Robust optimization for improving resilience of integrated energy systems with electricity and natural gas infrastructures
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Abstract
The integration of natural gas in electricity network requires a more reliable operating plan for increasing uncertainties in the whole system. In this paper, a three-stage robust optimization model is proposed for resilient operation of energy system which integrates electricity and natural gas transmission networks with the objective of minimizing load curtailments caused by attacks. Non-convex constrains are linearized in order to formulate the dual problem of optimal energy flow. Then, the proposed three-stage problem can be reformulated into a two-stage mixed integer linear program (MILP) and solved by Benders decomposition algorithm. Numerical studies on IEEE 30-bus power system with 7-node natural gas network and IEEE 118-bus power system with 14-node natural gas network validate the feasibility of the proposed model for improving resilience of integrated energy system. Energy storage facilities are also considered for the resiliency analysis.
Keywords
Resilience Robust optimization Integrated energy systems Natural gas networks Energy storage systems1 Introduction
In recent years, renewable energy generation gains rising attention due to the lack of traditional resources. However, the electricity quality and the reliability of power grids are significantly affected by the intermittency and instability of renewable energy resources such as wind and solar. By contrast, natural gas is a more stable and reliable sort of resource which can provide continuous energy for both gas and electricity loads by gas-fired generators [1].
Due to the clean, efficient and high-quality characteristics of natural gas, it has been widely used as the main energy resource in some areas and the coordinated operation of natural gas and electricity system has been researched in many previous studies [2]. A basic model for integrating natural gas and electricity networks is presented in [3], which shows the fundamentals of natural gas network and describes the constraints for the energy transmission between electricity and gas systems. In [4], a steady state power flow model is presented for solving the combined optimization operating problem of different energy facilities based on the new concept of energy hubs. A decomposition method is applied to solve the security-based model proposed in [5] for the solution of SCUC problem considering natural gas transmission system. In [6], an mixed integer linear program (MILP) method is presented to formulate the optimal power flow in multi-carrier energy systems. The non-convex constraints of natural gas transmission system are linearized, so that the problem is reformulated as an MILP problem which can be solved by traditional optimization methods.
However, the operating stability and reliability are still affected by increasing uncertainties in energy systems. Disruptions in an energy system are sometimes inevitable, uncontrollable and unpredictable [7]. As service industry, the energy system must guarantee the continuity of energy supply for customers. Therefore, improving resilience of energy systems is of vital importance. Resilience is defined as the ability to provide and maintain an acceptable level of service in the face of faults and challenges to normal operation [8]. The contingencies and challenges for services range from natural disaster to terrorist attacks. As a role of defender, the system itself will take preventive measures for attacks before disruptions occur and respond to the damage after attacks. The resiliency analysis in electricity system has been studied by several researchers. In [9], a resiliency-oriented microgrid optimal scheduling model is proposed for minimizing the load curtailments when the service of main grid is interrupted. A non-cooperative game-theoretic framework is presented in [10] to study the strategic behavior of microgrids. The framework incorporates economic factors and stability and efficiency of microgrids, which is solved by fully distributed phasor measurement unit (PMU)-enabled algorithm to ensure the resiliency of the proposed method. In [11], a resilient distribution network planning problem is presented and formulated as a two-stage robust optimization model to minimizing the system damage by coordinating the hardening and distributed resource allocation. In [12], a distribution system operating method by microgrid formation after natural disaster is proposed to restore important loads from power outage. A tri-level optimization model for electric power system defense is presented in [13] which can identify critical elements in power grid to defend against unpredictable attacks. A column-and-constraint generation algorithm is applied to solve a two-stage robust optimization problem in [14]. In [7], a risk assessment model is proposed to determine potential vulnerabilities of power system and provide feasible plans for enhanced protections according to the budgets for power grid construction.
