with Line 2–3 added using a scenario pool with 1,225 power injection scenarios sampled from the power injection space. Based on these results, and depending on costs and other grid expansion objectives, planners may favor the addition of Line 1–3 if it is cheaper or Line 1–2 even if in case it is more expensive if additional capacity margins are desired.

where P min and P max are the vector of bus power injection limits, (1) is the power balance equation, and (2) constrain P s within limits. Points in the P-space hyperplane are projected into the line power flow space, f-

Fig. 1. The constrained DC power flow model interpreted as a vector mapping from the power injection space (P-space) to the line power flow space (f-space) showing (a) the set of valid power exchanges and (b) the infeasible subset resulting in grid congestion. 3

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pool of scenarios defined. Since the model uses a scenario-based representation of the power injection space and infeasible set, we first need to sample a pool of power injection scenarios from the whole power injection space. That is, discrete values of power injections must be assigned to each bus subject to (1)–(2). The sampled scenarios comprise the scenario pool S with |S | scenarios where |·| gives the cardinality of the set. Choosing the appropriate number of scenarios is particularly challenging because it requires a balance between adequate space representation and problem tractability. It is an important and sizeable research problem that we place outside our scope. Some possibilities include the following approaches. Probability distribution models can be used to generate S if historical data is available and correlation models can be assumed to hold in the future. Market simulation tools can also be used to generate likely scenarios. Otherwise, combinatoric sampling, if tractable, or targeted or worst-case sampling become viable options when the following conditions apply: (a) historical data is not available to develop trend and correlation models, (b) historical trend and correlation assumptions are deemed unlikely to hold in the future, (c) planners would like to explore solutions that offer some degree of hedging against previously unobserved trends, or (d) market simulation models are deemed unreliable or intractable. Even if historical data and market simulation models are available, sampling from previously unobserved scenario subspaces may still be necessary to check for inadequacy against these scenarios, and if so, to inform planners and stakeholders of the existence of such inadequacies. In the case studies, we use combinatoric sampling for a small test system and targeted sampling for a medium-sized test system for illustration purposes. We also use clustering techniques to reduce |S | for tractability. 3.5. Optimization model After constructing the scenario pool, the following MILP model can be used to find grid expansion options that minimize the size of the infeasible set represented by the number of scenarios in S that result in congestion:

Objective min |I | =

∑ zs s

(5)

Subject to Fig. 2. Line loading range diagrams for different grid expansion options for the three-bus three-line grid.

Pis −

∑ ∑ {fijks j∈V

s + f¯ijk } = 0, ∀ i ∈ V , ∀ s

k

(6)

3.4. Optimization framework for reducing inherent grid inadequacy

s fijk = Bijk (θis − θjs ), ∀ ijk ∈ L , ∀ s

(7)

As the example in the previous subsection illustrates, different grid expansion options result in different inherent grid inadequacy characteristics. The small example allows an exhaustive enumeration and assessment of available options but this is generally intractable for larger problems. Here, we present an optimization framework that can help address this problem. Unlike other works reviewed in Section 2 that minimize cost or maximize social welfare, the optimization framework that we propose finds grid expansion options by directly minimizing inherent grid inadequacy. We do this by minimizing the size of the power flow infeasible set. The infeasible set, however, is generally high-dimensional and difficult to characterize. In Part I, we proposed three approaches to measure the size of the infeasible set relative to the power injection hypersurface: one uses a scenario-based metric and the other two use metrics derived from lower-dimensional projections of the P-space and f-space hypersurfaces. Here, we adopt a robust-like approach that minimizes the number of infeasible scenarios in a scenario-based representation of the power injection space. We call the approach robustlike to describe infeasibility for a minimal subset and differentiate it from robust approaches that usually connotes feasibility for the entire

s f¯ijk − B¯ijk (θis − θjs ) − M1 x 'ijk ≤ 0, ∀ ijk ∈ L¯, ∀ s

(8)

s f¯ijk

(9)

− B¯ijk (θis − θjs ) + M1 x 'ijk ≥ 0, ∀ ijk ∈ L¯, ∀ s

s f¯ijk − M1 x ijk ≤ 0, ∀ ijk ∈ L¯, ∀ s

(10)

s f¯ijk

+ M1 x ijk ≥ 0, ∀ ijk ∈ L¯, ∀ s

(11)

s s fijk − Cijk − yijk ≤ 0, ∀ ijk ∈ L , ∀ s

(12)

s s − fijk − Cijk − yijk ≤ 0, ∀ ijk ∈ L , ∀ s

(13)

s s f¯ijk − C¯ijk − y¯ijk − M1 x 'ijk ≤ 0, ∀ ijk ∈ L¯, ∀ s

(14)

