# Linearized model for optimization of coupled electricity and natural gas systems

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## Abstract

In this paper a combined optimization of a coupled electricity and gas system is presented. For the electricity network a unit commitment problem with optimization of energy and reserves under a power pool, considering all system operational and unit technical constraints is solved. The gas network subproblem is a medium-scale mixed-integer nonconvex and nonlinear programming problem. The coupling constraints between the two networks are nonlinear as well. The resulting mixed-integer nonlinear program is linearized with the extended incremental method and an outer approximation technique. The resulting model is evaluated using the Greek power and gas system comprising fourteen gas-fired units under four different approximation accuracy levels. The results indicate the efficiency of the proposed mixed-integer linear program model and the interplay between computational requirements and accuracy.

## Keywords

Electricity system Natural gas system Mixed-integer (non) linear programming Extended incremental method Outer approximation## 1 Introduction

The importance of electric gas units (mainly combined cycle gas turbines, CCGTs) has been increased during the recent years due to their attractive features as flexible resources in view of the increasing renewable energy sources (RES) penetration. This is attributed to their increased efficiency, small environmental footprint and high flexibility.

The popularity of CCGTs has given rise to common energy infrastructure considerations by regulators and system analysts, identifying the strong interdependence between the electricity and gas system in technical, economic and operational terms.

Despite their common nature as energy transmission systems, the operation of the natural gas system is extremely complex, employing a large-scale, highly nonconvex and nonlinear problem structure (comprising a group of nonlinear algebraic equations), which can be modeled as a nonconvex mixed-integer nonlinear program (MINLP) [1]. Compared to this, the electricity problem, which is a unit commitment problem with optimization of energy and reserves, can be modeled as a mixed-integrater program (MIP) [2].

A review of the developed nonlinear models for combined consideration of the electricity and gas systems is given in [3] and [4]. In all models, there is a nonlinear coupling constraint linking the electricity system with the gas system.

The majority of the research in the gas system focuses on simplifications of the nonlinear equations that govern the physics of gas flow as well as the general representation of the technical components (e.g. compressor model) of the gas transmission system. A current useful and efficient way to alleviate the burden of the nonconvex MINLP gas problem is the application of mixed integer linear programming techniques for piecewise linearization. In this way, the nonlinear formulation is approached by approximated linearized functions that render the problem as a MIP, which enables the computation of global optima within fast execution times. The most comprehensive article that studies the advantages and drawbacks of various MIP formulations for the gas optimization problem is [5], where different piecewise linearization approaches are tested for different test cases in both dynamic and stationary conditions in order to derive conclusions about the more accurate and faster approximations. Based on the results, the authors argue about the effectiveness of the incremental method and further apply it in [6] and [7] in order to linearize the transient flow equations for a combined electricity-gas model that is applied on the Belgian high-calorific gas network, where the effects of initial linepack in system operation are examined. A simpler linearization approach can be seen in [8] and [9] for a stochastic unit commitment and a combined optimal power flow model, respectively. Simple electricity-gas conversion factors are applied to all studies and the compressor machine modeling is also reduced to a compression factor. The incremental method is again applied in [10] for stationary gas flow within a large decomposition framework.

- 1)
The extended incremental method is used for the linearization of the gas physics [11]. This linearization method builds on the simple incremental method, exhibiting enhanced features and advantages as compared to standard piecewise linear approximation methods.

- 2)
The nonlinear and nonconvex compressor operating range is linearized by an outer approximation approach. This approach often yields a rather detailed approximation (as opposed to simple linear approximations typically used in the literature), while keeping the computational overhead relatively low compared to the piecewise linearization of multivariate nonlinearities with the extended incremental method.

- 3)
The coupling constraints (originally quadratic functions) are linearized with the extended incremental method as well, whereas in the literature a simple linear conversion ratio is applied.

Literature overview for linearized models

Reference | Gas physics | Compressor/ | Configuration |
---|---|---|---|

coupling | |||

Incremental | Single linear | Transient | |

Piecewise | Single linear | Stationary | |

Wang et al. [10] | Incremental | Single linear | Stationary |

Current | Extended | Linearized | Stationary |

Incremental | Range/extended | ||

incremental |

Section 2 describes the nonconvex MINLP, whereas Sect. 3 presents the corresponding linearized model (MIP). In Sect. 4, several computations are presented and discussed afterwards. Lastly, in Sect. 5 the computational results are summurized.

## 2 Problem statement

The nonconvex MINLP is composed of an electricity (MIP) and a gas model (nonconvex MINLP) which are described in Sects. 2.1 and 2.2, respectively. The coupling of these models via nonlinear functions is explained in Sect. 2.3.