There are few work which has been done to analyze resilience of integrated energy system which includes electricity, gas and other forms of energy. As the operational feature of natural gas system is different from that in power grids and the operating conditions have impacts on electricity networks, the resiliency analysis is an essential topic to be researched. In [15], a methodology is proposed to locate the most vulnerable components to make sure the resilient operation of multiple energy carrier microgrids when terrorists attack the network. The model is formulated as a bi-level optimization problem to solve the optimal operation for multi-energy microgrids in consideration of security and resiliency. In [16], a methodology is proposed to identify and protect vulnerable components of integrated electric and gas infrastructures. The resilience is guaranteed by solving a tri-level optimization problem. A mixed integer linear programming and nested column-and-constraint generation algorithm is applied to solve the proposed model. A novel mixed integer linear programming for security-constrained power and gas flow is presented in [17]. The proposed model allows the integrated system operates in both normal and contingency conditions with the least violations. In [18], three models are proposed for identifying optimal energy flow solvability to ensure secure operating conditions with corrective controls.
In this paper, a coordinated operation model of energy system which integrates electricity and natural gas infrastructures is formulated. The model includes constraints for both electricity and natural gas transmission networks. Then, a three-stage robust optimization algorithm is presented to solve the defender-attacker-defender problem of the integrated energy system when contingencies occur. The constraints and operational features of natural gas system and the coupling of electricity and natural gas networks make it more complicated to settle the plans for both attackers and defenders. In the first stage, as defender, the integrated energy system must make plans for network enhancement to minimize the damage caused by unpredictable attacks. In the second stage, attackers will attack vulnerable components of electricity and natural gas networks to cause maximum damage to the entire energy system. In the last stage, defender responds to the results of disruptions, which is formulated as optimal energy flow of integrated electricity and natural gas system. As some of the constraints of electricity and natural gas transmission networks are nonlinear, Taylor series expansion algorithm is applied to realize the linearization. Then, the proposed three-stage robust optimization problem is reformulated into a two-stage optimization problem by the application of duality theory. The two-stage problem can be solved as an MILP by decomposition algorithms.
- 1)
A nested Benders decomposition algorithm is applied to solve the proposed defender-attacker-defender problem. It is more effective to be applied for large-scale problems.
- 2)
Resiliency of integrated energy systems is analyzed in the presence of energy storage system. By adding proper electricity and gas storage facilities, reliability of integrated energy system is re-analyzed.
Compared with [15], we incorporate the hardening plans before attacks, which the resilience of the whole system can be better improved. That is to say, a defender-attacker defender model is presented in comparison with the attacker-defender model in [15]. Compared with [16], we apply a nested Benders decomposition algorithm to solve the nested two-stage proposed model. Moreover, Taylor series expansion is applied to linearize the quadratic polynomials of cost functions and the gas flow square for gas pipelines and compressors. In addition, energy storage systems are also considered in the end of this paper to analyze the resilience of integrated energy systems.
The remainder of this paper is organized as follows. Section 2 describes the mathematical formulations for the optimal energy flow and the three-stage robust optimization model of the integrated electricity and natural gas system. Section 3 provides the solution methodologies to linearize and decompose the presented three-stage problem. Section 4 presents and discusses the numerical results of the proposed method and analyze the impact of the presence of energy storage devices. Finally, the conclusion is given in Section 5.
2 Mathematical formulation
In this section, a power flow model for coordinated electricity and gas network is presented. Based on this integrated system, a robust optimization model is applied for resiliency analysis. Any contingency occurred in either electricity system or gas network may cause coordination problems or safety and stability problems of both systems. Models are presented to figure out the optimal defense and operation plans against contingencies.
2.1 Integrated electricity and gas network optimal flow model
It is a simplified form of expression which can be found in [5].
Equations (6) and (7) are the power balance formulation; (8)–(12) represent the constraints of node voltage, generator output and lines transmission limits.
Equations (13) and (14) are respectively the gas supply and gas pressure limits.
