Stochastic AC Transmission Expansion Planning: A Chance Constrained Distributed Slack Bus Approach With Wind Uncertainty

Integrating more renewable energy sources into the grid leads to a more vulnerable power system and challenges for power system planners. This paper proposes a probabilistic overload constraint based AC transmission expansion planning model. In terms of economic dispatch, generation adequacy and network constraints are assessed using a probabilistic participation factor power flow. Generation participation factors, which are handled in the AC power flow equations, are used to manage mismatch power that comes from uncertainties as well as transmission loss of the system in a distributed slack bus concept. Combinations of load, wind and N-1 contingency uncertainties are evaluated with Monte Carlo Simulation and the loading limits for the existing and candidate lines and violations on bus voltages are enforced as probabilistic constraints using chance constraint programming. The investment decisions taken under various operating conditions are re-evaluated under a new set of wind power uncertainty that differs from those utilized in the optimization process, and the resulting TEP decisions are analyzed from a risk perspective. The proposed method allows for the uncertain operating conditions to be easily and accurately adapted to the ACTEP optimization. The optimization results show that the participation factor power flow is a promising tool for evaluating probabilistic operating conditions, which can be used to explore the possibility of violations under different uncertainty conditions.


Initial number of lines between buses i and k n ik
Integer number of new lines added between buses i and k n ik Maximum number of added lines between buses i and k