−

s f¯ijk

∑

s − C¯ijk − y¯ijk − M1 x 'ijk ≤ 0, ∀ ijk ∈ L¯, ∀ s

s yijk +

ijk ∈ L

θrs 4

= 0, ∀ s

∑ ijk ∈ L¯

s y¯ijk − M2 z s ≤ 0, ∀ s

(15)

(16) (17)

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∑

S . There are differences however, aside from the full decoupling of the scenario identification process discussed above. In [20], investment and renewable curtailment cost is minimized subject to a chance constraint that limits the prevalence of curtailment scenarios. In the proposed approach, we do the opposite case of finding the minimal infeasible subset subject to investment limits. The proposed model can be converted to a similar framework adapted in [20] by converting (5) into a constraint and converting (18) into the objective. The same conversion is not as straightforward using the chance-constrained model in [20]. While both approaches relaxes the strict robustness requirement in TNEP models such as in [19], the proposed approach provides a different relaxation that allow planners to find other solutions. In practice, generation and load rescheduling is commonly used to extend grid capability to operate beyond its inherent inadequacy limits. As such, using a strict inherent inadequacy criterion that does not allow for generation or load rescheduling is conservative. However, having solutions from models such as in Section 3.5 available is potentially useful. For example, these solutions can provide a benchmark of what is achievable in terms of providing fair access to market participants. More rigorous assessment models in succeeding planning stages can then be used to reveal whether it is desirable to meet this benchmark subject to circumstantial criteria, conditions, and policy targets specific to each grid.

x ijk ≤ x max

ijk ∈ L¯

(18)

x 'ijk = 1 − x ijk , ∀ ijk ∈ L¯

(19)

x ijk , z s ∈ {0, 1}

(20)

s s yijk , y¯ijk

(21)

≥0

where I is the scenario-based representation of the infeasible set, |I | is the size of I , z indicates whether a scenario is infeasible, s = 1, 2, …, |S | indexes the scenarios in S , Pi is the power injection in bus i, θi is the bus angle in bus i, V is the set of buses, x max is the maximum number of lines that can be installed, and M1 and M2 are large enough constants. For existing lines, fijk is the power flow from bus i to bus j in the kth parallel circuit, Bijk is the line susceptance, Cijk is the line capacity, yijk indicates the amount of line overload, and L is the set of existing lines. For candidate lines, the corresponding variables are f¯ijk , B¯ijk , C¯ijk , y¯ijk , and L¯ respectively and x ijk indicates whether a candidate line is installed or not. Eq. (6) enforces the power flow balance at each bus, (7) models the power flow in an existing line, (8), (9) is the counterpart of (7) for installed candidate lines, (10), (11) forces the power flow to zero for uninstalled candidate lines, (12), (13) models the amount of overload in existing lines, (14), (15) is the counterpart of (12), (13) for installed candidate lines, (16) models the existence of line congestion in a scenario, (17) sets the bus angle of a chosen reference node r to zero, (18) limits the number of candidate lines that can be installed, (19) relates x ijk to its logic inverse, (20) forces x ijk and z s to be binary, and (21) s s enforces yijk and y¯ijk to be non-negative.

4. Case Study: 6-bus test system 4.1. Overview As in Part I, we use the 6-bus test system to illustrate the method in a simple case study. The six bus test system initially has five interconnected buses and one bus that needs connection. Power can be injected in three and drawn from five buses. These power injections can represent contributions from either conventional or renewable generation, energy storage, or conventional or dynamic loads. There are fifteen candidate lines and a maximum of four new lines can be installed in a right of way. We adapt the network and loading data in [21] to be able to compare results later where the loads can vary by up to 105% of the original values in [13]. We construct three scenario pools S1, S2 , and S3 with |S | = 100, 500, and 1000 respectively for optimization purposes and a much larger pool S4 with |S | = 5, 098, 771 for comparing the quality of the solutions obtained. To compare solutions, we take the ratio of the number of infeasible scenarios in a pool with the pool size:

