Energy Hub Management with Intermittent Wind Power

Abstract

The optimal energy management in energy hubs has recently attracted a great deal of attention around the world. The energy hub consists of several inputs (energy resources) and outputs (energy consumptions) and also some energy conversion/storage devices. The energy hub can be a home, large consumer, power plant, etc. The objective is to minimize the energy procurement costs (fuel/electricity/environmental aspects) subject to a set of technical constraints. One of the popular options to be served as the input resource is renewable energy like wind or solar power. Using the renewable energy has various benefits such as low marginal costs and zero environmental pollution. On the other hand, the uncertainties associated with them make the operation of the energy hub a difficult and risky task. Besides, there are other resources of uncertainties such as the hourly electricity prices and demand values. Hence, it is important to determine an economic schedule for energy hubs, with an acceptable level of energy procurement risk. Thus, in this chapter a comprehensive multiobjective model is proposed to minimize both the energy procurement cost and risk level in energy hub. For controlling the pernicious effects of the uncertainties, conditional value at risk (CVaR) is used as risk management tool. The proposed model is formulated as a mixed integer nonlinear programming (MINLP) problem and solved using GAMS. Simulation results on an illustrative test system are carried out to demonstrate the applicability of the proposed method.

Appendices

Appendix-I

1.1 Scenario Reduction Technique

Suppose that the original set of the scenarios is denoted by $\varOmega_{J}$ and we want to reduce the number of scenarios to $N_{{\varOmega_{S} }}$. Hence, scenario reduction proposes a method for selection of a set, i.e., $\varOmega_{S}$, with the cardinality of $N_{{\varOmega_{S} }}$, from $\varOmega_{J}$. The number of the reduced scenarios should be selected in a way that the computation burden reduced while not drastically reducing the accuracy [25]. The scenario reduction technique used in this chapter can be carried out using the following steps [40]: [step. 1]

• 1. Construct the matrix containing the distance between each pair of scenarios $c(w,w^{\prime})$

• 2. Select the fist scenario $w_{1}$ as follows:

  $$ w_{1} = { \arg }\left\{ {\mathop {min}\limits_{{w^{\prime} \in \varOmega_{J} }} \sum\limits_{{w \in \varOmega_{J} }} {\pi_{w} c(w,w^{\prime})} } \right\} $$
  $$ \varOmega_{S} = \left\{ {w_{1} } \right\},\varOmega_{J} = \varOmega_{J} - \varOmega_{S} $$

• 3. Select the next scenario to be added to $\varOmega_{S}$ as follows:

  $$ w_{n} = \arg \left\{ {\mathop {\hbox{min} }\limits_{{w^{\prime} \in \varOmega_{J} }} \sum\limits_{{w \in \varOmega_{J} - \left\{ {w^{\prime}} \right\}}} \pi_{w} \mathop {\hbox{min} }\limits_{{w^{\prime\prime} \in \varOmega_{S} \cup \left\{ w \right\}}} c\left( {w,w^{\prime\prime}} \right)} \right\} $$
  $$ \varOmega_{S} = \varOmega_{S} \cup \left\{ {w_{n} } \right\},\varOmega_{J} = \varOmega_{J} - \varOmega_{S} $$

• 4. If the number of selected set is sufficient then end and go to step 2; else continue.

• 5. The probabilities of each nonselected scenario will be added to its closest scenario in the selected set.

• 6. End.

Appendix-II

2.1 Pareto Optimality

Assume $F(X)$ is the vector of objective functions, and $H(X)$ and $G(X)$ represent equality and inequality constraints, respectively. A multiobjective minimization problem can be formulated as follows [41]:

$$ \begin{aligned} \hbox{min} F\left( X \right) &= \left[ {f_{1} \left( X \right), \ldots ,f_{{N_{O} }} \left( X \right)} \right]\\ & \; {\text{Subject to:}}\\ \left\{ G\left( X \right) \right. &= \left. \bar{0},H\left( X \right) \le \bar{0} \right\}\\ X &= \left[ {x_{1} , \ldots ,x_{m} } \right] \end{aligned} $$

$X_{1}$ dominates $X_{2}$ if:

$$ \forall k \in \left\{ {1 \ldots N_{O} } \right\}f_{k} \left( {X_{1} } \right) \le\;f_{k} \left( {X_{2} } \right) $$

$$ \exists k^{\prime} \in \left\{ {1 \ldots N_{O} } \right\}f_{{k^{\prime}}} \left( {X_{1} } \right) <\;f_{{k^{\prime}}} \left( {X_{2} } \right) $$

Any solution which is not dominated by any other is called to belong to a Pareto optimal front which is referred to as the first Pareto front or optimal front or nondominated front.

Appendix-III

3.1 Fuzzy Satisfying Method

Fuzzy satisfying (or max(min)) method is a popular technique for selection of the best solution among the obtained $N_{p}$ Pareto optimal solutions [20]. Suppose we have a problem with $N$ objectives to be minimized. The linear membership function for the $n$th solution of the $k$th objective function is defined as [42]:

$$ \begin{gathered} \mu_{k}^{n} = \left\{ {\begin{array}{*{20}c} 1 & {f_{k}^{n} \le f_{k}^{ \hbox{min} } } \\ {\frac{{f_{k}^{\hbox{max} } - f_{k}^{n} }}{{f_{k}^{\hbox{max} } - f_{k}^{\hbox{min} } }}} & {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} f_{k}^{ \hbox{min} } \le f_{k}^{n} \le f_{k}^{ \hbox{max} } } \\ 0 & {f_{k}^{n} \ge f_{k}^{ \hbox{max} } } \\ {} & {} \\ \end{array} } \right. \hfill \\ k = 1, \ldots ,N,{\kern 1pt} n = 1, \ldots ,N_{p} \hfill \\ \end{gathered} $$

where $f_{k}^{ \hbox{max} }$ and $f_{k}^{ \hbox{min} }$ are maximum and minimum values of the objective function $k$ in solutions of Pareto optimal set. $\mu_{k}^{n}$ represents the optimality degree of the $n$th solution of the $k$th objective function. The membership function of $n$th solution can be calculated using the following equation:

$$ \begin{gathered} \mu^{n} = min\left( {\mu_{1}^{n} , \ldots ,\mu_{N}^{n} } \right) \hfill \\ n = 1, \ldots ,N_{p} \hfill \\ \end{gathered} $$

(48)

The solution with the maximum weakest membership function is the best solution. The corresponding membership function of this solution ($\mu^{ \hbox{max} }$), is calculated as follows:

$$ \mu^{\text{max}} = max\left( {\mu^{1} , \ldots ,\mu^{{N_{p}}}} \right) $$

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