I. INTRODUCTION
As the conventional power plants are retired, more renewable energy sources (RES) will be linked to the grid due to environmental targets and market incentives. Wind turbines and solar panels are becoming more affordable, which means they may now compete with fossil fuels. However, integrating large amounts of variable RES makes the system more vulnerable and poses challenges for power system planners. Especially, load growth, unavailability of network assets and the availability of RES power need to be considered in the longer term to maintain supply reliability and minimize operational risks [1]. Long-term planning benefits from integrating renewable energy sources, but the stochastic nature of these resources is affecting generation dispatch and power flow patterns in unpredictable ways. Furthermore, including network constraints and RES placement has a substantial impact on network expansion and introduces new obstacles on decision-making such as operating cost, power quality, power imbalances and the impacts on transmission planning [2], [3]. Therefore, it's essential to properly address network behavior to assess the impact of uncertainty and develop reliable and economically viable investment decisions.
There are several approaches to solve stochastic transmission expansion planning (TEP) problem in the literature. The key distinction between these models is in how uncertainty is described and how an optimization with uncertainty is approached. Traditional stochastic TEP problem requires a scenario based or chance constrained model that obtains investment decisions under uncertain operating conditions. These models require the uncertainty scenarios to be modeled through probability density functions (PDF). Risk of overlimit can be obtained from these PDF and it can be included either in the objective function or the constraints of the planning optimization. In this way, an acceptable level of violation probability is considered, which intuitively relaxes the hard constraints imposed on TEP optimization.
Stochastic TEP optimization requires solving a set of equations that represent network change under uncertain scenarios. One of the methods used to model a stochastic TEP optimization is to partition the problem into investment and operation sub-problems. The investment part is an optimization that minimizes the total investment cost, while the operation part ensures the probabilistic operational constraints of transmission loading, generation, and voltage magnitude. Therefore, obtaining the system behavior under uncertain operating conditions is a challenging sub-problem for an adequate and secure operation. The sub-problem of TEP optimization can be handled in two ways: the overload minimization approach [4] or the load curtailment minimization approach [5]. The first one reduces the loading of overloaded lines by investment, and the second one quantifies how much of the load will be curtailed so that the network power balance is achieved without violating the loading constraints.
In power system analysis, the overloading information can be easily implemented by power flow studies. Besides, instead of satisfying all network constraints for all scenarios, the risk of violation can be handled using probabilistic constraints in the optimization.
In the recent literature, stochastic TEP problem can be solved either robust optimization [6]- [8], two or multi-stage stochastic optimization [9]- [13] and chance constrained programming [4], [5], [14]- [19]. In this study, we work with chance constraints under TEP optimization, which ensures the system constraints will be satisfied with a prespecified probability. In recent literature, chance constraints are widely applied in stochastic TEP optimization. In [4], a scenario-based approach for load and wind uncertainty is analyzed using probabilistic power flow in MCS, and chance constraints on line loading constraints are handled in TEP optimization. In [5], load, wind, and N-1 contingency uncertainties are analyzed through scenarios with an acceptable probability of load curtailment. In [4], [5], both authors consider a scenario-based approach for handling uncertainty, and the resulting optimization is solved by probabilistic power flow without considering operational cost. However, in [5], the probability of having load curtailment is obtained through generation adequacy problems instead of the line overload probability found in probabilistic power flow. In [14], generation expansion and load uncertainty are handled using Monte Carlo Simulation (MCS), and the line capacity violations having an acceptable probability are modeled through chance constrained TEP optimization. In [15], a three-line overload risk index is considered under TEP optimization with load, wind, and N-1 contingency uncertainties, and the proposed formulation gives a comprehensive risk control model. In [16], an equivalent linear model for stochastic congestion management is proposed using chance constraints on transmission lines. In [17], the authors quantified the probabilistic load curtailment degree by a capped load curtailment probability, which is incorporated into the multistage TEP model. In [18], the authors propose a stochastic TEP framework to assess the impacts of wind power penetration and demand response which incorporates risk on load curtailment constraint. In [19], the authors propose a chance constrained TEP model that utilizes a large portion of wind power generation with a high probability. It is inferred from the recent literature that chance constraint programming is beneficial when the risk is incorporated into the long-term expansion studies.
From modelling perspective, TEP problem is either modeled by DC or AC network models. DC model formulation generally used to represent the network and the electrical quantities in the literature. In [20], the transmission network is represented by the DC model and TEP is solved considering multiple generation scenarios. In [21], the impact of wind and solar generation on TEP is evaluated with comparison of DC and AC optimal power flows models. In [22], a fourstage methodology using DC and AC power flow models is presented examining the effects of wind generation on the TEP model. In [23], a bi-level DCTEP model is presented, which prioritizes minimizing investment costs and maximizing system reliability while making maximum use of wind energy. In general, the nonlinearity of AC power flow equations in power system planning is not considered frequently due to its mathematical complexity. However, the AC power flow model has been attracted researchers to reformulate the nonlinear equations for a more accurate representation of power system operation. In [24], a TEP model that includes a linear representation of reactive power, off-nominal bus voltage magnitudes, and network losses is proposed. In [25], an analytical reformulation based on a partial linearization of the AC power flow equations is presented to maintain generation, power flows, and voltages within their bounds with a pre-defined probability. In [26], a long-term stochastic co-optimization of generation and transmission planning model is presented by using a linearized AC power flow model to include bus voltages, reactive power, and network losses. In another promising study [27], to reduce the computational effort for transmission network planning when using AC model, a meta-heuristic concurrent approach for network and reactive power planning is presented in which DC and AC model are used sequentially. In a similar context, a mixed AC and DC power flow model is proposed for transmission expansion planning in the presence of wind farms and a metaheuristic method is used to cope with complex models due to uncertainties of power networks and nonlinearities of AC power flow [28]. In another study, a strictly pure AC power flow model based on penalty functions is used to solve TEP problem by using a meta-heuristic technique without considering uncertainty and generation cost [29]. In [30], AC based TEP model is transformed into a MILP environment and the dimensionality, and the mathematical computation burden of this method are pointed out particularly when the size of the power system gets larger. For this reason, a two-stage optimization strategy is presented in which the DC model is used in the first stage to provide an initial estimation for the AC model to reduce the computational effort when N-1 safety constraints are included into AC model based TEP problem. In another TEP study, the exact AC power flow equations are used to include the effect of reactive power and the power system losses on the planning process [31]. In the context of planning, DC power flow is accurate enough to produce decisions for long term planning problem, although the planning results with AC power flow must then be evaluated [32]. In [33], an ACTEP model associated with reactive power planning is presented by considering reliability through expected energy not supplied index. In [34], a threestage procedure is presented to solve TEP with AC model based on generalized Bender's decomposition. In another study, a robust TEP with AC load flow is presented considering intermittent renewable energy generation and loads [35]. In [36], the Pareto optimization method is used to analyze the effect of the DC and AC network models and wind generation on TEP objectives and optimal solutions. In [37], investment cost of new lines and reliability cost are considered in the objective function of TEP problem with AC power flow under different uncertain load conditions.In this current paper, the AC network model is used in the operational sub-problem which handles probabilistic AC network constraints under TEP optimization.
As renewable energy becomes more integrated, the ability to adjust to variability becomes more important [5], [6], [21], [38], [39]. One way to respond to this variability is to change generator outputs depending on the imbalance power. Slack bus generator is a mathematical necessity and has a strong ability to deal with any deviations to produce a feasible result, which is not always possible in realistic operating conditions. Especially in the case of uncertainty, the slack bus absorbs all the uncertainties [40]- [42]. For example, replacing conventional generators with renewables that have lower voltage support has resulted in a larger risk of voltage instability as well as huge overloading levels in transmission lines. For this reason, treatment of slack bus is crucial in planning context. Distributed slack bus approach is an alternative to reduce the burden on slack bus. When it is adopted, the mismatch power and the transmission losses are assigned among conventional generators using participation factors [43]. Several studies in the literature have combined the participation factors with power flow studies to find an economical dispatch of generators. The pioneer work is presented in [44], in which the treatment of slack bus is introduced under decoupled economic dispatch using participation factor power flow. In [45], the generator contributions are derived by reformulating the power flow problem in terms of distributed slack bus concept. The net power mismatch resulting from each flow is a function of all flows present in the system where the power mismatch can be compensated for by allocating the net power imbalance among different generating units according to the participation factors. In [46], the participation factors are incorporated into the three-phase power flow equations of Newton Raphson formulation, and this distributed slack bus concept is applied to generator domains of unbalanced distribution systems to anticipate the growth of distributed generators. In [47], authors proposed a modified power flow approach in a sense of economic dispatch to remove the burden on the slack bus. In [48], a distributed slack bus formulation with an N-1 contingency is adopted in the probabilistic power flow to distribute the active power mismatch across all the thermal units in the system. In [49], a fast adjustment methodology is proposed for real-time economic dispatch problems considering distributed slack bus approaches with load, RES, and power loss considerations. Recently, in [50], authors re-examine the concept of distributed slack bus definition and compare the power flow solutions with a single slack bus model. It is evident that the distributed slack bus concept with the participation factor power flow is a promising tool for the evaluation of different operating conditions of the power system and it is also compatible with economic dispatch studies.
In this paper, we propose a stochastic AC transmission expansion planning with a chance constrained distributed slack bus approach. The generation adequacy and network constraints are evaluated using a probabilistic participation factor power flow in a sense of economic dispatch. It is worthy of note that the significant effect of slack bus is usually ignored to obtain a feasible result in the operational part of the optimization. When considered with RES, the slack bus absorbs the uncertainty in power networks. For this reason, the mismatch power from uncertainties and transmission losses is managed using generation participation factors combined with AC power flow equations. Combinations of load, wind, and N-1 contingency uncertainties are evaluated under MCS and the loading limits for the existing and candidate lines, as well as bus voltages, are enforced as probabilistic constraints using chance constraint programming. The following contributions have been made to this paper: • The probability density functions of line flow and bus voltages are obtained with combinations of wind, load and N-1 contingency uncertainties under nonlinear AC power flow equations.
• The generation adequacy is handled using a probabilistic participation factor power flow in a sense of economic dispatch under TEP problem.
• Chance constraint programming is applied through directly probability density functions of both bus voltage and line flow in TEP optimization.
• Distributed slack bus concept is adapted to the stochastic TEP by combining participation factors with nonlinear AC power flow equations through a modified Jacobian matrix under Monte Carlo Simulations. The remaining part of this paper is organized as follows. In Section II, uncertainty modelling, mathematical formulation of the participation factor power flow, and the stochastic TEP optimization framework is introduced. Section III gives the case studies implemented in the proposed TEP model for different risk levels, and the obtained plans are also evaluated under a new set of uncertainties to demonstrate the effectiveness and feasibility of the proposed framework. In section IV, discussion on the results is given. Lastly in section V, the conclusions are presented.