3.6. Model discussion Like other TNEP models, the number of continuous variables and the number of constraints are dependent on |V |, |S |, |L |, and |L¯|. But unlike other models, variables representing generator dispatch and load curtailment as well as constraints representing power balance and power injection limits are no longer needed because these are already captured when constructing the scenario pool. The introduction of line overload variables yijk and y¯ijk in (12)–(15) and the formulation of constraint (16) are unique to this model. This makes it possible to count the number of scenarios with network congestion and to use it subsequently in the objective function. However, this also increases the number of binary variables. Specifically, there will be |S | additional binary variables and |S | additional constraints representing (16). Since each binary variable adds to the chances of more branching during the MILP solution process, ample consideration should be made in choosing |S | to keep the resulting optimization model tractable while maintaining adequate space representation. One of the key features of the model is that the scenario identification step is fully decoupled from the optimization step as in [14]. The decoupling is only partial in [19,20] wherein only renewable and load variability are sampled and the uncertainty in market dispatch is ignored. Since the dispatch model is fully integrated in the optimization model in [19,20], the optimization process favors the dispatch of some generators over others in finding expansion options. Some planners who want to explore less discriminatory grids may thus find partial decoupling of scenarios ill-suited to their needs. Fully decoupling the scenario sampling step from the optimization process gives planners greater flexibility in defining the diversity of the scenarios represented subject to data availability, forecasting and model accuracy, and planner and stakeholder policy guidelines. Moreover, this can allow for a more transparent process wherein stakeholders know whether their interest for fair grid access are adequately represented in the scenario identification step, and if not, provide relevant feedback. The robust-like formulation is similar in idea to the chance-constrained formulation in [20] that allows for infeasibility in a subset of

SPIR = |I |/|S |

(22)

where SPIR stands for the Scenario Pool Infeasibility Ratio defined in Part I. In order to generate a high-fidelity sample of the power injection space, we use combinatoric sampling to construct S4 by specifying positive power injection levels for the three generating buses from [0, 5, 15, …, 85, 95, 100%] and negative power injection levels for the five load buses from [0, 5, 15, …, 85, 95, 105%]. We generate all power injection combinations and allow positive power injection setpoints to vary by up to ± 5% to balance the total load. Only the scenarios that satisfy the power balance equation are chosen as part of S4 . The smaller pools are obtained from S4 using the mini-batch k-means clustering algorithm [22] by specifying the desired number of clusters. Initial assessment using S4 shows that SPIR is 38.5% for the solution obtained using a deterministic approach that uses only the peak load dispatch and 43.0% for the solution given in [21] that uses a robust approach that determines the worst-case scenario within the defined uncertainty intervals. This means that in the absence of line switching or FACTS-based control, market clearing schedules that belong in the infeasible set need to be curtailed using a potentially more expensive generation dispatch or through load shedding. These results highlight the limitation of considering only the peak dispatch or assuming 5

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Table 1 Solutions found for different S and Σx, 6-bus. Σx

Case

SPIR

Cost*

Lines added 2–3

|S | = 100

|S | = 500

|S | = 1000

Deterministic Robust [21]

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 7 5

88.7% 83.0% 72.8% 55.0% 33.7% 26.0% 18.9% 14.9% 88.7% 80.3% 68.7% 55.0% 33.7% 26.0% 14.1% 8.4% 88.7% 80.3% 67.6% 55.0% 33.7% 21.4% 12.8% 6.7% 38.5% 43.0%

30 60 80 100 148 191 239 282 30 50 70 110 148 191 226 251 30 50 70 110 148 196 270 269 200 130

1 1 1 1 1 1 1 1 1 1 1

2–6

3 3 4

1 3 3 3

3

1

1 1 1 1 1 1

3–6

4–6

1

1 2 2 2 2

5–6

1 1 2

1

Fig. 4. Cost values corresponding to the optimal SPIR solutions.

1 1

1 1 1 1 1 1

3–5

2 4

1 1 1 1 1 1

1 1 1 1 1 1 1 1

1

2

2

3 2 1 1 1

Likewise for seven lines added, installing Line 3–6 or both Lines 3–6 and 5–6 contribute to improving SPIR by 50.8–66.8% for a 13.0–35.0% increase in cost relative to the deterministic solution as highlighted in the large circle. The best solution obtained has a SPIR of 6.7% with eight lines added including all six lines identified in other solutions. This is a reduction of 82.7% compared to the deterministic seven-line solution at just 34.5% added cost. As with other scenario-based approaches such as [14,15,19], and [20], the solutions obtained from the proposed approach can be sensitive to the scenario pool chosen. In the case studies tested for the 6bus test system, increasing |S | led to solutions with increasingly lower SPIR. This observed solution sensitivity to the scenario pool is inherent to scenario-based approaches. To account for this effect, planners may opt to construct multiple scenario pools then find the solution for each as in [19]. Planners may then consider for further investigation either high-ranking or all solutions obtained.

1

1 2 1 1

2 2 2 2 2 3

1

2 1

* in thousands of USD.

generator dispatch flexibility as in [21] in finding grid expansion options. By ignoring the uncertainty in bid-based dispatch, the solutions may not give the least discriminatory access to market participants. This presents an opportunity to find other solutions that have a lower SPIR that planners may consider to minimize the need for rescheduling or congestion management interventions, to reduce operation costs, or to promote market fairness and competition, albeit at an added investment cost.

