A Gradient-Based Economic Model Predictive Controller with Zone Control Scheme Applied to Natural Gas Pipeline Networks

Introduction

In the last decade, natural gas has become an essential resource in the world energy matrix because of its lower price, more accessible transport scheme, and lower environmental impact than other fossil fuels [1]. Within these advantages, the transport of this fuel by pipeline stands out. Since the natural gas pipeline network (NGN) guarantees this fuel’s continuous access, they decrease the dependency on natural gas from logistic problems [2]. However, NGN has a high cost to maintain the flow of gas in long-distance pipes, mainly due to the compression systems, which can consume about 3 % to 5 % of natural gas transported through centrifugal compressors [3].

In addition to the high operational cost involved, another challenge related to NGNs is ensuring natural gas demand in a volatile market condition with the higher efficiency possible. Inside the natural gas market rules, the operation of natural gas is susceptible to its price, energy demand, and industrial requirements. On the other hand, transporting the gas over long distances consumes part of natural gas to recompress itself gas. Thus, according to the natural gas demand, the best condition to operate the NGN can continuously change [4].

Another issue associated with the NGN process is the safety of compression systems. Compressors are essential equipment in NGN, so that centrifugal compressors are, particularly, the most abundant in such a process. However, centrifugal compressors have operational constraints that, when violated, can lead the process to fail. The two most important constraints are the surge operating condition and the stonewall one. The surge constraint is undoubtedly the most relevant safety issue in centrifugal compressors, and in NGN operational schemes, it is always a critical constraint to be considered and addressed [4, 5, 6]. The motivations for this are: in the surge phenomenon, a low gas flow changes its direction, returning to the compressor suction, causing the oscillation of flow, and leading the process to an unstable condition; the other one is the fact that the surge constraint matches with the most isentropic efficiency of the compressor, what means to operate the process near the surge condition can be economically beneficial, reducing the amount of energy to compress the gas [7].

Therefore, in this environment where it is necessary to reduce the process cost while maintaining the process in safe conditions and attending the market oscillations, the advanced control schemes have been highlighted as a proper way to manage the process. The Model Predictive Control strategy (MPC) has been presented as a suitable and systematic mechanism to operate the NGN optimally. Such an approach is based on forecasting plant behavior through a dynamic model, so evaluate the best movements to lead the system to the operational target. The MPC schemes for NGN are in progress, initiated at the beginning of the century with the work by Zhu, et al. [8]. However, as far as we know, NGN-oriented MPC schemes use dynamics models with quite restrictive assumptions, such as a simple gas behavior [4] and isothermal flow [1, 9]. Such premises result in simplified concentrated parameters-based models to describe the pipelines, and it can lead to a considerable mismatch between the plant and the model used in MPC. The last works with MPC focused on NGN have considered two modeling approaches: the phenomenological approach [9] and data-driven approaches such as an artificial neural network – ANN [1]. In summary, these approaches are based on either first-principle models, which ignore the thermal conditions (by considering an inappropriate isothermal behavior in compression system) or, when using an ANN-type data-driven scheme, a new training step with a particular frequency is required. To circumvent this, this paper proposes using a phenomenological approach that accommodates the energy balance and equation of state- based non-ideal gas behavior for providing an NGN model based on algebraic-partial differential equations (PDAE).

In this work, the rigorous phenomenological model for NGN will be incorporated into the nonlinear model of the plant, and this model and surge prevention constraint will feed a successive linearization-based adaptive MPC scheme, coupled with an infinite horizon predictive control strategy (IHMPC) [10]. The proposed NGN-oriented MPC controller consists of a one-layer economic scheme based on the gradient of economic function [11]. The resulting control law explicitly considers the surge avoidance issue via slacked constraints, guaranteeing the feasibility of the controller by using adequately slack variables [12, 13]. Also, it considers zone tracking scenarios [14] rather than set-point tracking in the proposed control method to provide additional degrees of freedom while accommodating the practical industrial case in a condition with a lower economic cost, and it avoids numerical issues to find the controller solution.

