6.2.1 Regulated Versus Deregulated

In a regulated environment, the power system has a vertically integrated structure. The system operator owns all transmission and generation assets. Its aim is to provide electrical energy to consumers in the most economical manner, while maintaining a certain level of security and reliability. In such an environment, the transmission and generation expansion planning are conducted together centrally, with the purpose of minimizing the overall system cost.

The separation of ownership of transmission and generation assets after deregulation complicates the planning process. The objective of long-term planning in the new environment is seen differently, depending on the perspective of the entity that is performing the assessment. The generation units seek to maximize their own profit. The TSO strives to maintain the reliability and security of the grid, at the same time providing nondiscriminatory access to electricity markets for all market players. The TSO also tries to find effective solutions that meet society’s needs and promote sustainability.

In the deregulated power system, TSOs have limited information on the future development plans of the generation units. Therefore, uncertainty has become a major element in the decision-making process of TSOs, especially with respect to the choice of capacity. For transmission corridors, operation issues such as the acceptance level of congestion and the structure of the markets have become the main elements of long-term transmission planning. The situation is even more complicated, as the expansion decisions made by one entity will affect the benefits and so the decisions to be made by other partners. For example, building a new interconnector that connects the Netherlands to Denmark will induce significant changes in market prices, power flows, and the revenues of all the interconnectors that are connected to either system. As a result, after the establishment of the competitive environment the traditional TEP frameworks are no longer viable. It is, therefore, necessary to review both the ways, in which the problems are formulated and the algorithms are used to solve them [9–13]. Today, new expansion investments must be evaluated for their possible economic implications (on the market prices), as well as their social implications (on the behavior of transmission investors and, from there, on the competitive markets [14]). “Cooptimization” or “anticipatory transmission planning” are the computer-aided decision-support tools that consider generation dispatch and investment decisions on transmission capacities, congestion, and from there, the network investments. The anticipatory TEP enables transmission investors to evaluate how different network configurations will change investment and operational decisions to be made by generation investors. The cooptimization TEP can be an effective tool for regulators, as well as investors, to better understand various risks, benefits, and costs when assessing resource options, and to identify improved integrated solutions [15].

6.2.2 Centralized Versus Decentralized Decision-Making Process

Developing power system infrastructure is an expensive and time consuming process. It is normally conducted over the course of several years. From an economic perspective, the cash flow of a transmission expansion project consists of a large initial investment followed by transmission revenues and minimal operating and maintenance costs that occur every year over the lifetime of the project. Therefore, planning analysis generally compares alternatives that have different costs/revenues at different points in time. In this section, the “time-value of money” concepts which are widely used in investment planning analysis are discussed.

Financing the investment through external resources is a common practice, especially in power system planning projects. Therefore, it is more sensible to base the analysis on loan payments, rather than investment costs from a commercial perspective. However, when considering the system-wide, social perspective, direct investment costs are used instead in the cash flow calculations. Transmission infrastructure can exist as either a merchant system or as a regulated resource infrastructure. In the case of merchant interconnectors, the transmission investment is a profit-driven exercise. In this regard, the transmission revenue generated every hour would be considered as the source for investment recovery.

In the deregulated electricity industry, two major paradigms of decision-making processes in transmission planning can be identified

Regulated investment is a common practice in Europe. Under this scheme, TSOs are promoted to invest, build and operate the interconnector [16]. Their task includes the construction, maintenance and operation of the interconnectors [17]. The cost of new investments is normally financed through regulated transport tariffs. The TSOs must agree upon the rate and obtain the regulator’s approval for implementing the tariff. Therefore, the TSOs need to convince their respective regulators that the extra investment(s) in the new capacities are socially beneficial, so that they obtain their approval and may procure the investment.

In addition to the regulated tariffs, a part of the investment may be recovered through transmission revenues the TSO receives from (implicitly) auctioning the interconnector capacities. According to Article 6(6) of Electricity Regulation (EC) No. 1228/2003, the TSO may utilize any revenues resulting from the allocation of the transmission interconnector only for the following purposes:

Large price differences between electricity market areas create incentives for market parties to invest in new interconnectors to explore financial potentials and capture transmission revenues. These private interconnectors are referred to as “merchant interconnectors” and are considered as a commercial alternative to the regulated TSO investments.