2.2 Robust optimal operation model for resilient energy systems
As the attacks on integrated energy system are random and unpredictable, the complex nature of this problem makes robust optimization be the most suitable method that can take account of inherent randomness and uncertainties [11]. This robust optimization problem aims to find the optimal enhancement and dispatch plans according to the worst case caused by attackers.
In this section, we analyze the model from inner minimum problem to outer minimum problem.
2.2.1 Defender response model
where I and M represent the sets of load curtailment nodes for electricity and gas network respectively; \(\tau_{i}^{elec} \;{\text{and}}\;\tau_{m}^{gas}\) denote the coefficients of economic loss for electricity and gas network respectively. Equation (24) is the simplified objective function of (5).
2.2.2 Attacker interdiction model
It is obvious that this model is a two-stage optimization problem.
2.2.3 Defender reinforcement model
3 Solution methodology
The formulated model in Section 2.2 is a nested two-stage robust optimization problem which is shown in Section 2.2.3. We apply a nested Benders decomposition algorithm to solve this problem.
3.1 Master problem
Solving the master problem needs a set of network reinforcement plan H and the worst case under attack which can be derived from the subproblem.
3.2 Subproblem
As the objective and constraints in (4), (28)–(30) and (46) are nonlinear, we will find ways to linearize them.
As mentioned in [6], the domain of \(f_{mn}\) is divided into (NL)^{2} grids. NL is the number of segments which is decided by the accuracy requirement. Selection of neighbor points and approximation error analysis are also given in [6], which will not be repeatedly explained here.
The dual maximization problem is linear and can be solved by an MILP solver.
3.3 Solution step for nested two-stage robust optimization algorithm
- 1)
Step 1: Initialization of variables. Set iterations \(K \to 0,\,LB \to - \infty ,\,UB \to \infty .\)
- 2)
Step 2: Solve the subproblem with given reinforcement plan \(H^{ * }\). Get the objective maximum economic loss \(obj_{SP}^{{{\rm max}} }\) and the optimal attacking plan \(A^{ * }\) for the worst case. Set \(UB \to {\text{Min}}(UB,obj_{SP}^{{{\rm max}} } ),\,\xi_{c} \to UB - LB,\,K = K + 1.\)
- 3)
Step 3: Solve the master problem with attacking plan \(A^{ * }\) and optimal system operation variables R derived from Step 2. Get the minimum economic loss \(obj_{MP}^{{{\rm min}} }\). Set \(LB \to {\text{Max}}(LB,obj_{MP}^{{{\rm min}} } ),\,\xi_{c} \to UB - LB\). Update lines reinforcement plan \(H^{ * }\).
- 4)
Step 4: If \(\xi_{c}\) satisfies convergence condition, stop the process. Otherwise, return to Step 2.
4 Case study
To show the performance of our proposed optimization model for integrated electricity and natural gas energy system, we apply two testing systems which are IEEE 30-bus power system with 7-node natural gas system and IEEE 118-bus power system with 14-node natural gas system.
4.1 IEEE 30-bus system
This case is based on the modified IEEE 30-bus power network and 7-node natural gas system.