II. METHODOLOGY
In this paper, a combined Monte Carlo Simulation/probabili stic AC power flow analysis with distributed slack bus concept is first presented to handle the probability density functions of line flow and bus voltage in chance-constrained transmission expansion planning. In this context, a two-level optimization model is proposed for the TEP problem. The representation of the proposed TEP framework is summarized in Fig. 1. As it can be seen in Fig. 1 while the upper level deals with probabilistic constraints and minimization of line investment plans, the lower level is devoted to obtaining probability density functions to be used in chance constrained programming considering simultaneous wind, load, and N-1 contingency uncertainties. The proposed model requires an iterative VOLUME 10, 2022 process where AC power flow analysis is performed to characterize of uncertainty through probability density functions. However, while balancing powers in the conventional power system analysis, the slack bus absorbs uncertainty [40]- [42]. In other words, the single slack bus concept required in the conventional pure power flow analysis is inadequate to represent power system behavior under MCS. Besides, a power mismatch due to uncertainty should be allocated to some dispatchable generators in an economical manner [43]. In that respect, we need a modified power flow analysis that handles both burden on slack bus and the economic scheduling of generators under uncertainty to obtain probability density functions for the chance constrained TEP problem. Therefore, the generation participation factors, and AC power flow equations are combined through a modified Jacobian matrix to form a distributed slack bus concept at the lower level of TEP problem. In this way, the generation adequacy is simply handled using a probabilistic participation factor power flow in a sense of economic dispatch under TEP problem.

A. UNCERTAINTY MODELLING
The uncertainty scenarios generated in the MCS consist of the combination of load, wind speed and N-1 contingency. Load uncertainty is modelled through normal distribution N(µ, σ 2) with µ is the mean value and σ 2 is the variance. Wind uncertainty is generated using the Weibull distribution according to its scale and shape parameters. The power characteristics of the wind turbine are determined according to the wind speeds and the probability density function of the wind power output is obtained by using (1).
where Pr is the rated value for wind power [4], [5]. Transmission availability is determined by the forced outage rate (FOR), which is sampled from a standard uniform distribution to determine the failure of transmission lines. The outage rate for transmission lines is assigned as 1%. The selection of the contingency set is based on the lines that connect either the wind turbines or the weaker parts of the network. For simplicity, all uncertainties are considered independently.
In the operational sub-problem of the optimization, 100 scenarios are evaluated according to probability distributions given above.

