A low-carbon oriented probabilistic approach for transmission expansion planning
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Abstract
Following the deregulation of the power industry, transmission expansion planning (TEP) has become more complicated due to the presence of uncertainties and conflicting objectives in a market environment. Also, the growing concern on global warming highlights the importance of considering carbon pricing policies during TEP. In this paper, a probabilistic TEP approach is proposed with the integration of a chance constrained load curtailment index. The formulated dynamic programming problem is solved by a hybrid solution algorithm in an iterative process. The performance of our approach is demonstrated by case studies on a modified IEEE 14-bus system. Simulation results prove that our approach can provide network planners with comprehensive information regarding effects of uncertainties on TEP schemes, allowing them to adjust planning strategies based on their risk aversion levels or financial constraints.
Keywords
Power system planning Emission reduction Dynamic programming Risk management1 Introduction
1.1 Transmission expansion planning
Transmission expansion planning (TEP) refers to comprehensive studies on determining the time, location, and type of adding new power transmission lines as well as the associated electrical components, in order to ensure the economic, secure, and reliable operations of a power system [1]. There are a variety of factors contributing to the necessity of TEP, including load growth, components’ entry or decommissioning, technological improvement, policy incentive, etc. [2]
Following the advent of electricity markets in many countries, power industry has been transformed from a vertically integrated and regulated utility to an unbundled and liberalized structure. This deregulation and restructuring has resulted in fundamental changes to the power system TEP practices [3, 4]. For instance, the emergence of various self-interested market participants such as brokers and independent power producers has made planning difficult due to conflicting objectives and uncertainties in a market environment [5, 6].
Moreover, the power sector is one of the biggest emission sources and should take a key responsibility in promoting the coordinated development among economy, energy and environment by mitigating carbon emission [7, 8]. This concern on carbon emission mitigation is driving the need to explore power system planning practices under the mode of low-carbon economy, e.g. network planning to facilitate the integration of renewable energy and/or to encourage clean power outputs. Therefore, power system planning becomes more complicated under the impacts of different emission reduction policies.
1.2 Carbon pricing policies
Some key carbon pricing policies in the world are summarized as follows. In 2003 in the U.S., carbon emission allowances could be traded among American corporations under a voluntary scheme on Chicago Climate Exchange. In October 2003, the EU parliament approved a new emission trading scheme in order to meet its commitment made in the Kyoto Protocol. Following this, an EU emission trading scheme (ETS), which was the largest multinational, greenhouse gas emission trading scheme in the world, was enforced in February 2005. Under the EU ETS, a specific allowance for emission was allocated to each EU member and any excessive allowance could be sold to whom was in need of allowances. Some allowances were permitted to be transferred between countries through joint implementation (JI) or clean development mechanisms (CDM), but these transfers should be validated by the United Nations Framework Convention on Climate Change (UNFCCC). This type of EU ETS is also called cap-and-trade. In April 2007, an emission price of $15 NZ per carbon equivalent was implemented. In September 2008, the New Zealand ETS was legislated, and it adopted all free allocation without caps. In July 2010, India introduced a nationwide carbon tax of about $1.07 US on coal, as coal was a major fuel resource for power generation in India. In November 2011, China had a pilot test of carbon trading in seven provinces and a national trading is expected to start in 2016. In July 2012, the Australian government introduced a carbon price of $23 AU per ton of emitted carbon equivalent, but this carbon price has been phased out.
1.3 Review of TEP
Conventional TEP models are formulated as minimizing investment costs of adding new lines. The widely used deterministic and static TEP model in a regulated power utility is defined as [6, 9, 10]:
To obtain the value of r_{i}, usually an optimal power flow (OPF) model is employed to reschedule power generations and alleviate violations of network constraints. The objective of this OPF model is to minimize the total load curtailment [11]. Note this penalty factor for r_{i} helps the optimization process in (1) to find an economical planning solution without a loss of load.
However, the above model is more suitable for a vertically regulated power system, as they do not take into account the market interactions among various stakeholders, e.g. congestion costs caused by different marginal generation costs [12]. By contrast, the main objective of TEP in a new deregulated environment is to provide all stakeholders with nondiscriminatory access to cheap, secure and clean energy resources, subject to reliability and other criteria [13].
1.4 Contributions of this paper
- 1)
A chanced constrained load curtailment index is proposed;
- 2)
A risk based probabilistic TEP model is proposed with the consideration of planning uncertainties;
- 3)
A novel hybrid solution algorithm in an iterative process is proposed to solve the formulated multistage programming problem.