Merchant investment opens the way for profit-motivated investors to participate in electricity transmission projects (which has been so far considered a natural monopoly). Some argue that it is the only way to address the perceived problem of underinvestment in transmission system [18]. Moreover, in the absence of political willingness to increase the transmission tariffs for making new investments, merchant investment can be an option for transmission capacity expansion [19].

Merchant interconnections differ fundamentally from the regulated interconnectors in at least the three following points:

Regulated TEP is the state of practice. However, in recent years, decentralized merchant investment is penetrating some liberalized markets (e.g., Cross Sound Cable Interconnector [22], Path 15, TransBay cable and Green Line in the United States [23], Basslink in Australia [24], and Brit-Ned, Swe-Pol and Baltic Cable in Europe [25]). It is a move toward ending the transmission monopoly that prevailed for decades. For the moment, merchant transmission is viewed as a solution for large and risky investments, as would be the case for meshed offshore grids [23,25].

From an economic perspective, a major obstacle toward wider adoption of merchant transmission is the difficulty in determining the economic benefits and identifying the stakeholders, who may benefit from the facility. From a legal perspective, a good regulatory regime is required that provides opportunities for merchant investors to participate in transmission projects, when they are the most viable option.

By 2015, the EU had granted authorization to few merchant projects, such as BritNed, Estlink, NorGer, France-Angleterre (IFA), 3 the East West Cables, and a new interconnector between Austria and Italy. Merchant transmission is a new practice in Europe. The introduction of the merchant transmission investment has created new challenges for the regulators, especially from an institutional level [25,26]. In this regard, the centralized (regulated monopoly) and decentralized (merchant transmission) network management approaches result in distinctly different expansion decisions and grid designs. Shrestha et al. [26] proved that in theory (under restrictive mathematical assumptions) that the centralized expansion approach results in socially optimal network capacity, which ensures maximum benefit to society (i.e., electricity consumers and producers). They also show that in contrast to the centralized approach, monopolistic merchant entrants under a decentralized system will always result in underinvestment in transmission capacities. However, enforcing proper competition in transmission investments or implementation of transmission revenue rights can make it plausible to achieve near socially optimal grid expansion, even under decentralized expansion [27].

6.2.3 Deterministic versus Nondeterministic Methods

There is a significant level of uncertainty involved in the future development of generation, transmission, demand, and electricity markets. The uncertainty stems from the nature of various technical, environmental, economic, and social/regulatory factors.

In general uncertainty sources can be divided into two groups

There are two types of transmission planning algorithms for dealing with the uncertainties: Deterministic and nondeterministic. In the classic deterministic approaches, planning is performed for a reduced number of operating states that represent the worst case conditions, for example, peak load or generation out of service [30–32]. The planner seeks to find the most essential reinforcements that are required to attain planning objectives (a certain economic goal and/or reliability level) and, at the same time, operate within the acceptable operating range for transmission equipment. The advantage of this approach is that it gives a unique set of investment decisions.

The disadvantage of deterministic approaches is that uncertainties in various parameters, such as prices and load forecast, location of new power plants in the future and others, are not included in solving the problem [33]. Consequently, deterministic models are very likely to miss the whole picture, as the model looks at only a few snapshots of future scenarios and, therefore, their final results usually lack robustness. For example, by looking only at peak-load hours, the planning analysis fails to account for operating conditions of major interest that can happen during offpeak hours, in combination with high production from RES.

An alternative approach is to use scenario planning. In scenario planning, several scenarios are defined that represent future regulatory and economic conditions. For each scenario, an optimal transmission configuration is developed using deterministic TEP or a production costing-based comparison of predefined plans. Then, the resulted designs are aggregated and those that are attractive for most cases are identified as robust solutions [34].