B. OPERATIONAL SUBPROBLEM: PARTICIPATION FACTOR POWER FLOW (PFACPF)
In this paper, the generation adequacy and network constraints are evaluated using a probabilistic participation factor power flow in a sense of economic dispatch. The concept of slack bus is a necessity for finding a feasible solution in power flow analysis. It is required since transmission losses are not known as priori. Therefore, a generator is selected to operate as a reference to cover the losses in order to obtain a feasible solution. This is called as single slack bus model. In the economic dispatch, the difference between generation and demand refers to system power imbalance and the distributed slack bus concept is adopted with participation factors [44], [45], [48], [50]. In this way, the imbalance power is distributed among the conventional generators based on specified participation factors. In the uncertain operating conditions, the treatment of slack bus behavior is crucial. The uncertain characteristic of load and intermittent RES power deviate active and reactive power injections of buses, which directly change the power flow patterns and bus voltages throughout the system. In that manner, the conventional loadflow equations, which express a power balance at each system bus, are modified to include participation factors, to distribute the real power mismatch to each generator. The following section provides the information to find optimal operating points for generators and the participation factors.
Traditional economic dispatch problem [44], [47], [49], [51], [52] finds the real power generation for each plant providing the minimum generation cost that satisfies the power balance equation for the network. Lagrange multiplier method is typically used to solve such problems and the equal incremental cost is denoted as λ which is also the Lagrange multiplier of the Lagrange function is given in (2) [53]- [54].
where C i (P Gi ) is the total generation cost and C i is the generation cost equation of each plant which is a quadratic function of generator active power C i = c i + b i P Gi + a i P 2 Gi . In the traditional formulation total system generation is equal to total load and loss of the system. However, in our study, transmission losses and mismatched power from uncertainties are represented as P IMB . The derivatives which are given in (3) and (4) with respect to equal incremental cost and generator output are solved together to find the coordination equation for all generators.
It is inferred from the solution given in (5) that generator output is a function of λ. The analytical solution can be obtained for λ by substituting the (5) into (4). Consequently, the equal incremental cost λ found in (6) is the optimal scheduling of the generation.
56800 VOLUME 10, 2022 The first part of the (6) is the scheduled generator powers when mismatch does not consider in the formulation. In the second part, a portion of mismatch power is added to the first part to satisfy the power balance in the system. In the proposed model, the generator output which is used in power flow analysis is determined using (7), where the P Gi_upd is equal to the optimal scheduling of generation λ. The division of the mismatch power is calculated through the participation factors which is determined by using the generator cost function as in (8).
The participation factors calculated in (8) are all positive, which means that all participating generators will increase its scheduled power when the net power imbalance is positive. Additionally, the sum of participation factors for all generators must be added to one. In our study, the participation factors and nonlinear AC power flow equations are handled together with a modified Jacobian matrix. Therefore, to find the state vectors V and θ, the injected power equations for each bus are written as in equations (9) and (10).
The specified (known) parameters for each bus are calculated in (11) and (12). The updated generator output and participation factors are determined by (7) and (8) which can be used in the modified power flow solution. Subsequently, the mismatch power for each bus is determined by the (13) and (14) as follows: The modified Jacobian matrix consist of derivate vectors with participation factors are given in (15).
The size of the Jacobian matrix is N × N , where the N is the total number of buses in a power system. An artificial variable is introduced in the mismatch vector. Proposed modified power flow solution continues until the convergence tolerance is achieved for the iterative solution. The final calculated state vectors V and θ are used to determine power flows which is required in the TEP optimization. The generators are also bounded by their limits and the bounds are checked after power flow solution. If the generator exceeds its limit, the output power of the generator is fixed at its limit, and then participation factor power flow is re-applied to the case with that fixed generator outputs.
In the TEP optimization, the uncertainty scenarios are evaluated with participation factor power flow under MCS, and the PDFs of line flows and voltage magnitudes are determined to be used in chance constraint framework. The chance constrained method is one of the promising approaches for solving optimization problems under various uncertainties. A probabilistic or chance constraint is expressed as an inequality in generic way in the optimization process whose to find the best solution which maintain feasible with probability at least 1 − ε (β), for a given violation probability of ε. In this study, the probabilities are calculated based on the calculation of not violation probability which is required to be more than a pre-specified confidence level. The not violation probabilities are calculated using probability density functions and these probabilities guide the planning optimization by constraint checking mechanism. Consequently, PDF of the complex power flows are constructed after the MCS with the given formulation in (16) to (27). The superscript c means the contingency condition parameters in each Monte Carlo sample.
S c,to ik = (P c,to ik ) VOLUME 10, 2022

C. SECURITY CONSTRAINED STOCHASTIC ACTEP FORMULATION
In this part, the mathematical formulation of the stochastic TEP model with probabilistic participation factor power flow is given. The operational sub-problem given in the previous section is applied to each candidate network infrastructure to simply obtain the probabilistic constraints in both base and contingency operating conditions. The power network is represented by the AC model, and the mathematical formulation for the chance constrained AC TEP optimization is given as follows: The above TEP optimization finds the minimum investment decision that satisfies the given operational constraints on line flow and voltage. The objective function in (28) is a cost function that minimizes the investment cost for transmission lines that satisfies the probabilistic over limits for base and N-1 contingency conditions for each network decision. The equation set from (29) to (31) gives the base case whereas (32) to (34) give the contingency conditions. The parameters with superscript c show the modified variables when a transmission line (i,k) is outage. Lastly, bounds of the network expansion variables is given in (35).

D. FLOWCHART OF THE ACTEP PROBLEM
The mathematical formulation of ACTEP is a mixed-integer non-linear problem, and it is solved using the genetic algorithm (GA) solver. As a metaheuristic optimization technique, genetic algorithms are of great interest for solving the TEP and many other problems [4], [27], [55]- [58]. In this study, the main problem for GA is the investment optimization that minimizes the total investment cost subject to probabilistic line loading and voltage constraints and the number of invested lines. In the constraints of GA, the participation factor power flow is adapted under MCS to find the output power of conventional generators in each of the sample scenarios. The distributed slack bus concept is utilized in the AC power flow equations with generator participation factors. Subsequently, chance constraint programming is applied to the PDF of the state vectors to find the overload probabilities of network constraints. The resulting probabilistic constraints trigger the TEP investment decision according to prespecified risk levels instead of satisfying the hard constraint. The flowchart of the proposed TEP framework is summarized in Fig. 2, and the solution process is described as follows: • Block 1: At this stage, load, wind power, and N-1 contingency data are prepared using the distributions given in Section-A. The acceptable level of non-overload probability is predefined for the network constraints. The parameters for GA, such as population size, generation, crossover, mutation, and elite count are specified, and lastly, the initial population for the evaluation is created.
• Block 2: The population vector, which consists of various TEP infrastructures, is applied to the system one at a time. Line impedance and the bounds of transmission lines are updated according to the candidate TEP infrastructure.
• Block 3: MCS begins by sampling various operation scenarios to apply the uncertainty data to the system. Subsequently, scheduling of the generators is calculated, and the participation factor power flow is applied to the candidate infrastructure to find the state vectors Pg, Qg, |V|, θ and line power flows. The formulation for this block is given in Section-B. This part is evaluated for both base and contingency case conditions. Convergence of the MCS is checked based on a pre-specified sampling number.
• Block 4: Using the chance constraint formulation, the PDF of the state vectors is constructed, and the overload probability of each line and bus voltage is determined. This part is also evaluated for both base and contingency case conditions.
• Block 5: Transmission line loading and voltage magnitude constraints are checked, and the objective function is calculated. If the constraints are not satisfied, a new infrastructure is generated by selection, crossover, and mutation operations and by re-applying the procedure from block (2) to (5).
• Block 6: The optimization is terminated if the maximum stall generation limit is checked, resulting in the final investment decision. The selection, crossover, and mutation operations generate a new infrastructure if the termination does not satisfy. The best population with the least fitness value must be reached to find an optimal investment decision.