The Proposed Gradient-Based Economic MPC for Natural Gas Networks

The proposed model predictive controller inherits properties from Alvarez, et al. [11], whose proposal is to couple the control actions and the economic target in the same optimization layer, namely economic MPC based on the gradient of an economic function. In order to accommodate the practice case, a zone control scheme [14] will be incorporated into the proposed control law. Therefore, the resulting control law of this proposal aims to solve the following optimization problem:

1 2 2 2 | , , , , 1 0 min , u m ∞

− + + + ∆ = = = − + ∆ + − ∑ ∑ Q R P u y y y k k i k sp k k m e k e k i i V f t ∞ , k sp k (25)

subject to the proposed natural gas network model, linearized at each sampling instant k, and , , ≤ ≤ min max y y y sp k (26) | , 1,2, , , + ≤ ≤ = … min max u u u k j k j m (27) | , 1,2, , 1, + ∆ ≤∆ ≤∆ = … − min max u u u k j k j m (28) | , 0, 1,2, , , + − + ≤ = …   surge k surge Q Q ck j k c a j m (29) Where y is the process variable (in this work, they will be the pressure of gas in consumer nodes), u is the manipulated variables (in this work, they will be the compressor speed and the mass flow rate in recycle valve), fe is the economic function, te is the economic target (profit set-point). For the decision variables, the trend of control actions are T T T T m   ∆ = ∆ ∆ ∆   Λ u u u u , and the sp y is the process variable setpoint. Q and R are the weighting matrices for the process variables, control actions, and P economic functions, respectively. c Q is the flow rate in compressors, and ,surge c Q is the flow rate in the surge line, and surge a is the safety factor of the surge prevention line, assumed as 1.1 based on Ghazanfarihashemi, et al. [20]. The surge line has been evaluated by the approach presented by Meira, et al. [21] by using the phenomenological approach [7] given by,

1 2 ( ) , , 0, η ∂ = ∂   surge c c Q N Q (30) where η is the isentropic efficiency of the compressor. Thus, the solution of the expression (29) given a compressor speed (N) will result in the ,surge c Q - the flow rate in the surge condition, which will be used as a constraint of the control scheme, as the surge prevention constraint, , 1.1 0, − ≤   surge c c Q Q (31) where c Q is the volumetric flow rate of the gas in compressor. In some works, such constraint also can be rewritten as a relative surge line distance (SLD) [22], , 100 10 . − = × < −   Q Q % % Q surge , surge SLD c c (32)  c On the other hand, the economic function fe has been evaluated by the sum of power consumed in centrifugal compressors (Pot), given by,

, , , 1 , , = = = ∑   e c i axial i i c i i f Pot m w N Q (33)

( ) 3 where c m is the mass flow rate in the compressor. As this function and the NGN model are nonlinear, evaluating at each time step k by the controller can demand much time. Because of that, so this work opts by linearizing the NGN space-discretized model, along with the economic function and the surge line.

For the NGN modeling, after linearization, it was built a state-space model as proposed in Odloak, et al. [10], by similarity transformation, which allows the evaluation of states in the infinite horizon-based MPC scheme proposed here, namely:

I x x B u F x x B

0         = +               ∆    B

s s s k k k st st st st k k (34)

k x y I Ø x [ ]   =      C

s k k st k (35)

k where st x is of the vector of the stable states, sx is the vector of the integrating states introduced by incremental form of inputs, responsible for the offset eliminating in the control law. The auxiliaries matrices st F , s B , and can be evaluated by steps presented in Odloak, et al. [10].

For the economic function, it must be evaluated in k + ∞step, besides that, due to the expression (33) being a nonlinear equation, it must be successively linearizated to apply in the cost function of IHMPC, evaluated as:

( ) ( ) , , , 1| , | , ∞ ∞ + + − + = + ∇ + ∇ T T

u y u y e k e k e k k m k e k k k f f f f (36) where , 1 e k f − is the economic function in step 1 k −, and e e f f ∇ ∇ y u is the gradient of the economic function for the process and manipulated variables, respectively.

Thus, the cost function of the proposed MPC can be obtained by adapting the approach of Odloak, et al. [11], and it can be rewritten as:

1 2 2 | , 1 0

m ∞ − ∑ ∑ R Q

y y u V

= − + ∆ k k i k sp k k m i i + + = =

(37) ( ) ( ) T T

2 u y f f f t + + − ∇ + ∇

u y P

, , 1| , | ,

e k e k k m k e k k k e k ∞

+ − + The infinite sum in (37) can then be evaluated as follows:

m s st k i k sp k k i k sp k k i k sp k k i k i i i ∞ ∞

2 2 2 + + + + = = = + − = − + − + Ψ ∑ ∑ ∑ Q Q Q m y y y y x y x | , | , | , | 1 1 1 .