4.3. Solution time Fig. 5 shows the solution times in seconds in logarithmic scale for the different optimization case studies in Table 1. The models were solved using Gurobi 8.0.1 [23] using the Artemis High Performance Computing system [24] with CPU cores that can clock up to 2.60 GHz. Depending on the model size, the test cases used 1–32 cores and 0.1–32.3 GB of memory. Solution times increased exponentially with x max as the number of possible combination of line additions balloon. Solution times also increased by about an order of magnitude each with an increase in |S | from 100 to 500 to 1,000. While some cases took a long time to solve to optimality, solution times are still practicable in a long-term expansion planning setting where there is a more relaxed time constraint. Depending on available computing resources, planners may need to empirically determine suitable values of x max and |S | for tractability, perform the optimization into smaller chunks by incrementally adding a few new lines, or terminate the optimization earlier without waiting for optimality.

4.2. Optimization results Table 1 shows the optimization results for the three scenario pools used and up to eight new lines added. Figs. 3 and 4 graph the SPIR and cost values in the table for ease of comparison. Despite the availability of fifteen distinct candidates, only six are commonly chosen as part of the solution. The proposed model identified lines 3–6 and 5–6 in multiple cases which are not part of the solutions given by either deterministic or robust approaches. By ignoring these options, grid planners can miss out on improving grid access fairness. For five lines added for example, Line 3–6 is chosen instead of a third Line 4–6 in the robust solution. This achieves a 21.6% improvement in SPIR for just 13.8% increase in cost as highlighted in the small circle in Figs. 3 and 4.

Fig. 5. Solution times (in log scale) for the different optimization case studies in Table 1.

Fig. 3. Optimal SPIR values as a function of lines added and |S |. 6

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Table 2 Comparison of feasible sets: five-line solution from robust-like approach vs. robust solution. % of scenarios in S4

Description

54.3% 12.0% 2.7% 31.0%

feasible in both feasible only in the robust-like solution feasible only in the robust solution [21] infeasible in both

Table 3 Solutions found for different S and Σx, 118-bus. Case

Σx

Lines added

SPIR

Time, (s)

|S | = 100

0 1 2 3 4

– 30–38 30–38, 65–68 8–30, 30–38, 65–68 8–30, 30–38, 65–68, 94–100

39.2% 26.5% 24.8% 22.1% 21.1%

20.97 39.29 439.63 727.99 1,395.63

case study however, further assessments are needed to determine whether the benefits of reduced inherent grid inadequacy outweigh the costs of adding new infrastructure, among other considerations.

4.4. Comparing alternatives Showing that a grid expansion option has the least inherent inadequacy level relative to other options for the same investment cap or number of lines added is often not enough to make a case in practice. Other criteria must be considered such as reliability, operational efficiency, economic feasibility, and alignment with policy targets. Finding grid expansion options is only an initial step and should be followed by a process of evaluating and comparing alternatives using multiple assessment methods and feedback from stakeholders. One such criteria for comparing alternatives is to compare the infeasible sets spanned by different competing solutions. Table 2 shows the proportion of scenarios in S4 uniquely spanned by the robust solution and the unique five-line solution obtained using the robust-like approach. It also shows the proportion of scenarios feasible and infeasible in both. The robust solution uniquely hedges against 2.7% of the scenarios in S4 while the proposed robust-like solution uniquely hedges against 12.0%. To decide between the two options, the likelihood of future scenarios and the risk of failing to hedge against such scenarios must be assessed. For example, there would be a stronger case for the robust-like solution if future power exchanges are more likely to fall within the space the solution uniquely spans and there are cost-effective risk mitigation measures available for the infeasible set it spans. The converse is true for the robust solution otherwise. Assessing which scenarios are more likely than others, however, is a challenging exercise because of the inherent uncertainty in predicting the future. This will inevitably depend on data availability, stakeholder input, and expert judgment specific to each grid.