III. SIMULATION STUDIES AND RESULTS
The proposed stochastic TEP framework is implemented in the MATLAB environment and tested on IEEE Reliability Test System [59] and IEEE 118 Bus test system [60]. The comparison of power flows that consider single and distributed slack bus approaches is analyzed in the first stage of the study using cumulative probability distribution functions of line loading and voltage violations for the base and contingency networks. In the second stage, TEP investment results are obtained under different uncertainty conditions. The final stage involves testing the investment plans obtained under a new set of uncertainty conditions, examining the investment plans with respect to various confidence levels.

A. PROBABILISTIC COMPARISON OF CONVENTIONAL AND PROPOSED PARTICIPATION FACTOR POWER FLOW
This study is conducted to show the overload probability and cumulative distribution function (CDF) comparison between conventional AC power flow (ACPF) and the proposed participation factor power flow (PF-ACPF) on the IEEE RTS 24 bus network. In the ACPF, the scheduled generator power for generators is determined using in (7) to make a fair comparison between two models. The uncertainty data for the test network is given as follows. The mean value of the load is increased 25% from its original value. σ /µ ratio of load is considered 20%. Weibull characteristic of wind speed is scale=7, shape=2.5 and the turbine characteristic is as follows Vci=4, Vr=10, Vco=22 m/s. Wind turbine is located at buses 3, 4, 5 and 6 which is given in Fig. 3 and its rated capacity is 50 MW for each turbine. L4-9 is considered as the N-1 contingency of the system. MCS is used to obtain the PDF of line loading and voltage magnitudes and the violation probabilities for each line (L) and voltage magnitude (V) is given in Table 1. It is observed that lines connected to slack bus generators are significantly overloaded in the ACPF model. This is due to the difference in generation dispatch for each model.  In ACPF, the slack bus generator (G1) is responsible for the mismatched power in the network. However, in PF-ACPF, all active power mismatch is distributed in the conventional generators using participation factors. In this way, distributing imbalance among generators reduce the violation on the lines connected through slack bus generator. The results shows that the biggest differences between two power flows occur VOLUME 10, 2022 when the system has stressful uncertain operating condition. In particularly, the violation probabilities of lines 2-4, 2-6 and voltage magnitudes at bus 3, 4, 5 and 6 differs due wind turbine connections. It is also observed that the system is heavily loaded from the buses 6 to 15 so that in the most stressful scenarios the ACPF supplies the load by slack bus generator. When the system has an outage condition, the overload information is important for dispatch procedure. It is observed in Table 1 that the overload probabilities of lines are increased for both models. However, the overload probability of the lines connected to slack bus is less in PF-ACPF model due to distribution of mismatch power. As an example, the CDF of power flow on line 1-3 and voltage magnitude at bus 4 for without N-1 contingency are given in Fig. 4. In Fig. 4a, the capacity of this line is 175 MVA and it is observed that the violation probability of the ACPF higher than the PF-ACPF due to single slack bus consideration. On the other hand, in Fig. 4b, the violation probability of voltage magnitude at bus 4 has extended to different margins. Line 1-3 and voltage magnitude at bus 4 for with N-1 contingency is given in Fig. 5. The loading margin line 1-3 is different for both models. The result showed that using the single slack bus method to derive violation probability values directly in the chance-constraint optimization does not accurately reflect the probability of violation in a realistic way. For this reason, in our study, the simple computational capability of power flow is adapted to the TEP with uncertain operating conditions. It is crucial to assess this behavior in a planning context since it may lead to different investment decisions.

B. IEEE 24 BUS TEP DECISIONS AND COSTS SATISFYING DIFFERENT NOT-OVERLOAD PROBABILITIES
In this section, the proposed model is applied to the IEEE RTS 24 bus network with respect to different not-overload probability consideration. The uncertainty data for the test network given in Section-A is used in the optimization. Lines L3-9, L4-9, L5-10, L6-10, L8-9, L11-13, L12-13, L15-21, L16-19, and L18-21 are selected as N-1 contingency candidates and FOR of each line is assigned from a standard uniform PDF to define its unavailability. In the upper level of the optimization, there are thirteen variables for IEEE 24 bus test system. The GA parameters for the IEEE 24 bus test network is assumed as population size is 200, total number of generations is 100 and the crossover fraction is 0.8. We consider four candidate lines for each possible corridor and three case studies are established as follows: • Case-1: Considering only load and wind uncertainty • Case-2: Considering only load and N-1 contingency uncertainty • Case-3: Considering load, wind, and N-1 contingency uncertainty where the confidence levels are chosen between 85 to 100%. The resulting TEP investments for case studies are given in Table 2. An example for the convergence of the proposed method for TEP-0.95 in case-3 is shown in Fig. 6.
It has been observed that the proposed model provides alternative TEP decisions to the planner by choosing a pre-specified risk level. As the non-overload probability β decreases less expensive TEP decisions are obtained for all case studies. The comparison between case-1 and case-3 shows that, the inclusion of N-1 contingency increases the investment cost significantly. However, as the risk decreases, the difference between the investment cost of the expansion plan decreases. It is observed that adequate configurations can be obtained with using PF-ACPF with and without considering N-1 contingency. The security consideration increases the cost and the number of lines invested through conventional generators. Particularly, lines 1-2, 10-11 and 20-23 are the most invested lines in each risk level.  Besides it also observed that, the cost of difference between two risk levels both in cases 1 and 3 decreases as the planner chooses to take risk in planning decision.
Additionally, the comparison between case-2 and case-3 show that the integration of wind turbines increases the investment cost in each risk level. Network is heavily loaded in the lower part and with the inclusion of wind turbine connection both line loading and voltage magnitude violations are observed. For example, when β = 85% TEP plan of case-1 is investigated with a new set of uncertainty data, lines 2-4, 5-10 and 12-23 takes risk up to 11% however the major violation is observed on voltage magnitudes at bus 6 and 8 especially in N-1 contingency constrained network consideration.
As the non-overload probability β increases both line loading and voltage violation probability is reduced with additional investments. It is also concluded that with a small additional investment in the system, the probability of line loading and voltage magnitude could be kept at a satis-factory level considering wind power uncertainty. Another interesting observation can be seen on the TEP plans where β = 90%. In case-2, lines 12-23 and 1-2 takes risk up to 10% and the voltage at bus 8 has a violation probability of 5% and 9% for both base and contingency network consideration. When the wind uncertainty is considered in case-3, lines 1-2 and 20-23 are significantly affected both line loading and voltage violations. With the addition of 7 transmission lines in case 3, the network can accommodate the increased planning risk due to wind uncertainty to meet the same standard of system adequacy and security with minimal budget. The expected generation cost for each plan is also calculated for each case study. It varies between 15015 $/h -15083 $/h, 30804 $/h -31033 $/h and 29939 $/h -30091 $/h for different β in Case-1, Case-2, and Case-3, respectively. Expected generation cost for each plan increases as the beta decreases. Yet, the most crucial point in the proposed model is to rationally model PDFs of voltages and line flows in response to uncertain operating conditions with distributed slack bus. Distributed slack bus concept is mainly used to ensure the distribution of mismatch power under uncertainty operating conditions in an economical manner. In other words, the probabilistic constraints guide the line investment optimization with a sense of economic dispatch.