(38) In order to bound the infinite sum associated with the artificial integrating states, a terminal equality constraint must be imposed to the control optimization problem:

| , 0. + − = x y s k m k sp k (39)

On the other hand, as st x are stables, they yield a convergent term, in such a way that:

∞

+ + = = ∑  Q Q x x st st k i k k m i

2 2 | 1 , (40)

where Q is the terminal weighting matrix obtained from the solution of the Lyapunov equation:

( ) ( ) Q Q Φ − Φ =   T T F F F Q F st st st st st st (41) Since an equality constraint can compromise the feasibility of the controller, it is common to use slack variables to soften it. Such a strategy may also be used to soften the surge prevention constraint, as Meira, et al. [22] proposed. Thus, the final form of the proposed MPC controller results in

1 2 2 2 | , , , , , 1 0

m m s st k k i k sp k y k k m k m i i −

Results And Discussion

The case simulation study is represented in Figure 2. The economic function considered here is to minimize the power consumption of the compression system. At the same time, the pressures in consumer nodes are the controlled variables, which will be accommodated by the zone tracking scheme instead of set-point tracking one, while the rotational frequency of compressors and mass flow rate of the recycle valve, in compression nodes 1, 2 and 3, will be the manipulated variables of the controller. Also, the economic target te in (42) will be set at zero, given that the aim is the lowest energy consumption, even if such value is unreachable. Furthermore, all compression system operations must meet the surge prevention line to maintain the process’s safety. The nonlinear plant model will be considered the PDE discretization by using the numerical solution of the CasADi framework according with Figure 2.

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In the simulation reference of the zones related to the pressures in consumer nodes ( ,1 ,2 and f f P P ) are limited to the maximum value of 5.0 MPa, while the minimum value will change throughout the simulation case, the operational conditions are presented in Table 2. Also, according to the expression (32), SLD will be set up as lower than -10% to maintain the system in safe operating conditions. The tuning parameters of the controller, obtained from a sensitivity analysis, are summarized in Table 3, the sampling time for the controller actions considered here is 300s.

The simulation results are detailed in Figure 4 to Figure 7. These show that the proposed controller succeeded in leading the ,1 ,2 and f f P P to their predefined zones in all scenarios of zone changes. Concerning the manipulated variables, the rotational frequencies of the compressors were successfully modulated to accommodate the minimal power consumption of these pieces of equipment. At the same time, it is possible to notice that the mass flow rates in recycle valves in Compression Systems 2 and 3 were only used to meet the surge prevention constraint, given that, contrarily, their values are returned to zero to avoid increasing the energy consumption in compression systems. However, for Compression System 1, the mass flow rates in the recycle valve have been used in almost all simulations, once the compression process is near the surge prevention line, as demonstrated in Figure 6. Another prominent controller feature is its capacity to deal with surge prevention via slacked SLD constraints, thus ensuring the feasibility of the MPC controller. As shown in Figure 6, the controller could meet the target at lower -10% SLD and used only the slacks in a few time steps in compression system 1, immediately after perturbations. Due to the large weighting values on surge S , slacks are used only when necessary, and as a consequence, the controller can bring back the system rapidly at the maximum target of -10% SLD when this constraint is violated. This process behavior can indicate a good performance of the controller to seek the minimal economic cost condition, once the poins near the surge line (SLD = 0) have high isentropic efficiency [7].

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Legend: (

) pressure, ( $$ - \bigcirc $$

$$ \mathbf {3} $$

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)limits of pressure zones. Figure 4: Pressure in the consumer nodes.

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) the compression system 3. Figure 5: Response of manipulated variables for each compression system.

Finally, regarding the behavior of the energy consumption of compression systems throughout the zone tracking scenarios, it is noticeable that the controller can accommodate the compression system at steady states with minimal power consumption possible, after zone changes (Figure 7), implying the minimal cost associated with the operation of the compressors.

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) the compression system 3. Figure 6: Response of surge line relative distances.

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Conclusion

This work proposes a practical economic MPC scheme for natural gas networks (NGN), aiming to reduce the operational cost of compression systems. At the same time, the proposed controller incorporates softened surge prevention and zone control schemes in its control law. By considering a non-isothermal flow and non-ideal gas behavior to phenomenologically model NGN, the proposed adaptive strategy based on successive linearization of the nonlinear model and surge prevention constraint was responsible for providing fast computational responses to solve the resulting control law. The results showed that the controller could minimize the energy consumed in compression systems and successfully avoid unsafe operating conditions within the zone tracking simulated scenarios. These characteristics proved to be a one-layer strategy-oriented NGN management scheme that can ensure control performance, operational safe and economic competitivity. As directions of future perspectives of the work, one will seek to extend the proposed controller to distributed-cooperative MPC schemes in such a way as to treat more extensive and interconnected NGN.

Conflicts of Interest

(if applicable)

Acknowledgements

The authors would like to thank CNPq under grant 408339/2021-7.

Funding

FAPESB (Brazilian research agency of the State of Bahia) financially supported this work under grant BOL2333/2016.