6. Conclusion and future work Power injection scenarios will increasingly become more diverse in the future as more intermittent generation, energy storage, and dynamic loads get integrated to a market-based grid. Grid planners will need new tools to identify grid expansion options that account for this increased diversity. In this work, we reviewed existing approaches on modeling power injection diversity in grid expansion planning, noted their limitations, and identified emerging ideas to deal with this challenge. One such idea is the identification of less inherently inadequate grid structures that can accommodate diverse power exchanges without the need for generator rescheduling, load curtailment, or other power flow control interventions. In Part I, we defined and characterized inherent grid inadequacy using the size of the power flow infeasible set. Here, we proposed a robust-like approach to minimize the size of the power flow infeasible set by minimizing the number of infeasible scenarios in a scenariobased representation of the power injection space. We presented a mixed-integer linear program like existing TNEP models but can count and minimize the number of infeasible scenarios. Test results showed that the model can identify grid expansion options distinct from existing deterministic and robust approaches. The solutions found have reduced inherent grid inadequacy that is better than other solutions, albeit at an extra investment cost. Identifying such options give planners alternatives to consider especially when improving grid access fairness is an important objective. In the future, planners and researchers may find the study of inherently inadequate grids as important on its own or as a means towards more detailed models. However, there is little research on this field. This work adds to existing limited literature but more needs to be done. Other ways of measuring and reducing inherent grid inadequacy need to be explored. If a scenario-based approach is used, as is the case in this work, appropriate sampling approaches and efficient optimization algorithms need to be developed. Risk analysis methods need to be designed to quantify the risk of failing to hedge against families of infeasible power injection scenarios. Ways to supplement the performance of inherently inadequate grid structures with power flow control interventions such as topology, susceptance, power angle, or direct line flow control also need to be explored. Extension of the proposed approach to distribution grid inadequacy assessment and expansion planning will also be interesting future work.

5. Case study: 118-Bus test system We also perform tests on the 118-bus test system with 54 generating buses, 91 load buses, and 186 lines using the data in [25]. Since it will be intractable to define combinatoric generator and load levels as in the 6-bus case, we used a targeted sampling approach to define the scenario pool as in Part I. The process involves solving for the unconstrained loading range of each line, defining 100 loading levels for each line, and solving for the bus power injections that result in a given line loading level. This results in a scenario pool SA with |SA| = 18, 600 that we use for final inadequacy assessment. We then use k-means clustering to generate a smaller pool SB for optimization purposes with |SB| = 100. We identify 25 candidate lines from the original 186 by identifying bottlenecks comprised by lines with maximum feasible loading ranges at or within 10% of the line’s capacity as follows: 11–12, 23–24, 8–30, 26–30, 23–32, 30–38, 54–56, 64–65, 65–66, 65–68, 69–77, 68–81, 81–80, 77–82, 82–83, 92–93, 94–95, 82–96, 94–96, 80–98, 80–99, 94–100, 98–100, 99–100, and 17–113. Table 3 shows the SPIR and solution times for the 118-bus case studies for different values of x max . Without additional lines, 39.2% of the scenarios in SA need either generator rescheduling, load curtailment, or other congestion management intervention. This value can be reduced by almost half to 21.1% if four lines are added. This reduction translates to less rescheduling, load curtailment, or other interventions. It also contributes to less discriminatory grid access. As with the 6-bus

Funding The DOST-ERDT Faculty Development Fund of the Republic of the Philippines finances the Ph.D. degree program of A.E.Tio. They are not involved, however, in the study design, results analysis and interpretation, document writing, and the decision to submit this article for publication.

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CRediT authorship contribution statement

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Adonis E. Tio: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Visualization, Writing - review & editing. David J. Hill: Conceptualization, Validation, Resources, Writing - review & editing, Supervision. Jin Ma: Validation, Resources, Writing - review & editing, Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement A.E.Tio acknowledges the DOST-ERDT Faculty Development Fund of the Republic of the Philippines for sponsoring his degree leading to this research. The authors also acknowledge the University of Sydney HPC service at The University of Sydney for providing high-performance computing access and the Sydney Informatics Hub, a Core Research Facility of the University of Sydney, for providing technical training and assistance in using the HPC cluster. References [1] Velasquez C, Watts D, Rudnick H, Bustos C. A framework for transmission expansion planning: a complex problem clouded by uncertainty. IEEE Power Energ Mag 2016;14(4):20–9. [2] Thiam FB, Demarco CL. Transmission expansion via maximization of the volume of feasible bus injections. Electr Power Syst Res 2014;116:36–44. [3] Tee CY, Ilić MD. Toward valuing flexibility in transmission planning. In: Chen H, editor. Power Grid Operation in a market environment. Wiley Online Library; 2016. p. 219–49. [4] Li J, Liu F, Li Z, Shao C, Liu X. Grid-side flexibility of power systems in integrating large-scale renewable generations: A critical review on concepts, formulations and solution approaches. Renew Sustain Energy Rev 2018;93(March):272–84. [5] Rudnick H, Barroso L, Magazine E. A framework for transmission expansion planning, 2016;14(4):12–8. [6] Billinton R, Li W. Reliability assessment of electric power systems using Monte Carlo

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