The purpose of nondeterministic planning methods is to better capture uncertainties associated with the analysis of the random and nonrandom factors in the future scenarios. This can be done by considering various operating snapshots (multitime period formulation), to which a probability of occurrence is assigned. Note that in many cases, the probability denotes a degree of importance of the operating states that are considered. The nondeterministic approaches include: Information gap decision theory [35], probabilistic (Monte Carlo simulations [36], point estimate [37], scenario-based modeling [38]), interval-based analysis [39], robust optimization [40], hybrid possibilistic-probabilistic approach (fuzzy-scenario [41,42], fuzzy Monte Carlo [43,44]), or combination of them [39,45].

6.2.4 Static versus Dynamic Methods

Depending on the implementation horizon, TEP problems are classified into static and dynamic problems. In static TEP problems, the solution provides an optimal planning, assuming that system expansions are implemented instantly at a certain point in the future. The optimal design can include grid topology, transmission capacities, or merely a set of possible candidate reinforcements to attain a particular objective (e.g., minimize cost and maximize benefit) [6–48].

In reality, transmission network developments take place gradually in multiple development stages, because: (1) Building large infrastructures is costly and time consuming; (2) other parts of the power system and electricity markets develop in a gradual manner; (3) there can be disruptive delays that happen due to unforeseen technical and legal complications; and (4) the transmission development plan can adapt to aforementioned changing circumstances (closed loop rather than open loop design). Therefore, in addition to the optimal grid design, time restrictions should ideally be included in the analysis. Any static model will fail to provide information regarding the timing of the project development steps. In Section 4.3, a static framework for TEP in the North Sea is introduced.

There are three planning horizons: Short term (1–5 years), medium term (up to 10 years), and long term (planning horizon longer than 10 years) [49]. Dynamic TEP (DTEP) takes into account temporal continuity of expansion projects. Note that unlike the models discussed in [7,50], the term “dynamic” does not merely refer to a series of statically built-up plans. The optimal plan includes development strategy and timing considerations, in addition to the sizing and placement of the assets. Although DTEP is computationally intensive [7], it usually leads to more economically efficient grid design and development strategy [51].

6.4.1 Formulation of Transmission Expansion Planning Problem

In this section, weighted STEP frameworks that take into account the probability of occurrence of various system states (${\omega }^t$). The formulation carries the advantage of including all operating states relevant to the design, but compresses their representation.

Consider a power system with $n({\mathrm{\Omega }}_z)$ buses indexed $i=\mathrm{1,2},\dots ,n({\mathrm{\Omega }}_z)$. ${\mathrm{\Omega }}_z$ is the set of all buses. Each bus is assumed to be a price area of producers and consumers that participate in a zonal competitive market. For a hybrid AC and DC system, the STEP problem takes on the form

$\max \mathrm{\Omega }=\varphi -\psi$ (6.1)

(6.1)

Subject to

$\sum _{g\in G_i}{P}_{G_g}^t-\sum _{d\in D_i}{P}_{D_d}^t=P_i^t,\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.2)

(6.2)

$P_i^t=\sum _{j\in {\mathrm{\Omega }}_z}{f}_{\mathit{ij},\mathrm{AC}}^t+\sum _{j\in {\mathrm{\Omega }}_z}{f}_{\mathit{ij},\mathrm{DC}}^t,\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.3)

(6.3)

$P_i^{\min ,t}\le P_i^t\le P_i^{\max ,t},\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.4)

(6.4)

$f_{\mathit{ij},\mathrm{AC}}^t={N_{\mathit{ij},\mathrm{AC}}·f}_{\mathit{ij},{\mathit{cbl}}_{\mathrm{AC}}}^t,\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.5)

(6.5)

$f_{\mathit{ij},\mathrm{DC}}^t={N_{\mathit{ij},\mathrm{DC}}·f}_{\mathit{ij},{\mathit{cbl}}_{\mathrm{DC}}}^t,\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.6)

(6.6)

$f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{AC}}}^t\le f_{\mathit{ij},\mathrm{AC}}^{t,\max }·w_{\mathit{ij},\mathrm{AC}},\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.7)

(6.7)

$f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{DC}}}^t\le f_{\mathit{ij},\mathrm{DC}}^{t,\max }·w_{\mathit{ij},\mathrm{DC}},\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.8)

(6.8)

