Online Gradient Descent for Grid Regulated Power Point Tracking Under a Highly Fluctuating Weather and Load

Depending on the level of penetration from inverter-based resources and the stability and reliability of the grid, countries have various regulations and standards for the operation of PV systems under different conditions. Each operational constraint imposes different constraints that vary with time, weather conditions, load, and other operational factors. The operational constraints are as follows [5, 7]:

1. 1.

   MPPT control zone (unconstrained): MPPT operation is implemented here, in which there are no operational constraints. The purpose of the algorithm in this zone is to extract the maximum available power from the PVs.

2. 2.

   Power reserve constraint (delta power constraint): A delta power constraint is used to limit the active power from a PV to a desired constant value in proportion to the maximum available power. This operational zone is vital as a regulating reserve.

3. 3.

   Ramp rate constraint: A ramp rate constraint is used to limit the power ramp rate by which the active power can be changed

4. 4.

   Power limiting control (absolute power constraint): An absolute power constraint is typically used to protect the power system against overload in critical situations.

The maximum Power point tracking domain is older and hence more extensively explored than Flexible Power Point Tracking (FPPT). Therefore, to prove that FPP and MPP tracking need online optimization, MPPT literature will be considered as it is richer and better documented. There are numerous MPPT algorithms; each has its strengths and drawbacks. However, the majority of the literature only tests these algorithms under rapidly changing weather conditions (mainly irradiance) and only focuses on accuracy and convergence time. It is worth mentioning that irradiance variability is around $80\mathrm{~{}W}/\mathrm{m}^{2}$ in less than 2 seconds [8], [9] and varies more with larger time steps. An $80\mathrm{~{}W}/\mathrm{m}^{2}$ change can contribute to a $6\%$ loss in the power extracted if the operational point remains unchanged. In addition to the weather, the load is continuously changing, affecting the optimal operation point. The algorithms in literature usually consume more time to converge to the maximum power point, which is summarized in Table 1[10]:

Therefore, as can be noted from Table 1, since the convergence time is less than the time it takes for the environmental conditions to change, there is a pressing need for an online algorithm.

Let $N:=\{1,2,\ldots\}$ and $N_{0}:=N\cup\{0\}$. In this problem, $F$ is considered the overall power in which $F:=\left\{f_{t}:R^{n}\mid k\in\right.$ $\left.N_{0}\right\}$ represents a family of continuous functions and where $X:=\left\{X_{t}\subset R^{n}\mid t\in N_{0}\right\}$ are closed, non-empty sets that capture the possible time-varying constraints, and where $n$ represents the number of PV modules that are connected to the grid. The algorithm is required to generate solutions to the following sequence of optimization problems as in equation $(2)$, using iteration of the form shown in equation (3):

Where the operator $G_{t}:R^{n}\rightarrow R$ uses first-order information and knowledge of the feasible set $X_{t}$ at the current time step.

In this work, uniform irradiance across the PV panels is assumed. Hence, it is safe to say that attention is restricted to families of locally smooth, strongly convex functions with Lipschitz gradient and convex constraints, and hence the following assumptions:

Theorem 1 (Smoothness): Every $f\in F$ is twice continuously differentiable. This indicates that the power is an exponential function that is known to be smooth and is not globally Lipschitz continuous

Theorem 2 (Uniform strong convexity): It is well documented and known in the literature that a PV under uniform irradiance has one unique maximizer (minimzer) and, therefore, is strongly convex.

Assumption 1 (Convex Constraints): Every $X\in X$ is closed, convex and non-empty.

This work employs the online projected gradient descent algorithm to solve the problem presented in equation (2), which applies a specific choice for $G_{t}$ that is shown in equation (3). Under assumption 1, $f_{t}$ is guaranteed to have a unique global minimizer over $X_{t}$.

In this work, The following metrics will be used to evaluate the performance of the algorithm:

1. 1.

   Tracking Error: $\left\|x_{t}-x_{t}^{*}\right\|$, which represents the difference between the estimated voltage and the actual voltage at MPP.

2. 2.

   Instantaneous regret$f_{t}\left(x_{t}\right)-f_{t}\left(x_{t}^{*}\right)$ which represents the difference between the output power and the maximum available power at time $t$ and hence it represents the power lost. In the context of FPPT, the instantaneous power is greater than zero.

3. 3.

   Average Dynamic Regret$\sum_{t=1}^{T}f_{t}\left(x_{t}\right)-\sum_{t=1}^{T}f_{t}\left(x_{t}^{*}\right)$ which represents the average power between the algorithms resulting output power and the maximum power available. It is useful to compute the dynamic regret as if multiplied by the operational time, representing the overall energy loss; therefore, it is important to compute the dynamic regret.

4. 4.

   Static Regret: $\sum_{t=1}^{T}f_{t}\left(x_{t}\right)-\min_{x\in X}\sum_{t=1}^{T}f_{t}\left(x_{t}^{*}\right)$ Since there exists a tracking algorithm that is called constant voltage which fixes the operational voltage at a specific value and does not change it. Therefore it is important to compute the static regret.

The instantaneous regret is defined as in equation (4):

First of all, bounds for the unconstrained case (MPPT control) are considered. The guarantee that $\nabla f_{t}\left(x_{t}^{*}\right)=0$ when $X_{t}=R^{n}$ leads to the inequality in (5):

Consequently, instantaneous regret is bounded as following when $\alpha=\frac{2}{L+\mu}$ which is the value used to minimze the upper bound, then:

The previous error bounds apply equally well to constrained and unconstrained problems. However, the presence of constraints means that extra consideration should be considered. The bounds in the previous subsection do not apply as the minima does not have to be a stationary point.

Finite time feasibility: the following assumption of finite time feasibility guarantees the feasibility of the projected gradient descent as follows:

Where $D<+\infty$

Bounds on the temporal variability:

Proof: To show the computation of the bound, starting from (derived in the previous problem):

Where $f_{t}$ is strongly convex with $\mu$.

Now substituting $x=x_{t+1}^{*}$ and using the fact that $\nabla f_{t+1}\left(x_{t+1}*\right)=0$ and adding it to the above, the equation becomes:

And assuming that:

Then the bound is:

Uniform Lipschitz cost for such $G\geq 0$, there exists: For the pair $F,X$ there exists $G>0$ where

This work proposes using the online optimization method, Online Projected Gradient Descent (OPGD). First, the sensors in the system take different relative measurements at each time; irradiance, temperature, and load-related values. As in (1), the power equation can be calculated with sufficient measurements. Thus, the gradient for the voltage can be calculated to get the optimal operating voltage. Finally, the DC-DC converter’s duty cycle is calculated according to the optimal value that equalizes the load resistance as seen by the source resistance to the source resistance and thus achieving maximum power transfer. Figure 3 illustrates the flow of the algorithm. Online projected gradient descent will be executed with a constant step size of $\alpha=\frac{2}{10}$.

Perturb and Observe (P&O) is the most popular and widely used tracking algorithm for PV systems due to its relatively good accuracy, ease of implementation, and being PV module independent. This work compares the performance of online gradient descent to the widely used P&O as a benchmark.

MATLAB Simulink is used as a simulation tool to build the following system configuration shown in figure 4.

It is worth mentioning that for flexible power point tracking, the same controller will be used but with constraints on the operating conditions and thus has the same block diagram as above.

Table 2 summarizes the characteristics of the PV modules used.

Finally, the algorithms were tested under a highly fluctuating irradiance to test their ability to track the optimal operating point. The code and simulation files are available at this link.