### 2.1 Electricity problem

- 1)
Unit start-up/desynchronization constraints.

- 2)
Unit minimum up/down time constraints.

- 3)
Logical status of unit commitment constraints.

- 4)
Unit power output normal and AGC limits.

- 5)
Unit reserve capability constraints.

- 6)
Unit ramp-up/down constraints.

- 7)
The power balance equations.

- 8)
System reserve requirements.

Due to the presence of binary variables for unit commitment and automatic generation control (so-called AGC) status, the electricity problem constitutes a MIP model. The objective value is defined as the electricity system cost \(C_{\mathrm{e}}\) for all dispatch periods \(t\in T\).

### 2.2 Gas problem

The optimization of gas transport networks has been recently described in [1] and in the survey article [12]. The main objective is to find cost optimal operations, which stick to the required pressure and flow bounds. As gas mainly flows from higher to lower pressures, it is required to install compressor machines in order to increase the gas pressure again. These compressions can be controlled by the transmission system operator, which adds discrete aspects to the problem among others. The physics of gas transport are driven by differential equations, which are simplified herein for the description of a steady-state problem. In this subsection, the gas model is analytically described. It should be noted that in all descriptions of Sect. 2.2 the dispatch period index is deliberately ignored.

The gas network is modeled via a directed graph \(G=(V,A)\) with node set \(V\) and arc set \(A\). The set of gas nodes \(u,v\in V\) is partitioned into the set of entry nodes \(V_+\) (where gas is supplied), the set of exit nodes \(V_-\) (where gas is discharged) and the set of inner nodes \(V_0\) (without any supply or discharge). Entries can be LNG (liquefied natural gas) entries \(V_{\text {lng}}\) or standard pipeline entries \(V_{\text {ent}}\). Exits can be electric gas loads (power plants) \(V_{\text {e}}\) or non-electric gas loads \(V_{\text {ne}}\). The arcs are divided into pipes \(A_{\text {pi}}\), control valves \(A_{\text {cv}}\) and compressor machines \(A_{\text {cm}}\).

The isothermal case assumes constant temperature \(T\), which is used for that model. Now, (8)–(10) are combined for the calculation of \(z_a\). \(z_a = z(p_{m, a}^{}, T), \forall a\in A_{\text {pi}}.\)

Moreover, so-called short pipes are defined as auxiliary network elements. They are used to model multiple entries at a single entry node or to model multiple exits at a single exit node and can be interpreted as pipes \(a= (u, v)\in A_{\text {pi}}\) with length \(L_{a} = 0\) and therefore from (6), \(p_{u}^{} = p_{v}^{}\) can be concluded.

### 2.3 Coupling of gas and electricity problem

## 3 Linearized formulation

Two linearization techniques are used to tackle the nonlinearities of (27)–(30). The nonlinear functions in (6), (16), and (26) are linearized with the so-called extended incremental method [11]. The nonlinear and nonconvex set of (15) is linearized with outer approximation constraints [16].

Both techniques ensure the relaxation property, which means that each solution to the nonlinear model (27)–(30) is also feasible for the linearized model. This is important as pure approximations tend to be infeasible even if the underlying nonlinear model is feasible [17].

### 3.1 Extended incremental method

First of all, it should be mentioned that the incremental method as basis of the extended incremental method is just one of various methods to linearize nonlinear functions. MIP-relaxations may be formulated with alternative methods, for instance with a convex combination model that introduces only a number of extra binary variables and constraints that is logarithmic in the number of breakpoints [18]. However, the incremental method is used in this paper, since it performs best for gas transport problems [5, 17, 19].

The extended incremental method works by first introducing a new finitely bounded variable for each nonlinear term, and by then computing a piecewise linear approximation of it. The approximation is constructed such that an a priori given upper bound on the approximation is satisfied, while introducing as less breakpoints as possible. Using the known values of the resulting approximations error, a MIP-relaxation model for the nonlinearities is derived.

To be more precise, for a constraint of the form \(\varvec{c}^\mathrm {T}\varvec{x} + f(y) = 0\), with \(f:\mathbf R \rightarrow \mathbf R\) nonlinear, and finitely bounded \(y \in [y_{\min },y_{\max }]\), a new variable *z* is introduced, reformulating the constraint as \(\varvec{c}^\mathrm {T}\varvec{x} + z = 0\), and a piecewise linear approximation \(\varPhi (y)=z\) of *f*(*y*) over \([y_{\min },y_{\max }]\) with \(\max _{y_{\min } \le y \le y_{\max }} |\varPhi (y) - f(y)| \le \varepsilon\) is computed.