$\begin{array}{l}{f}_{\mathit{ij},{\mathit{cbl}}_{\mathrm{AC}}}^t-g_{\mathit{ij},\mathrm{AC}}[{(v_{i,\mathrm{AC}}^t)}^2-v_{i,\mathrm{AC}}^t·v_{j,\mathrm{AC}}^t·\cos ({\delta }_i^t-{\delta }_j^t)]+b_{\mathit{ij},\mathrm{AC}}{v}_{i,\mathrm{AC}}^t·v_{j,\mathrm{AC}}^t·\sin ({\delta }_i^t-{\delta }_j^t)\le \\ (1-w_{\mathit{ij},\mathrm{AC}})·M,\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}\end{array}$ (6.9)

(6.9)

$f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{DC}}}^t-g_{\mathit{ij},\mathrm{DC}}[{(v_{i,\mathrm{DC}}^t)}^2-v_{i,\mathrm{DC}}^t·v_{j,\mathrm{DC}}^t]\le (1-w_{\mathit{ij},\mathrm{DC}})·M,\forall i,j\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.10)

(6.10)

$N_{\mathit{ij},\mathrm{AC}}\ge 0,\forall i,j\in {\mathrm{\Omega }}_z$ (6.11)

(6.11)

$N_{\mathit{ij},\mathrm{DC}}\ge 0,\forall i,j\in {\mathrm{\Omega }}_z$ (6.12)

(6.12)

$0\le P_{G_g}^t\le P_{G_g}^{\max ,t},\forall g\in G_i,\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.13)

(6.13)

$0\le P_{D_d}^t\le P_{D_d}^{\max ,t},\forall d\in D_i,\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.14)

(6.14)

${\delta }_i^t=0,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.15)

(6.15)

${-\delta }_{\mathit{ij}}^{\max }-(1-w_{\mathit{ij},\mathrm{AC}})·M\le {\delta }_i^t-{\delta }_j^t\le {\delta }_{\mathit{ij}}^{\max }+(1-w_{\mathit{ij},\mathrm{AC}})·M,\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.16)

(6.16)

$v_{i,\mathrm{DC}}^{\min }\le v_{i,\mathrm{DC}}^t\le v_{i,\mathrm{DC}}^{\max },\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.17)

(6.17)

$v_{i,\mathrm{AC}}^{\min }\le v_{i,\mathrm{AC}}^t\le v_{i,\mathrm{AC}}^{\max },\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.18)

(6.18)

$w_{\mathit{ij},\mathrm{DC}}\in \left\{0,1\right\},\forall i,j\in {\mathrm{\Omega }}_z$ (6.19)

(6.19)

$w_{\mathit{ij},\mathrm{AC}}\in \left\{0,1\right\},\forall i,j\in {\mathrm{\Omega }}_z$ (6.20)

(6.20)

The first term in the objective function (6.1),

$\varphi =\sum _{t\in {\mathrm{\Omega }}_{\mathcal{O}}}\left[\sum _{i\in {\mathrm{\Omega }}_z}\sum _{d\in D_i}{P}_{D_d}^t·{\lambda }_{D_d}^t-\sum _{i\in {\mathrm{\Omega }}_z}\sum _{g\in G_i}{P}_{G_g}^t·{\lambda }_{G_g}^t\right]·{\omega }^t$ (6.21)

(6.21)

presents the social welfare over the whole planning horizon. The second term in the objective function,

$\psi =\frac{1}{2}\sum _{i\in {\mathrm{\Omega }}_z}\sum _{j\in {\mathrm{\Omega }}_z}[w_{\mathit{ij},\mathrm{AC}}(k_{\mathit{ij}}^{\mathrm{AC}}+k_{\mathit{ij}}^{\prime \mathrm{AC}}·N_{\mathit{ij},\mathrm{AC}}·f_{\mathit{ij},\mathrm{AC}}^{t,\max }·L_{\mathit{ij}})+w_{\mathit{ij},\mathrm{DC}}(k_{\mathit{ij}}^{\mathrm{DC}}+k_{\mathit{ij}}^{\prime \mathrm{DC}}·N_{\mathit{ij},\mathrm{DC}}·f_{\mathit{ij},\mathrm{DC}}^{t,\max }·L_{\mathit{ij}}\left)\right]·n({\mathrm{\Omega }}_{\mathcal{O}})$ (6.22)

(6.22)

denotes the total investment cost of building AC and DC transmission infrastructures. Note that the investment cost includes both investment and installation costs as proposed in Ref. [79].