Units outputs for electricity network
Unit | Node No. | P_{ G } (MW) |
---|---|---|
G1 | 1 | 38.18 |
G2 | 2 | 66.38 |
G3 | 13 | 26.37 |
G4 | 22 | 10.00 |
G5 | 23 | 12.84 |
G6 | 27 | 39.04 |
Gas outputs for natural gas network
Supplier | Node No. | Output (kcf) |
---|---|---|
S1 | 7 | 4144.74 |
S2 | 6 | 4124.67 |
Gas loads for natural gas network
Loads | Node No. | L_{gas} (kcf) |
---|---|---|
L1 | 1 | 1075.47 |
L2 | 1 | 4000.00 |
L3 | 3 | 607.56 |
L4 | 3 | 2000.00 |
L5 | 2 | 456.37 |
Reinforcement plans for integrated energy system
Lines | HL_{elec} | HL_{gas} | Worst case | Load curtailments | ||
---|---|---|---|---|---|---|
Electricity | Gas | Electricity (MW) | Gas (kcf) | |||
0 line | None | None | 2–4, 6–8, 15–18, 10–22, 27–28 | 1–2 | 115.1 | 4764 |
1–2 | 1–3 | 99.7 | 2908 | |||
1 line | 27–28 | None | 6–9, 8–28, 10–22, 15–23, 29–30 | 1–2 | 91.8 | 4644 |
1–2 | 1–3 | 76.4 | 2747 | |||
2 lines | 10–22, 27–28 | None | 4–6, 8–28, 19–20, 15–23, 29–30 | 1–2 | 89.4 | 4542 |
1–2 | 1–3 | 74.0 | 2610 | |||
3 lines | 15–23, 10–22, 27–28 | None | 1–3, 3–4, 12–13, 12–14, 24–25 | 1–2 | 81.6 | 4458 |
1–2 | 1–3 | 66.0 | 2499 | |||
4 lines | 1–3, 15–23, 10–22, 27–28 | None | 3–4, 4–6, 12–13, 12–14, 24–25 | 1–2 | 77.5 | 4392 |
1–2 | 1–3 | 62.0 | 2412 | |||
5 lines | 1–3, 3–4, 15–23, 10–22, 27–28 | None | 1–3, 3–4, 15–23, 10–22, 27–28 | 1–2 | 73.0 | 4346 |
1–2 | 1–3 | 57.5 | 2351 |
Congestions caused by attacks for 6 reinforcemet plans
Plan No. | B_{cg} | S_{ij} (MW) | S_{lim} (MW) |
---|---|---|---|
1 | 8–28, 6–28, 10–21 | 40.83, 42.62, 39.69 | 32, 32, 32 |
21–22, 15–23, 22–24 | 61.06, 23.14, 28.49 | 32, 16, 16 | |
24–25, 25–27 | 26.7, 28.9 | 16, 16 | |
2 | 6–8, 6–28, 10–21 | 42.81, 37.54, 38.28 | 32, 32, 32 |
21–22, 22–24, 23–24 | 58.36, 26.35, 26.19 | 32, 16, 16 | |
3 | 6–8, 6–28, 10–21 | 42.56, 37.29, 33.88 | 32, 32, 32 |
21–22, 22–24, 23–24 | 53.64, 25.89, 25.42 | 32, 16, 16 | |
4 | 21–22 | 111.8 | 32 |
5 | 21–22 | 89.4 | 32 |
6 | 21–22 | 71.6 | 32 |
4.2 IEEE 118-bus system
This case is studied to test the validity of proposed method in large-scaled systems. It is based on a modified IEEE 118-bus power network and 14-node natural gas system.
From the results shown in Figs. 8 and 9, we can see that two gas pipelines hardened and three gas pipelines hardened has nearly the same gas and power load curtailments. However, compared with none gas line hardened and one gas line hardened, load curtailments obviously decrease. Similarly, 7–10 power transmission lines hardened also have nearly the same results. As the costs will increase if more lines are hardened, 7 power transmission lines and 2 gas pipelines hardened is the most economic and effective plan to protect integrated energy system from attacks.
From the results for this large-scale system, we can also draw that our proposed model and method is feasible to improve and analyze resilience of integrated energy systems. Optimal plan for hardening power and gas lines can be derived from simulation results, which breaks the traditional view that resilience of system will improve as long as the number of hardened lines increases.
Computation time for C&CG in [16] and the proposed method in the case of IEEE 30-bus system
Number of hardened lines | Computation time (s) | |
---|---|---|
C&CG | Proposed method | |
1 | 55 | 61 |
2 | 184 | 190 |
3 | 321 | 303 |
4 | 753 | 628 |
5 | 1008 | 836 |
Computation time for C&CG in [16] and our proposed method in the case of IEEE 118-bus system
Number of hardened lines | Computation time (s) | |
---|---|---|
C&CG | Proposed method | |
1 | 311 | 288 |
3 | 1021 | 893 |
5 | 4737 | 3929 |
10 | 19,210 | 15,647 |
4.3 Impact of energy storage system
The concern on energy storage technologies is rapidly increasing due to high penetration of renewable and intermittent energy plug-in. As an effective manner to improve energy utilization efficiency, energy storage systems can solve the problem of mismatch between energy supply and demand side in time and space [25].