C. ASSESSMENT OF IEEE 24 BUS NETWORK TEP DECISIONS UNDER DIFFERENT UNCERTAINTY SCENARIOS
In this section, the obtained three TEP plans in Case-3 are evaluated under different wind speed conditions to find the probabilities of line overloading and voltage violation. Load and N-1 contingency is sampled from the same uncertainty set that was used in the optimization procedure whereas wind speed scenarios are sampled from four different Weibull  characteristic. The scale and shape parameters of different Weibull characteristics of wind speeds are given, and the wind power PDF of each wind cases are given in Fig. 7. Low, medium, extremely high, and high wind speeds were assigned to cases ranging from 1 to 4. Table 3 shows the probability of line loading and voltage magnitude violations under various wind speed conditions when Case-3 TEP decisions are implemented in the network. The analysis of different wind speeds clearly shows that the probability of voltage violation increases, particularly for lower risk and wind speed conditions. In wind case-1 contingency constrained test results, the voltage at bus 6 has a 16% probability of overload, and the voltage at bus 8 has a 10% and 5% probability of overload at higher risk levels. The bus 8 has reached its overload limit. For each risk level in wind case-2, voltage at bus 6 is violated its limit beyond its risk level in contingency constrained network. To address voltage security issues, two capacitors with capacities of 5 and 6 MVAR are added to buses 6 and 8, respectively. After that, the same testing procedure is applied to the TEP plans and the resulting violation probabilities are given in Table 4. It is observed that the voltage magnitude violations at buses 6 and 8 have fallen under acceptable levels. Based on the test results, we can determine which lines will be overloaded for a given uncertainty scenario with different wind speed considerations. However, this test can be done repeatedly to examine the range of overload for each line and to investigate the impact of both load and wind speed characteristics on system power flow. In this way, the probabilistic behavior of uncertainties will be better reflected in the overload assessment. Therefore, 50 MCSs were computed using the uncertainty characteristics of wind speed, load, and N-1 as in the optimization. In this way, the overload probabilities are given as an interval for each line, and the uncertainty assessment can be handled comprehensively. As an example, the violation probability margins of line voltage magnitude at bus 6 for contingency constrained network is given in Fig. 8. It is observed that voltage magnitude at bus 6 has different probability margins due to the combination of a new set of uncertainties. Particularly in varying wind speed conditions, voltage magnitude at bus 6 is prone to have risk of violation; so that the overload margin information becomes critical for guiding planners' decision. In Fig 8a, the outliers of the overload probability increase by up to 11-12% at lower wind speeds (wind-1 and wind-2). However, the mean overload probability for each TEP decision is almost below its pre-specified not overload probability. The overload probability margin observed in wind case-3 is lower than in other wind cases for each TEP decision since it has the highest wind speed characteristic. There is no violation on TEP-0.95 decision; so, it is not given. The violation probability margins of voltage magnitude at bus 6 are shown in Fig. 9 with and without capacitor addition. Without the addition of a capacitor, the outliers for overload probability increase to 18% for the higher wind speed condition in Case 4, and to 20% for the lower wind speed condition. After the capacitor is added, the overload probability margins for each wind case are determined to be less than the accepted confidence level. Consequently, based on the test results it is observed that the overload probability margin provides a detailed information for assessing the overload and violation for each TEP decision, which would be valuable when considering various risk levels in TEP optimization.