In the formulation above, ${\mathrm{\Omega }}_t$ is a set of all operating hours included in the planning horizon. ${\mathrm{\Omega }}_Z$ is the set of indexes of all price zones. $G_i$ is the set of indexes ($g$) of all generating units in zone $i$. Likewise, $D_i$ is the set of all indexes ($d$) of the demands located in zone $i$.

Eq. (6.2) enforces power balance at every operating scenario. Eq. (6.3) defines the power injection $P_i^t$ of zone $i$ into the rest of the system. Constraint (6.4) enforces power injection constraints, and $P_i^{\min ,t}\le 0\le P_i^{\max ,t}$. Constraint (6.4) limits the combined power import/export of every country over AC and DC connections to certain range. Note that the value of $P_i^{\min ,t},P_i^{\max ,t}$ might be determined based on security provisions that every price zone considers.

In the context of this work, the AC interconnector connects zone $i$ and $j$ is assumed to compose of a number $N_{\mathit{ij},\mathrm{AC}}$ of identical parallel AC cables $f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{AC}}}^t$, each with conductance $g_{\mathit{ij},\mathrm{AC}}$ and susceptance of $b_{\mathit{ij},\mathrm{AC}}$. In a similar manner, the DC interconnector connecting zone $i$ and $j$ is assumed to composed of $N_{\mathit{ij},\mathrm{DC}}$ of identical parallel DC cables $f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{DC}}}^t$, each with conductance $g_{\mathit{ij},\mathit{dc}}$. Eqs. (6.5) and (6.6) define the power flow over respectively AC and DC interconnectors connecting zone $i$ and $j$.

Eqs. (6.7) and (6.8) express the Kirchhoff’s Voltage Laws as disjunctive constraints for the candidate cables, for AC and DC technology, respectively. Note that $w_{\mathit{ij},\mathrm{AC}}$ and $w_{\mathit{ij},\mathrm{DC}}$ are integer variables and take a one value for integer that exist. For AC interconnectors that exist, $w_{\mathit{ij},\mathrm{AC}}=1$ constraint (6.7) limits the power flow of AC cables, to the maximum capacity. Likewise, for DC interconnectors that exist $w_{\mathit{ij},\mathrm{DC}}=1$ and (6.8) limits the power flow of DC cables to the maximum capacity. For solutions with near-zero $N_{\mathit{ij},\mathrm{AC}}$ and $N_{\mathit{ij},\mathrm{DC}}$ values, $w_{\mathit{ij},\mathrm{AC}}=w_{\mathit{ij},\mathrm{DC}}=0$ and therefore the constraints become inactive.

In a similar manner, for $w_{\mathit{ij},\mathrm{AC}}=1$ and constraint (6.9) becomes an equality constraint equivalent to $f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{AC}}}^t-g_{\mathit{ij},\mathrm{AC}}[{(v_{i,\mathrm{AC}}^t)}^2-v_{i,\mathrm{AC}}^t·v_{j,\mathrm{AC}}^t·\cos ({\delta }_i^t-{\delta }_j^t)]+b_{\mathit{ij},\mathrm{AC}}{v}_{i,\mathrm{AC}}^t·v_{j,\mathrm{AC}}^t·\sin ({\delta }_i^t-{\delta }_j^t)=0$. For $w_{\mathit{ij},\mathrm{DC}}=1$, constraint (6.9) becomes an equality constraint equivalent to $f_{\mathit{ij},{\mathit{cbl}}_{\mathrm{DC}}}^t-g_{\mathit{ij},\mathrm{DC}}[{(v_{i,\mathrm{DC}}^t)}^2-v_{i,\mathrm{DC}}^t·v_{j,\mathrm{DC}}^t]=0$. For $w_{\mathit{ij},\mathrm{AC}}=w_{\mathit{ij},\mathrm{DC}}=0$, Eqs. (6.9) and (6.10) do not constrain the power flow in the left-hand side since M is an adequately large positive number.