*n*intervals, subdivided by breakpoints \(y_{\min }=b_0< b_1< \dots <b_{n-1}=y_{\max }\), the overall MIP-relaxation model of the constraint \(\varvec{c}^\mathrm {T}\varvec{x} + f(y) = 0\) is reformulated as follows:

*i*of an interval with \(0< \delta _i < 1\). In this case,

*y*lies within the \(i^\mathrm{th}\) interval. Assuming \(e=0\), the point (

*y*,

*z*) would lie on the \(i^\mathrm{th}\) line segement of the piecewise linear function. However, with \(e \in [-\varepsilon , \varepsilon ]\), the point (

*y*,

*z*) lies within a box of height \(2\varepsilon\) that encloses the graph of the piecewise linear function. Since \(\varepsilon\) has been chosen such that also the graph of the approximated nonlinear function is contained in the boxes, model (31) yields a relaxation.

The extended incremental method is used for the nonlinear terms in (6), (16), and (26). In order to rewrite (16) as a sum of univariate nonlinear expressions, we plug in Eq. (14) and apply the standard reformulation \(xy = -((x - y)^2\,{-}\,x^2 - y^2)/2\) for bilinear expression two times, before the extended incremental method is applied. It should be also noted that that after linearization, (26) is still a product of a binary variable \(u_{u}^{t}\) and a newly defined continuous variable according to (31). This resulting nonlinear term can be linearized without an error by the well-known bigM method.

### 3.2 Outer approximation

Outer approximation is a well-known linearization method for convex nonlinear programs [16]. In this paper, it is used for the linearization of (15), although a nonconvex operating range is involved. The idea is to compute a convex envelope of the operating range and to add tangential hyperplanes to the envelope during optimization. More details on this technique can be found in [1].

## 4 Case study

The developed model is tested on the Greek power system together with the Greek natural gas transmission system. The description of these networks is given in Sect. 4.1, while the computational results are presented in Sect. 4.2. First, the linearized formulation of Sect. 3 is tested in Sect. 4.2.1. Afterwards, the results are compared with a single electricity model in Sect. 4.2.2, where the gas network is not taken into account.

### 4.1 Real-world test system

Greek power system generating units’ data with unit type, number of unit types, installed capacity, and marginal cost range

Unit type | Number | Capacity | Marginal cost |
---|---|---|---|

(MW) | (€/MWh) | ||

Lignite units | 16 | 4302 | 43.18–51.20 |

CCGTs | 11 | 5026 | 76.30–85.00 |

OCGTs | 3 | 147 | 150 |

Hydro units | 15 | 2997 | 90 |

RES | 50 | 4191 | – |

Data of Greek gas network with 134 nodes and 133 arcs

134 nodes | 133 arcs | ||
---|---|---|---|

Type | Quantity | Type | Quantity |

Pipeline entries | 2 | Pipes | 86 |

LNG entries | 1 | Short pipes | 45 |

Non-electric | 34 | Control valves | 1 |

Gas loads | |||

Electric gas | 11 | Compressor | 1 |

loads | machines | ||

Inner nodes | 86 |

### 4.2 Computational results

All computations have been performed on a computer with an Intel i7-4770 CPU with 4 cores running at 3.4 GHz each, and 16 GB of main memory.

The electricity model was developed with GAMS 24.4.6 [22], while the linearized gas model was built with the C++ software Framework LaMaTTo++ [23]. GAMS was used to combine these models and to linearize the coupling. As MIP solver, Gurobi 6.0.4 was used [24]. A MIP gap tolerance of 0.01‰ was selected. Moreover, a time limit of 1 h was set as the electricity unit commitment problem has to be solved in 1 h.

As instances, the days with highest non-electric gas loads \(\sum _{t\in T}\sum _{u\in V_{\text {ne}}}d_{u}^{t}\) out of all days of every quarter of the years 2013–2015 were chosen yielding in total twelve instances.

Approximation accuracy levels

Label | \(\varepsilon _1\) | \(\varepsilon _2\) |
---|---|---|

A | 1.5 | 1 |

B | 2.5 | 10 |

C | 5.0 | 100 |

D | 15.0 | 1000 |

Moreover, preprocessing according to chapter 6 in [1] is used for the gas problem. These methods tighten the bounds of flow variables and reduce the number of linearized functions. If a squared variable \(x^2\) occurs in the problem without the use of the variable *x* itself, then \(x^2\) can be used without a linearization and *x* can be calculated a posteriori.