D. IEEE 118 BUS TEP DECISIONS AND COSTS SATISFYING DIFFERENT NOT-OVERLOAD PROBABILITIES
In this section, the proposed model is applied to the IEEE 118 bus network with respect to different not-overload probability consideration. The uncertainty data for the test network is given as follows: The mean value of the load is increased 20% from its original value. σ /µ ratio of load is considered 5%. Weibull characteristic of wind speed is scale=7, shape=2.5 and the turbine characteristic are as follows Vci=4, Vr=10, Vco=22 m/s. Wind turbine is located at buses 43,44,45,47,48,50,51,52 and 53 its rated capacity is 10 MW for each turbine. Lines L43-44, L44-45, L45-46, L46-47, L47-49, L48-49, L49-50, L51-52, L52-53, L53-54, L50-57, and L51-58 are selected as N-1 contingency candidates and FOR of each line is assigned from a standard uniform PDF to define its unavailability. In the upper level of the optimization, there are thirteen variables for IEEE 118 bus test system. The GA parameters for the test network is assumed as followingly: Population size is 100, total number of generations is 200 and the crossover fraction is 0.8. We consider three candidate lines for each possible corridor and only the case study-3 which considers all uncertainties are established. The confidence levels are chosen as 80, 90 and 100%. The resulting TEP investments for case this study is given in Table 5. An example for the convergence of the proposed method for TEP-0.80 is shown in Fig. 10.
It has been observed that the proposed model provides alternative TEP decisions for a large power system by choosing a pre-specified risk level. As the non-overload probability β decreases less expensive TEP decisions are obtained for all risk levels. The main reason for investment is the overloading in transmission lines. A similar test procedure is also applied VOLUME 10, 2022  to the IEEE 118 bus test system. Weibull characteristic of different wind speed cases for testing procedure is given followingly: • Wind Case-1: scale=6, shape=3 • Wind Case-2: scale=7, shape=2.5 • Wind Case-3: scale=10.4, shape=5.4 The wind power considered for the three wind cases is low, medium, and high wind speed conditions, respectively. The violation probability margins of lines for the TEP plans 0.80 and 0.90 risk levels are given in Fig. 11 and Fig. 12.
It is observed that the overloaded lines have different probability margins due to the combination of new set of uncertainties. The outliers of the overload probability increase up to 20% and 14% for the TEP-0.80 and TEP-0.90 decisions, respectively. However, the mean overload probability for each TEP decision is almost below its pre-specified not overload probability for each wind case study. The overload probability margin observed in wind case-3 is very low than in other wind cases for TEP decision since it has the highest wind speed characteristic. There is no any violation on voltage magnitude so, it is not given. Since there is more investment in TEP-0.90 plan line 70-71, 71-72 and 70-75 has no violation probability. The expected generation cost of TEP plans is also calculated as 14838 $/h for the TEP-0.80 and TEP-0.90 plans and 14822 $/h for the TEP-1.00 plan, respectively. The results show a similar trend with previous studies that the expected generation cost is higher in risk taking TEP plans.