Constraints (6.11) and (6.12) enforce the number of AC and DC cables respectively to be a positive real number. Eq. (6.13) states that each generator must produce below its capacity. Eq. (6.14) states that power consumed by each consumer must be a value between zero and its capacity. Eq. (6.15) sets the angle of the reference zone to zero. Constraint (6.16) enforces the fact that the angle difference between the two ends of a line cannot exceed a certain level. Note that $v_i^t$ is per-pole line-to-ground voltage of DC converters at each end of the bipolar interconnector. Due to operating limits, the DC voltage of the converters is bound between 0.9 p.u. and 1.1 p.u. As power flow of each DC cable is precisely controlled through voltage control of the converters, zonal voltages are considered independent decision variables. Therefore, Eq. (6.17) ensures that the voltage of the DC converters at the gate remains within acceptable operational range. Eqs. (6.18) and (6.19) define the integer variables $w_{\mathit{ij},\mathrm{AC}}$ and $w_{\mathit{ij},\mathrm{DC}}$ as discussed above.

Only the steady state operating conditions are considered. Thus, the power system is assumed to be dynamically secure. That is, the DC converters are utilized with a robust control system (e.g., using voltage margin method [4] or voltage droop control [80]) which maintains transient stability of the system after a disturbance. The results of the optimization formulation (6.1)–(6.17) include an angle, a voltage (independent optimization variables), and an injection (dependent optimization variable) for each converter, which represent its steady-state operating point. Therefore, the optimization model determines the reference values of the voltage angles, DC-voltage as well as the supply/demand bids of the generation unites and consumers in each zone, for each operating state. In addition, the optimal solution includes grid topology and transmission capacities to meet the objective function. The optimization problem (6.1)–(6.17) is a multiperiod, nonlinear, nonconvex, mixed-integer and large-scale optimization problem. Therefore, it has a high degree of complexity and is computationally very intensive to solve.

6.4.2 Simplifying the Problem Formulation

6.4.2.2 Continues Social Welfare

As nonlinear, step-wise, aggregated supply–demand curves are more difficult to work with, we assume that aggregated supply–demand curves are linear function of power generation and consumptions. The linear supply and demand curves are presented in Fig. 6.1 and read respectively as follows:

${\lambda }_{G_g}^t=a_{G_g}^t·P_{G_g}^t+b_{G_g}^t,\forall g\in G_i,\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.23)

(6.23)

${\lambda }_{D_d}^t={-a}_{D_d}^t·P_{D_d}^t+b_{D_d}^t,\forall d\in D_i,\forall i\in {\mathrm{\Omega }}_z,\forall t\in {\mathrm{\Omega }}_{\mathcal{O}}$ (6.24)

(6.24)

where $a_{G_g}^t$ and $a_{D_d}^t$ are the slope of the supply and demand curves, $b_{G_g}^t$ is the lowest price producers require before participating in any production activity, and $b_{D_d}^t$ is the maximum price the consumers are willing to pay for the electricity.

This assumption allows determination of the changes in social welfare (i.e., incremental social welfare) as a quadratic function of power injection of each zone in the rest of the system (see Fig. 6.2) and, therefore, makes the problem quadratic and so, easier to solve. It also enables implicit modeling of the operation of the onshore electricity markets and their responses to the power trades.

Note that one could use more complex assumptions (i.e., nonlinear supply/demand curves). Then one would need to define social welfare explicitly as a function of total quantities demanded and supplied. This makes the objective function nonsmooth and more difficult to solve. Note that, for such nonsmooth problem, the Karush–Kuhn–Tucker (KKT) optimality conditions cannot be applied, and therefore, the analytical solution is not accessible.

6.4.3 Static Transmission Expansion Planning

In this section, we present a weighted STEP framework that takes into account the probability of occurrence of various system states. This formulation carries the advantage of including all operating states relevant to the design, but compresses their representation⁵.