Open Access

This paper addresses long-term changes in solar irradiance for West Africa (3° N to 20° N and 20° W to 16° E) and its implications for photovoltaic power systems. Here we use satellite irradiance (Surface Solar Radiation Data Set-Heliosat, Edition 2.1, SARAH-2.1) to derive photovoltaic yields. Based on 35 years of data (1983–2017) the temporal and regional variability as well as long-term trends of global and direct horizontal irradiance are analyzed. Furthermore, at four locations a detailed time series analysis is undertaken. The dry and the wet season are considered separately. According to the high resolved SARAH-2.1 data record (0.05° x 0.05°), solar irradiance is largest (with up to 300 W/m² daily average) in the Sahara and the Sahel zone with a positive trend (up to 5 W/m²/decade) and a lower variability (< 75 W/m²). Whereas, the solar irradiance is lower in southern West Africa (between 200 W/m² and 250 W/m²) with a negative trend (up to −5 W/m²/decade) and a higher variability (up to 150 W/m²). The positive trend in the North is mostly connected to the dry season, while the negative trend in the South occurs during the wet season. PV yields show a strong meridional gradient with lowest values around 4 kWh/kWp in southern West Africa and reach more than 5.5 kWh/kWp in the Sahara and Sahel zone.

Solving differential-algebraic equations (DAEs) efficiently by means of appropriate numerical schemes for time-integration is an ongoing topic in applied mathematics. In this context, especially when considering large systems that occur with respect to many fields of practical application effective computation becomes relevant. In particular, corresponding examples are given when having to simulate network structures that consider transport of fluid and gas or electrical circuits. Due to the stiffness properties of DAEs, time-integration of such problems generally demands for implicit strategies. Among the schemes that prove to be an adequate choice are linearly implicit Rung-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes. Compared to fully implicit methods, they are easy to implement and avoid the solution of non-linear equations by including Jacobian information within their formulation. However, Jacobian calculations are a costly operation. Hence, necessity of having to compute the exact Jacobian with every successful time-step proves to be a considerable drawback. To overcome this drawback, a ROW-type method is introduced that allows for non-exact Jacobian entries when solving semi-explicit DAEs of index one. The resulting scheme thus enables to exploit several strategies for saving computational effort. Examples include using partial explicit integration of non-stiff components, utilizing more advantageous sparse Jacobian structures or making use of time-lagged Jacobian information. In fact, due to the property of allowing for non-exact Jacobian expressions, the given scheme can be interpreted as a generalized ROW-type method for DAEs. This is because it covers many different ROW-type schemes known from literature. To derive the order conditions of the ROW-type method introduced, a theory is developed that allows to identify occurring differentials and coefficients graphically by means of rooted trees. Rooted trees for describing numerical methods were originally introduced by J.C. Butcher. They significantly simplify the determination and definition of relevant characteristics because they allow for applying straightforward procedures. In fact, the theory presented combines strategies used to represent ROW-type methods with exact Jacobian for DAEs and ROW-type methods with non-exact Jacobian for ODEs. For this purpose, new types of vertices are considered in order to describe occurring non-exact elementary differentials completely. The resulting theory thus automatically comprises relevant approaches known from literature. As a consequence, it allows to recognize order conditions of familiar methods covered and to identify new conditions. With the theory developed, new sets of coefficients are derived that allow to realize the ROW-type method introduced up to orders two and three. Some of them are constructed based on methods known from literature that satisfy additional conditions for the purpose of avoiding effects of order reduction. It is shown that these methods can be improved by means of the new order conditions derived without having to increase the number of internal stages. Convergence of the resulting methods is analyzed with respect to several academic test problems. Results verify the theory determined and the order conditions found as only schemes satisfying the order conditions predicted preserve their order when using non-exact Jacobian expressions.

The need for innovation around the control functions of inverters is great. PV inverters were initially expected to be passive followers of the grid and to disconnect as soon as abnormal conditions happened. Since future power systems will be dominated by generation and storage resources interfaced through inverters these converters must move from following to forming and sustaining the grid. As “digital natives” PV inverters can also play an important role in the digitalisation of distribution networks. In this short review we identified a large potential to make the PV inverter the smart local hub in a distributed energy system. At the micro level, costs and coordination can be improved with bidirectional inverters between the AC grid and PV production, stationary storage, car chargers and DC loads. At the macro level the distributed nature of PV generation means that the same devices will support both to the local distribution network and to the global stability of the grid. Much success has been obtained in the former. The later remains a challenge, in particular in terms of scaling. Yet there is some urgency in researching and demonstrating such solutions. And while digitalisation offers promise in all control aspects it also raises significant cybersecurity concerns.