IV. DISCUSSION
The grid is becoming increasingly stressed with the integration of large-scale renewable sources. This brings with its risk and randomness which is inherently existing in power network. The current grid is not capable of accommodate with the large amount of uncertainty. Therefore, it is crucial to develop reliable and economically viable investment decisions. TEP is a complex problem that involves numerous levels of power system analysis studies. These studies include the operational behavior of system which changes stochastically with the integration of large-scale renewable sources. The problem is mainly provided by the generators to balance the load at minimal cost while meeting the operational limits. Generally, the economic dispatch is applied for a specified time steps. Renewable energy forecasts, such as wind farms, does not have a sufficient precision unless they are within a very short time span [49]. In that manner, the uncertainty in the system should be handled carefully to find the operational behavior of the system. A grid combined with renewable sources can result in power flows that exceed line capacity and voltage limits. When a line's capacity is exceeded, the probability of line tripping will also increase [61]. Therefore, the amount of overload level is a key to identify risk in planning studies under uncertainties. From the solution point of view, two main approaches are used to solve this problem under TEP. These are based on line overload probability [4], [14], [15] and load curtailment probability [5], [10], [17]. At the core of the problem, both approaches try to satisfy the future forecasted load under uncertain operating conditions. Yet, from modelling perspective first one requires a power flow analysis; the second one requires an optimization procedure called optimal power flow. Using optimal power flow seems to be more suitable to consider both technical and economic aspects of planning problem. However due to the complexity and computational burden of nonlinear equations; it becomes very difficult problem especially for large scale networks. Therefore, for this particular problem, the modified power flow analysis promises to find a secure and economical investment decision as well as offering the possibility to directly use the probability density functions of line flows and bus voltages instead of indirect means.
A probabilistic power flow approach is convenient when the uncertainty is handled in the operational problem. Generally, the steady-state power flow studies use the full AC power flow formulation to assess the losses and the reactive compensation requirements, both for the base and N-1 contingency conditions. However, the nonlinearity of AC power flow equations in power system planning is not considered frequently due to its mathematical complexity. The AC power flow model has been attracted researchers to reformulate the nonlinear equations for a more accurate representation of power system operation [24]- [37]. As far as the concerns on power loss, voltage and reactive power issues which require AC network modeling have been the subject of research together with the TEP problem. However, it is a complex problem which involves single/multiple objectives, integer, and discrete variables. When the operational problem is nonlinear, the solution space is likely to have more than one minimum. For this reason, it is even more difficult to achieve a feasible result within the specified time frame. Another important issue is that the probabilistic power flow suffers from the very existence of slack bus. The slack bus generator is a mathematical necessity for finding a feasible solution but with the stochastic operating conditions it absorbs uncertainty. This situation contradicts the studies of uncertainty. The output of slack bus generator is generally beyond its limits unless constraints are imposed on slack bus generator [40]. In that point, the imbalance power can be distributed among the dispatchable generators using participation factors. This methodology can be easily adopted in the existing power flow formulation with an artificially introduced parameter. This concept is called a distributed slack bus approach. In probabilistic power flow studies distributed slack concept is efficient, simple to implement and does not require an optimization to find the state vectors V and θ for different operating conditions. In this paper, the generator cost coefficients are used to find the optimal generator outputs based on equal incremental cost criterion under uncertain conditions. In that manner, the economic management of generators are obtained in a sense of economic dispatch. It is clear that, the proposed probabilistic model cannot be extended to an optimal power flow by simply introducing an artificial variable for imbalance power. However, the proposed method has the potential to be a suitable alternative for traditional single slack bus power flow analysis in associated with economic dispatch studies. Treatment of slack bus with modified power flow approaches proposed promising results for operational problem with the minor modifications on existing power flow studies [40], [41], [47], [62].
In this paper, uncertainties of wind, load and N-1 contingency are considered simultaneously to obtain the probability density function under nonlinear AC power flow with distributed slack bus concept. A method is proposed to combine over limit risk constrained stochastic TEP problem with participation factor power flow. The risk is defined as the non-over limit probability of each investment decision such that high risk level means risk-averse approach. It is observed from the results that when the risk level is increased, the number of candidate lines selected in the planning scheme and the total investment cost will also increase. Such an outcome is certainly within the expectation. However, maintaining line flows and bus voltages within their prescribed limits is the foremost operational criterion. Therefore, a risk-taking strategy is effective for reducing both number of invested lines and cost of investment. Another important observation is that when the system is heavily loaded and renewable integration VOLUME 10, 2022 is higher in weaker part of the network, there may also be a risk of voltage variations as well as line overloads. The location and size of voltage support must be carefully handled to maintain voltage profile within the specified limits. Thanks to AC network modeling, more realistic solutions can be achieved by adding voltages constraints to the investment optimization. To the best of author's knowledge, no prior work has utilized AC power flow analysis with distributed slack bus concept in chance constrained TEP problem. Therefore, it goes without saying that although power flow analysis is a powerful tool, it is a subject that is open to development when used in planning studies. Testing of TEP plans is another issue that has been overlooked in the literature. In this paper, the final stage of the study involves testing the investment plans obtained under a new set of uncertainty conditions, examining the investment plans with respect to various confidence levels allows, and presenting the benefits and drawbacks of the proposed method. As a result, the main differences between other existing studies and the method proposed in this paper are that exact probability density functions are obtained under nonlinear AC power flow with distributed slack bus concept to be used in chance constrained TEP problem by using a simple formulation and algorithm approach.
The potential limitation of the proposed method is that this model with nonlinear AC power flow needs a meta-heuristic method to perform the optimization process through probability density functions that used in the chance constrained optimization. As a metaheuristic optimization technique, genetic algorithms are of great interest for solving the TEP problem [4], [27], [55]- [57]. The proposed method utilizes the advantages of distributed slack bus-based AC power flow, meta-heuristic technique, and Monte Carlo Simulation, instead of using AC optimal power flow formulations integrated with TEP problem that may result in convergence problem. On the other hand, although the convergence of the methods to the global optimum solution cannot be guaranteed in nonlinear programming problems, heuristic optimization algorithms can produce promising results for the solution of difficult optimization problems such as AC-TEP [27]- [29], [33], [37], [63]. Although it is difficult to say that these algorithms are superior to each other, depending on the setup of the problems, it is difficult to say that metaheuristic methods do not have a chance of finding better solutions than those obtained with mathematical approaches based on decomposition techniques [55], [56].The main drawback of using a genetic algorithm to solve TEP optimization is that the time required give a decision scale exponentially with the increasing number of decision variables, population, and generation considerations. In our model, both systems are executed in MATLAB R2018a. The average CPU time for obtaining one TEP decision for the two test networks are recorded, for the simulations on IEEE 24 bus and IEEE 118 bus test system. Table 6 shows the average execution time for a single TEP decision that considers load, wind, and N-1 contingency uncertainties. It is observed that, as the system grows, the time required to solve the optimization problem also increases.
For future studies, this model can be extended in two different ways with some modifications. First, different uncertainties such as marketing uncertainties will be included in a deregulated environment considering generation and transmission company profits in a chance constraint programming. Additionally, this model can be transformed into dynamic programming with some modifications in nonlinear power flow equations. The potential limitation of the proposed method is that this model needs a metaheuristic method to proceed the optimization through exact probability density function to be fed chance constraints at the upper level. Since this model is not as fast as the DC or linearized AC model, studies on dynamic decisions will be made as a future study by making some compromises from the network model. At this point, this issue can be said as a shortcoming of the proposed method. Needless to say, power system planning is still an ongoing research area, and it is obvious that further work is necessary in terms of the decision-making in TEP problem under different uncertainty environment with riskbased approach.

V. CONCLUSION
Increasing rate of uncertainty in the power network complicates the decision-making process, both economic and technical standpoints. This paper proposes a full probabilistic chance constrained ACTEP framework based on participation factor power flow. The probabilistic participation factor power flow is evaluated load, wind, and N-1 contingency uncertainties under Monte Carlo simulation. A stochastic TEP optimization is implemented using genetic algorithm and the line capacity and bus voltage constraints are converted into probabilistic chance constraints. The final investment plans with different risk levels are evaluated in the various studies and tested under different wind speed considerations. In the case studies it is observed that as the operating conditions get stressful, especially for N-1 contingencies, the mismatch power comes from uncertainties absorbed by the slack bus itself. Therefore, the slack bus behavior should be carefully handled in operational sub-problem. The intermittent characteristic of renewable units and the contingency consideration brings additional stress that needs to be considered. Chance constraints of both base and contingency conditions are applied in the constraints of the TEP optimization to incorporate probabilistic risk for future planning decision. The results of the proposed TEP problem also shows that as the risk level increases more transmission investment is needed. In terms of computational effort, the use of PF-ACPF is promising to obtain the adequacy of the generators as well as consider the treatment of slack bus which affects the overall investment decision in the long-term transmission planning context.