2 Formulated probabilistic model
2.1 Define probabilistic load curtailment
In order to consider uncertainties in TEP, we propose a probabilistic formulation to reflect the level of load curtailment bounded by a threshold, r_{max}. This load curtailment threshold is common in industrial practice. For instance, in Australia, expected energy not supplied (EENS) should be less than 0.002% of total energy consumption, which is also the network planning criterion used by the Australian Energy Market Operator (AEMO) [14]. In our probabilistic approach, the loss of load item in (1) is replaced by a percentage of having load curtailment over the threshold, as given in (9) and (10).
When TEP takes into account a variety of uncertainties such as load and wind power output, a commonly used approach is using Monte Carlo (MC) simulations to sample uncertainty scenarios. After simulation stops, EENS is obtained as the average or mean of unsupplied power in the simulation after solving the optimal power flow problems. Planning schemes selected by this method are optimal in the statistical sense, leading to low-probability scenarios being discounted [6]. Moreover, in practice, TEP investment costs may be very high to guarantee that there is no excessive load curtailment at all times.
2.2 Carbon emission modelling
2.3 Uncertainties modelling
In this paper, uncertainties taken into account are wind power outputs, component working state of the power system, load growth and carbon price. Four types of probability density functions (PDFs) are used to model those uncertainties. Parameters of these PDFs can be obtained based on historical data. MC simulations are deployed to randomly generated scenarios composed by values from the four PDFs.
2.4 Proposed probabilistic model
A multi-stage probabilistic TEP model is formulated as minimizing the total cost with the chance constrained index ɛ_{t} proposed in Section 2.1, as follows.
Equation (21) is a model represented by net present value (NPV) with a discount rate γ. T is the total planning horizon. i, j are superscripts for buses. k, t are superscripts for load block and planning year respectively. The first term is the investment cost, and the second term is the chance constrained index at year t scaled by a penalty factor λ_{ɛ}. The third term is the operating cost of power generators, whose incremental cost of output is denoted by C_{Oi}. The total number of generators in the system is N_{G}. The fourth term is the cost of carbon emission, derived from carbon price and the net annual emission of power generators.
For completeness, we use a piecewise function to derive the quadratic losses in the DC power model, as given in (29)-(37). Note losses are not incorporated into the objective function. Instead, they are considered as the additional active power in (22) required from power generation, in order to satisfy the nodal balance due to the presence of losses.
3 Solution algorithms
Solution algorithms for TEP problems mainly fall into either mathematical programming classes or heuristic search classes. Mathematical programming methods have strict requirements on the model itself (e.g. the problem or the continuous relaxation of the problem should be convex) and can provide more clues on the quality of the final solution [22]. However, mathematical programming methods tend to be trapped by local optima in some cases. On the contrary, heuristic methods are suitable for stochastic global search, free from problem formulation difficulties and can escape from premature local optimal. The drawbacks of heuristic methods are: the quality of the solution cannot be guaranteed; and prohibitive computation efforts are required [22, 23, 24].
In this paper, the proposed TEP model is a dynamic optimization problem with a chance constrained reliability evaluation. To enhance the solution performance of the proposed model, a hybrid method based on decomposition is proposed as follows.
The overall stochastic programming problem can be divided into two subproblems: 1) the investment subproblem in the first and second terms of the defined objective; 2) the operating subproblem in the third and fourth terms of the defined objective. Note that the investment subproblem is targeted at a specific planning year, whereas the operating decisions are subject to the decisions from the investment subproblem and should be evaluated over multiple years thereafter [25].
- 1)
Determine the PDFs of wind speed, carbon price, load, and FOR based on historical data. Note that the PDF of wind power outputs is translated from the wind power curve in (15).
- 2)
MC simulations are deployed to generate values of wind power outputs, load, component availability, and carbon price from the PDFs in step 1).
- 3)
Initialize the population of the EA corresponding to the number of new added lines. Note that the quality of initial population has great impacts on the final solution. Therefore, in this process, uncertain features are neglected and some deterministic values are assigned. To be specific, carbon price is set to its mean value, wind power output is set to the installed capacity scaled by its capacity factor, all components are set as available, and load is set as (μ + 3σ).
- 4)
Using the initial values generated by step 3), denoted by \( \varvec{\eta}^{G} = \left\{ {\eta_{ij,0} ,\forall i,\,j \in N} \right\} \), apply the IP to generate local optima, i.e. subproblem 2 is solved with this generation. In this process, IP can solve the DC optimal power flow (OPF) for each individual in the population. Note if network violations are identified, a linear programming is required to minimize the total load curtailment, i.e. re-dispatching of generation is not based on bids or marginal costs. Then calculate the operating and carbon emission costs and add the two local optima to \( \varvec{\eta}^{G + 1} \). Also, based on identify network violations, find the probability of \( \Pr \left\{ {\sum\limits_{{i \in N_{L} }} {r_{i} } \le r_{\hbox{max} } \;} \right\} \), and calculate the probabilistic index of reliability.