Regarding the entries, two different scenarios were examined.

Case I For the first case, the historical gas entry distribution of Kipi, Sidirokastro and Revythousa from 2013 to 2015 was averaged. Therefore, Kipi and Revythousa provide upto 20% of the gas each and the remaining upto 60% are supplied at Sidirokastro. In the present model, the optimizer is allowed to deviate from this distribution by at most ten percent at each entry.

Case II Another case with closed LNG entry is implemented, due to its periodic maintenance or outage. The entry distribution of Kipi and Sidirokastro can be chosen freely by the optimizer in this case. However, it is required that at least 20% of consumed gas has to be supplied from Kipi.

#### 4.2.1 Linearized formulation

Average number of continuous variables, discrete variables, constraints, and average solution time. The numbers in parentheses denote the number of instances that hit the time limit

Accuracy | Case I | Case II | ||||||
---|---|---|---|---|---|---|---|---|

Continuous | Discrete | Constraints | Solution | Continuous | Discrete | Constraints | Solution | |

variables | time (s) | variables | time (s) | |||||

A | 136570.00 | 54913.58 | 186217.00 | 500.18 (0) | 134467.50 | 53826.33 | 183706.50 | 1762.09 (5) |

B | 81756.17 | 27506.67 | 108507.17 | 96.05 (0) | 80845.67 | 27015.42 | 107188.67 | 917.54 (2) |

C | 61030.83 | 17144.00 | 80413.83 | 26.57 (0) | 60696.33 | 16940.75 | 79671.33 | 516.91 (1) |

D | 54284.83 | 13771.00 | 71411.83 | 8.72 (0) | 54212.83 | 13699.00 | 70931.83 | 142.51 (0) |

Electricity costs \(C_{\mathrm{e}}\), compressor costs \(C_{\mathrm{c}}\), shedding costs \(C_{\mathrm{s}}\), and solution time for instance 18/02/2015

Accuracy | Case I | Case II | ||||||
---|---|---|---|---|---|---|---|---|

\(C_\mathrm {e}\) (€) | \(C_\mathrm {c}\) (€) | \(C_\mathrm {s}\) (€) | Solution time (s) | \(C_\mathrm {e}\) (€) | \(C_\mathrm {c}\) (€) | \(C_\mathrm {s}\) (€) | Solution time (s) | |

A | 4155106.61 | 1918.72 | 0.00 | 593.54 | 4652407.31 | 4563.49 | 900463.23 | 3600.00 |

B | 4155105.52 | 1587.34 | 0.00 | 166.86 | 4642512.18 | 4408.67 | 900418.07 | 3600.00 |

C | 4155095.11 | 1219.04 | 0.00 | 34.65 | 4642326.27 | 3856.23 | 899957.85 | 3600.00 |

D | 4155000.91 | 865.70 | 0.00 | 10.05 | 4642212.69 | 2912.82 | 895779.05 | 1089.00 |

Table 5 shows the growing amount of variables and constraints for tighter linearizations leading to higher solution times. While the number of variables and constraints only slightly deviates from the average within each row, the solution time is in general higher for higher non-electric gas loads. Moreover, Case II turns out to be harder than Case I, which can be seen from the average solution time and the higher number of instances that hit the time limit.

#### 4.2.2 Comparison with single electricity model

It can be seen that the average production of the electric gas loads is reduced for many dispatch periods for both cases. The sharpest decline takes place during the evening peak hours and is even more evident for Case II.

The reduced energy production can be attributed to the fact that the gas network cannot allocate the desired amount of gas to the electric gas loads according to the optimal solution of the single electricity model. The difference is more evident in Case II when the supply capacity of the gas network is further reduced.

Average increase of \(C_{\mathrm{e}}\) for both cases

Accuracy | Case I (€) | Case II (€) |
---|---|---|

A | 2536.80 | 118771.41 |

B | 2534.28 | 117864.75 |

C | 2530.43 | 117841.05 |

D | 2520.40 | 117779.13 |

## 5 Conclusion

- 1)
The increase of objective function costs and solution times with respect to tighter approximations for all instances examined.

- 2)
The appearance of shedded non-electric gas loads during certain instances, caused by the need to provide the demand of the electric gas loads, in order to maintain the electricity system balance.

- 3)
The effect of the gas system’s constraints on the electricity model, which was expressed in the increase of the electricity unit commitment cost.

## Notes

### Acknowledgements

This research has been performed as part of the Energie Campus Nürnberg. We also acknowledge funding through the DFG SFB/Transregio 154, Subprojects A05 and Z01.

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