The system discussed in this paper integrates electricity and natural gas. Therefore, energy storage systems may include both electricity and gas storage systems. In power system, storage system is an effective manner to adjust peak. Excess power is stored into storage devices at off-peak period and stored energy will be transported back to grid when load demand is at peak. Moreover, as the unpredictable and intermittent nature of wind, solar and other renewable energy generation have great influence on resilience of power system, storage devices are usually installed nearby for tracking load changes [25]. However, electricity has the feature of easy to transport but difficult to store. Therefore, large-scale electricity storage technology is still a challenge. Compared with electricity, gas has the feature of easy to store but difficult to transport. Natural gas transportation costs mainly depend on the volume of gas supply and transport distance [26]. In fact, natural gas consumers are usually far away from gas sources and the cost for transporting is quite expansive, as a result of which the storage facilities for natural gas is of vital importance.
Compared with Figs. 8 and 9, load curtailments in both electricity and natural system decrease apparently. With 7–10 electricity transmission lines and 2–3 gas transmission lines hardened, gas load curtailments are prevented thoroughly and electricity load curtailments reduced by nearly 80%. The energy storage system plays a significant role in improving resilience of integrated energy system.
In Section 2.1, we have stated that we assume natural gas system operates in steady state and line pack is ignored. Actually, large amounts of gas stored in pipelines, such as line-pack, can provide additional gas supply after the occurrence of contingencies. In this regard, the load shedding results obtained from steady-state model are conservative. Certain amounts of load shedding can be compensated by line-pack and hence can be avoided.
5 Conclusion
This paper proposes a robust optimization model for resilient operation of integrated energy system with electricity and natural gas infrastructures. The proposed model is formulated as a three-stage optimization problem which considers network reinforcement, damage caused by attackers and defenders response of both electricity and natural gas systems. Linearization technologies and decomposition algorithms are applied to reformulate and solve this defender-attacker-defender problem. Numerical results validate the effectiveness of our proposed model. Studies also point to the importance of energy storage systems in improving resilience of integrated energy system against contingencies.
In future works, we will focus on the resilience of energy systems which integrate other forms of energies and demand response management. And distribution networks or micogrids will also be concerned in future works.
Notes
Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 51577116).
References
- [1]Gabbar HA, Bower L, Pandya D et al (2014) Resilient micro energy grids with gas-power and renewable technologies. In: Proceedings of the 2nd IEEE conference on power engineering and renewable energy, Bali, Indonesia, 9–11 Dec 2014, 6 ppGoogle Scholar
- [2]Xu Y, Ma Y, Zhang X et al (2013) The optimization of the operation program for natural gas pipeline transmission and end segment storing gas. In: Proceedings 2013 international conference on mechatronic sciences, electric engineering and computer, Shengyang, China, 20–22 December 2013, 5 ppGoogle Scholar
- [3]Unsihuay C, Lima JWM, Souza ACZD (2007) Modeling the integrated natural gas and electricity optimal power flow. In: Proceedings of 2007 IEEE power engineering society general meeting, Tampa, USA, 24–28 June 2007, 7 ppGoogle Scholar