- 5)
Start mutation and recombination process. Integers representing adding or reducing lines will be generated and recombined.
- 6)
For each TEP scheme, the objective function in (21) is assigned as the fitness value. A newly generated child individual is compared with the parent, and replaces it if the fitness value of the child is smaller.
- 7)
Terminate the algorithm if the stopping criterion is satisfied, otherwise go back to step 2).
4 Case studies
In this section, a series of numerical experiments are undertaken to demonstrate the performance of the proposed model on the modified IEEE 14-bus system. The 14-bus system initially has five power generators, and 20 transmission corridors.
In our paper, the five power generators in the base case are assumed to be fossil-fuel-fired, and the carbon emission coefficients for them are set as 1.2, 1, 0.8, 0.8 and 0.6 tCO_{2}/MWh respectively. Other power generation types such as hydro power or nuclear power could be included as future works. Moreover, for simplicity, power generators are assumed to receive 80% of their emission allowances for free based on their emissions in the base year, and free allowances will decrease linearly each year to 30% in the last planning horizon. Note that allocating emission allowances is a complex issue involving political motivation. The assumption made in this paper regarding emission allocation is in accordance with the EU ETS practice, which can be found in [27].
The capacity of each candidate line is 100 MW, and up to four lines are allowed for each corridor. The investment cost is assumed to be 50 M$/100 km. We set the upper bound of load curtailment threshold as a percentage of the total annual energy demand, i.e. r_{max} = 0.1%. To make ɛ_{t} have a significant effect in the objective function, the penalty factor λ_{ɛ} is set big enough to ensure that planning schemes with considerable load curtailments will be eliminated in the optimization process during the heuristic search. The settings of α and λ_{ɛ} are 95% and $1 × 10^{9}.
Comparisons of deterministic and probabilistic TEP with different carbon pricing policies
Case No | Total cost (M$) | Invest. cost (M$) | Operating cost (M$) | Emission cost (M$) |
---|---|---|---|---|
Case 1: deterministic TEP with no emission price | 890.36 | 378.25 | 512.11 | 0 |
Case 2: deterministic TEP in mode 1 with $23/tCO_{2} | 1119.22 | 378.25 | 615.62 | 125.35 |
Case 3: deterministic TEP in mode 2 with $23/tCO_{2} | 1061.48 | 378.25 | 572.98 | 110.25 |
Case 4: probabilistic TEP in mode 2 | 1107.77 | 415.64 | 586.77 | 105.36 |
Final result against different carbon price characteristics
Lines added | Different carbon price characteristics ($/tCO_{2}) | α |
---|---|---|
η_{1–2,t=2} = 1; η_{4–10,t=2} = 1; η_{7–9,t=2} = 1; η_{6–11,t=3} = 2; η_{10–11,t=3} = 1; η_{12–13,t=3} = 1; η_{14–9,t=3} = 1; η_{2–3,t=4} = 2; η_{3–4,t=4} = 3; η_{1–2,t=5} = 1; η_{1–5,t=5} = 1; η_{5–6,t=5} = 4; η_{6–11,t=5} = 2; η_{12–13,t=5} = 1 | Mean is 23; std. is 3 | 0.9575 |
Mean is 23; std. is 5 | 0.9028 | |
Mean is 35; td. is 3 | 0.9329 | |
Mean is 35; std. is 5 | 0.8266 |
5 Conclusion
This paper has proposed a probabilistic TEP model with a chance constrained load curtailment index. Planning uncertainties such as wind power output, component availability, load, and carbon price are incorporated by a Monte Carlo simulation based approach. Our multi-stage planning objective is formulated as minimizing the total cost, including investment cost, operating cost, emission cost and a risk factor of load curtailment. For completeness, a piecewise approximation function is used to linearize the quadratic power losses. Meanwhile, a novel iterative solution algorithm combing heuristic search and mathematical programming is proposed to solve the formulated a dynamic optimization problem. The performance of the proposed approach is demonstrated by a modified IEEE 14-bus system. Simulation results have proved that our approach can give network planners an opportunity to trade-off between the overall cost and the probability of load curtailment in the presence of uncertainties. Our approach can also provide network planners with comprehensive information regarding effects of uncertainties on TEP schemes, allowing them to adjust planning strategies based on their risk aversion levels or financial constraints. Moreover, our approach can be used for renewable energy integration analysis in terms of long-term network planning. Therefore, our novel TEP approach is a risk-based, flexible decision tool, which is important for achieving low carbon economy through planning practices.
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