- [4]Geidl M, Andersson G (2007) Optimal power flow of multiple energy carriers. IEEE Trans Power Syst 22(1):145–155CrossRefGoogle Scholar
- [5]Liu C, Shahidehpour M, Fu Y et al (2009) Security-constrained unit commitment with natural gas transmission constraints. IEEE Trans Power Syst 24(3):1523–1536CrossRefGoogle Scholar
- [6]Shao C, Wang X, Shahidehpour M et al (2017) An MILP-based optimal power flow in multicarrier energy systems. IEEE Trans Sustain Energy 8(1):239–248CrossRefGoogle Scholar
- [7]Nezamoddini N, Mousavian S, Erol-Kantarci M (2017) A risk optimization model for enhanced power grid resilience against physical attacks. Electr Power Syst Res 143:329–338CrossRefGoogle Scholar
- [8]
- [9]Khodaei A (2014) Resiliency-oriented microgrid optimal scheduling. IEEE Trans Smart Grid 5(4):1584–1591CrossRefGoogle Scholar
- [10]Chen J, Zhu Q (2017) A game-theoretic framework for resilient and distributed generation control of renewable energies in microgrids. IEEE Trans Smart Grid 8(1):285–295CrossRefGoogle Scholar
- [11]Yuan W, Wang J, Qiu F et al (2016) Robust optimization-based resilient distribution network planning against natural disasters. IEEE Trans Smart Grid 7(6):2817–2826CrossRefGoogle Scholar
- [12]Chen C, Wang J, Qiu F et al (2016) Resilient distribution system by microgrids formation after natural disasters. IEEE Trans smart grid 7(2):958–966CrossRefGoogle Scholar
- [13]Yao Y, Edmunds T, Papageorgiou D et al (2007) Trilevel optimization in power network defense. IEEE Trans Syst Man Cybern Part C (Appl Rev) 37(4):712–718CrossRefGoogle Scholar
- [14]Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper Res Lett 41(5):457–461MathSciNetCrossRefMATHGoogle Scholar
- [15]Manshadi SD, Khodayar ME (2015) Resilient operation of multiple energy carrier microgrids. IEEE Trans Smart Grid 6(5):2283–2292CrossRefGoogle Scholar
- [16]Wang C, Wei W, Wang J et al (2017) Robust defense strategy for gas-electric systems against malicious attacks. IEEE Trans Power Syst 32(4):2953–2965CrossRefGoogle Scholar
- [17]Correa-Posada CM, Sanchez-Martin P (2014) Security-constrained optimal power and natural-gas flow. IEEE Trans Power Syst 29(4):1780–1787CrossRefGoogle Scholar
- [18]Chen S, Wei Z, Sun G et al (2017) Identifying optimal energy flow solvability in electricity-gas integrated energy systems. IEEE Trans Sustain Energy 8(2):846–854CrossRefGoogle Scholar
- [19]Mercado RR (2002) Natural gas pipeline optimization. Handbook of Applied Optimization. Oxford University Press, OxfordGoogle Scholar
- [20]Correa-Posada CM, Sánchez-Martín P (2015) Integrated power and natural gas model for energy adequacy in short-term operation. IEEE Trans Power Syst 30(6):3347–3355CrossRefGoogle Scholar
- [21]Alsac O, Stott B (1974) Optimal load flow with steady-state security. IEEE Trans Power Appar Syst PSA 93(3):745–751CrossRefGoogle Scholar
- [22]Ferrero RW, Shahidehpour SM, Ramesh VC (1997) Transaction analysis in deregulated power systems using game theory. IEEE Trans Power Syst 12(3):1340–1347CrossRefGoogle Scholar
- [23]Power system test case archive. http://www.ee.washington.edu/research/pstca/
- [24]Zhao L, Zeng B (2012) An exact algorithm for two-stage robust optimization with mixed integer recourse problems. http://www.doc88.com/p-9929434504490.html
- [25]Ma Y, Yang P, Zhou X et al (2016) Research review on energy storage technology. In: Proceedings of 2016 IEEE international conference on mechatronics and automation, Harbin, China, 7–10 August 2016, 6 ppGoogle Scholar
- [26]Shi G, Jing Y, Zhang X et al (2009) Prospects of natural gas storage and transportation using hydrate technology in China. In: Proceedings of 2009 4th IEEE conference on industrial electronics and applications, Xi’an, China, 25–27 May 2009, 5 ppGoogle Scholar
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